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A result on three solutions theorem and its application to p-Laplacian systems with singular weights

Eun Kyoung Lee1 and Yong-Hoon Lee2*

Author Affiliations

1 Department of Mathematics Education, Pusan National University, Busan, 609-735, Korea

2 Department of Mathematics, Pusan National University, Busan, 609-735, Korea

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Boundary Value Problems 2012, 2012:63  doi:10.1186/1687-2770-2012-63


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/63


Received:16 February 2012
Accepted:18 May 2012
Published:22 June 2012

© 2012 Lee and Lee; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper, we consider p-Laplacian systems with singular weights. Exploiting Amann type three solutions theorem for a singular system, we prove the existence, nonexistence, and multiplicity of positive solutions when nonlinear terms have a combined sublinear effect at ∞.

MSC: 35J55, 34B18.

Keywords:
p-Laplacian system; singular weight; upper solution; lower solution; three solutions theorem

1 Introduction

In this paper, we study one-dimensional p-Laplacian system with singular weights of the form where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M1">View MathML</a>, λ is a nonnegative parameter, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M3">View MathML</a> is a nonnegative measurable function on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M5">View MathML</a> on any open subinterval in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M4">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M7">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M8">View MathML</a>. In particular, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M2">View MathML</a> may be singular at the boundary or may not be in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M10">View MathML</a>. It is easy to see that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M11">View MathML</a>, then all solutions of (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M12">View MathML</a>) are in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M13">View MathML</a>. On the other hand, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M14">View MathML</a>, then this regularity of solutions is not true in general; for example, even for scalar case, if we take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M15">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M16">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M17">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M18">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M19">View MathML</a>, and the solution u for corresponding scalar problem of (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M12">View MathML</a>) is given by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M21">View MathML</a> which is not in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M13">View MathML</a>.

For more precise description, let us introduce the following two classes of weights;

We note that h given in the above example satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M24">View MathML</a> but <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M25">View MathML</a>. The main interest of this paper is to establish Amann type three solutions theorem [4] when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M26">View MathML</a> with possibility of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M25">View MathML</a>. The theorem generally describes that two pairs of lower and upper solutions with an ordering condition imply the existence of three solutions. Recently, Ben Naoum and De Coster [6] have proved the theorem for scalar one-dimensional p-Laplacian problems with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M28">View MathML</a>-Caratheodory condition which corresponds to case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M29">View MathML</a>; Henderson and Thompson [18] as well as Lü, O’Regan, and Agarwal [23] - for scalar second order ODEs and one-dimensional p-Laplacian with the derivative-dependent nonlinearity respectively; and De Coster and Nicaise [11] - for semilinear elliptic problems in nonsmooth domains. For noncooperative elliptic systems (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M30">View MathML</a>) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M31">View MathML</a> and Ω bounded, one may refer to Ali, Shivaji, and Ramaswamy [3]. Specially, for subsuper solutions which are not completely ordered, this type of three solutions result was studied in [26].

The three solutions theorem for our system (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M12">View MathML</a>) or even for corresponding scalar p-Laplacian problems is not obviously extended from previous works mainly by the possibility <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M25">View MathML</a>. Caused by the delicacy of Leray-Schauder degree computation, the crucial step for the proof is to guarantee <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M34">View MathML</a> regularity of solutions, but with condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M35">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M34">View MathML</a> regularity is not known yet. Due to the singularity of weights on the boundary, the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M34">View MathML</a> regularity heavily depends on the shape of nonlinear terms f and g. Therefore, the first step is to investigate certain conditions on f and g to guarantee <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M34">View MathML</a> regularity of solutions. Another difficulty is to show that a corresponding integral operator is bounded on the set of functions between upper and lower solutions in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M39">View MathML</a>. To overcome this difficulty, we give some restrictions on upper and lower solutions such that their boundary values are zero. As far as the authors know, our three solutions theorem (Theorem 1.1 in Section 2) is new and first for singular p-Laplacian systems with weights of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M40">View MathML</a> class.

To cover a larger class of differential system, we consider the systems of the form where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M41">View MathML</a> are continuous. We give more conditions on F and G as follows: (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M42">View MathML</a>) = For each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M43">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M44">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M45">View MathML</a> are nondecreasing in u.; (H) = There exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M46">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M47">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M48">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M49">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M50">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M51">View MathML</a>.; (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M52">View MathML</a>) = <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M53">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M54">View MathML</a>, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M55">View MathML</a>.. We now state our first main result related to three solutions theorem as follows. See for more details in Section 2.

Theorem 1.1Assume (H), (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M42">View MathML</a>) and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M57">View MathML</a>). Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M58">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M59">View MathML</a>be a lower solution and an upper solution and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M60">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M61">View MathML</a>be a strict lower solution and a strict upper solution of problem (P) respectively. Also, assume that all of them are contained in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M62">View MathML</a>and satisfy<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M63">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M64">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M65">View MathML</a>. Then problem (P) has at least three solutions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M66">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M67">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M68">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M69">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M70">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M71">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M72">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M73">View MathML</a>.

As an application of Theorem 1.1, we study the existence, nonexistence, and multiplicity of positive radial solutions for the following quasilinear system on an exterior domain: where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M74">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M75">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M76">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M77">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M78">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M3">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M80">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M8">View MathML</a>.

In recent years, the existence of positive solutions for such systems has been widely studied, for example, in [1] and [27] for second order ODE systems, in [3,7,9,10,13,14,16] and [8] for semilinear elliptic systems on a bounded domain and in [5,15,17] and [2] for p-Laplacian systems on a bounded domain.

For a precise description, let us give the list of assumptions that we consider. (k) = <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M82">View MathML</a>, where

; (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M84">View MathML</a>) = <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M85">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M86">View MathML</a>,; (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M87">View MathML</a>) = <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M88">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M89">View MathML</a>,; (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M90">View MathML</a>) = f and g are nondecreasing..

Condition (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M87">View MathML</a>) is sometimes called a combined sublinear effect at ∞ and simple examples satisfying (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M84">View MathML</a>) ∽ (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M90">View MathML</a>) can be given as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M94">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M95">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M96">View MathML</a>, and also

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M97">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M98">View MathML</a>.

Among the reference works mentioned above, Hai and Shivaji [17] and Ali and Shivaji [2] (with more general nonlinearities) considered problem (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M99">View MathML</a>) with case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M31">View MathML</a> and Ω bounded. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M34">View MathML</a> monotone functions f and g with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M102">View MathML</a> and satisfying condition (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M87">View MathML</a>), they proved that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M104">View MathML</a> such that the problem has at least one positive solution for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M105">View MathML</a>.

We first transform (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M99">View MathML</a>) into one-dimensional p-Laplacian systems (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M12">View MathML</a>) with change of variables <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M108">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M109">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M110">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M111">View MathML</a> where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M2">View MathML</a> is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M113">View MathML</a>

It is not hard to see that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M114">View MathML</a> in (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M99">View MathML</a>) satisfies (k), then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M2">View MathML</a> in (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M12">View MathML</a>) satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M118">View MathML</a>, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M3">View MathML</a>. Mainly by making use of Theorem 1.1, we prove the following existence result for problem (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M12">View MathML</a>)

Theorem 1.2Assume<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M121">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M3">View MathML</a>, (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M84">View MathML</a>), (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M87">View MathML</a>) and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M90">View MathML</a>). Then there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M126">View MathML</a>such that (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M12">View MathML</a>) has no positive solution for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M128">View MathML</a>, at least one positive solution at<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M129">View MathML</a>and at least two positive solutions for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M130">View MathML</a>.

As a corollary, we obtain our second main result as follows.

Corollary 1.3Assume (k), (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M84">View MathML</a>), (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M87">View MathML</a>) and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M90">View MathML</a>). Then there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M126">View MathML</a>such that (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M99">View MathML</a>) has no positive radial solution for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M128">View MathML</a>, at least one positive radial solution at<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M129">View MathML</a>and at least two positive radial solutions for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M130">View MathML</a>.

We finally notice that the first eigenfunctions of make an important role to construct upper solutions in the proofs of Theorem 1.2 and Theorem 1.1. This is possible due to a recent work of Kajikiya, Lee, and Sim [19] which exploits the existence of discrete eigenvalues and the properties of corresponding eigenfunctions for problem (E) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M121">View MathML</a>.

This paper is organized as follows. In Section 2, we state a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M34">View MathML</a>-regularity result and a three solutions theorem for singular p-Laplacian systems. In addition, we introduce definitions of (strict) upper and lower solutions, a related theorem and a fixed point theorem for later use. In Section 3, we prove Theorem 1.2.

2 Three solutions theorem

In this section, we give definitions of upper and lower solutions and prove three solutions theorem for the following singular system where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M41">View MathML</a> are continuous.

We call <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M142">View MathML</a> a solution of (P) if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M143">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M144">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M142">View MathML</a> satisfies (P).

Definition 2.1 We say that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M146">View MathML</a> is a lower solution of problem (P) if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M147">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M148">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M149">View MathML</a>

We also say that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M150">View MathML</a> is an upper solution of problem (P) if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M151">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M152">View MathML</a> and it satisfies the reverse of the above inequalities. We say that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M146">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M150">View MathML</a> are strict lower solution and strict upper solution of problem (P), respectively, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M146">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M150">View MathML</a> are lower solution and upper solution of problem (P), respectively and satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M157">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M158">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M159">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M160">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M50">View MathML</a>.

We note that the inequality on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M162">View MathML</a> can be understood componentwise. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M163">View MathML</a>. Then the fundamental theorem on upper and lower solutions for problem (P) is given as follows. The proof can be done by obvious combination from Lee [20], Lee and Lee [21] and Lü and O’Regan [22].

Theorem 2.2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M164">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M165">View MathML</a>be a lower solution and an upper solution of problem (P) respectively such that(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M166">View MathML</a>) = <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M167">View MathML</a>, for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M168">View MathML</a>..Assume (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M42">View MathML</a>). Also assume that there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M170">View MathML</a>such that(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M171">View MathML</a>) = <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M172">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M173">View MathML</a>, for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M174">View MathML</a>..Then problem (P) has at least one solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M142">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M176">View MathML</a>

Remark 2.3 It is not hard to see that condition (H) implies the following condition;

For each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M177">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M178">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M179">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M43">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M181">View MathML</a>.

Lemma 2.4Assume (H) and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M52">View MathML</a>). Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M142">View MathML</a>be a nontrivial solution of (P). Then there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M184">View MathML</a>such that bothuandvhave no interior zeros in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M185">View MathML</a>.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M142">View MathML</a> be a nontrivial solution of (P). Suppose, on the contrary, that there exist sequences <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M187">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M188">View MathML</a> of interior zeros of u and v respectively with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M189">View MathML</a>. We note that both sequences should exist simultaneously. Indeed, if one of the sequences say, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M190">View MathML</a>, does not exist, then assuming without loss of generality, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M191">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M192">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M184">View MathML</a>, we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M194">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M195">View MathML</a> by (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M52">View MathML</a>). From the monotonicity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M197">View MathML</a>, we know that v is concave on the interval. Thus v should have at most one interior zero in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M192">View MathML</a>, a contradiction. With this concave-convex argument, we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M199">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M200">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M201">View MathML</a> and if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M202">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M203">View MathML</a> are local extremal points of u and v on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M204">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M205">View MathML</a> respectively, thus both <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M202">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M203">View MathML</a> are in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M201">View MathML</a>. We consider the case that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M209">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M210">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M211">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M201">View MathML</a>. All other cases can be explained by the same argument. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M213">View MathML</a>, then by using Remark 2.3, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M214">View MathML</a>

(2.1)

and similarly,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M215">View MathML</a>

(2.2)

Therefore, it follows from plugging (2.2) into (2.1) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M216">View MathML</a>

(2.3)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M217">View MathML</a>, for sufficiently large n, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M218">View MathML</a>

This contradicts (2.3) and the proof is done. □

Theorem 2.5Assume (H) and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M52">View MathML</a>). If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M142">View MathML</a>is a solution of (P), then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M221">View MathML</a>.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M142">View MathML</a> be a nontrivial solution of (P). Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M223">View MathML</a> so that it is enough to show

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M224">View MathML</a>

We will show <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M225">View MathML</a>. Other facts can be proved by the same manner. Suppose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M226">View MathML</a>. By Lemma 2.4 and the concave-convex argument, we may assume without loss of generality that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M227">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M228">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M192">View MathML</a>. Then for given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M230">View MathML</a>, by the fact <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M231">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M3">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M233">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M234">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M213">View MathML</a>. Then integrating (P) over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M236">View MathML</a> and using Remark 2.3, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M237">View MathML</a>

(2.4)

where we use the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M238">View MathML</a> is decreasing since v is concave. From <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M239">View MathML</a> and (2.4), we know <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M240">View MathML</a>. This implies that conditions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M239">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M242">View MathML</a> are equivalent. From (2.4), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M243">View MathML</a>

Thus we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M244">View MathML</a>

Since ε is arbitrary, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M245">View MathML</a>

(2.5)

Using the fact <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M242">View MathML</a>, with same argument, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M247">View MathML</a>

(2.6)

On the other hand, we observe the inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M248">View MathML</a>

(2.7)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M249">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M250">View MathML</a>, we may choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M251">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M252">View MathML</a>

(2.8)

Integrating (P) over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M253">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M254">View MathML</a> and using Remark 2.3, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M255">View MathML</a>

here we use the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M256">View MathML</a> is increasing in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M257">View MathML</a>. Using (2.7), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M258">View MathML</a>

(2.9)

Integrating (2.9) over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M259">View MathML</a> with respect to s and using (2.8), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M260">View MathML</a>

(2.10)

Similarly, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M261">View MathML</a>

(2.11)

Adding (2.10) and (2.11), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M262">View MathML</a>

(2.12)

on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M257">View MathML</a>. From (2.5) and (2.6), we see that the right-hand side of (2.12) goes to zero as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M264">View MathML</a>. This is a contradiction and the proof is complete. □

Now, we consider the three solutions theorem for singular p-Laplacian system (P). For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M265">View MathML</a>, if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M266">View MathML</a>

then the zero of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M267">View MathML</a>, denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M268">View MathML</a> is uniquely determined by ν. Define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M269">View MathML</a> by taking

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M270">View MathML</a>

It is known that A is completely continuous [24]. Define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M271">View MathML</a> with norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M272">View MathML</a>. We note that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M273">View MathML</a>

(2.13)

If F and G satisfy condition (H), then for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M274">View MathML</a>, from Remark 2.3 and (2.13), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M275">View MathML</a>

This implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M276">View MathML</a> and by similar computation, we also get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M277">View MathML</a>. This fact enables us to define the integral operator for problem (P) and the regularity of solutions (Theorem 2.5) is crucial in this argument. Now, define an operator T by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M278">View MathML</a>

then we see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M279">View MathML</a> and completely continuous.

Lemma 2.6Assume (H), (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M42">View MathML</a>) and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M52">View MathML</a>). Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M146">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M150">View MathML</a>be a strict lower solution and a strict upper solution of problem (P) respectively such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M284">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M285">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M286">View MathML</a>. Then problem (P) has at least one solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M287">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M288">View MathML</a>

Moreover, for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M289">View MathML</a>large enough,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M290">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M291">View MathML</a>.

Proof Define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M292">View MathML</a> given by

and also define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M294">View MathML</a>

Let us consider the following modified problem We first show that there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M289">View MathML</a> such that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M142">View MathML</a> is a solution of (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M297">View MathML</a>), then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M298">View MathML</a>. In fact, every solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M142">View MathML</a> of (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M297">View MathML</a>) satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M301">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M302">View MathML</a>. From (H), (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M42">View MathML</a>) and the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M284">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M305">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M306">View MathML</a>

Similarly, we see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M307">View MathML</a> is bounded. Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M308">View MathML</a>, for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M289">View MathML</a>. Thus it is enough to show that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M310">View MathML</a>

Assume, on the contrary, that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M311">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M312">View MathML</a>

Then choosing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M313">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M314">View MathML</a>, we get the following contradiction:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M315">View MathML</a>

Now, assume <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M316">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M317">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M43">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M319">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M320">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M321">View MathML</a> and we get the same contradiction from the above calculation by using 0 instead of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M322">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M323">View MathML</a> case, we also get the same contradiction. Consequently, we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M324">View MathML</a>. The other cases can be proved by the same manner. Taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M325">View MathML</a>, we see that every solution of (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M297">View MathML</a>) is contained in Ω. We now compute <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M327">View MathML</a>. For this purpose, let us consider the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M328">View MathML</a> defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M329">View MathML</a>

Then it is obvious that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M330">View MathML</a> is completely continuous. We show that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M331">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M332">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M333">View MathML</a>. Indeed, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M334">View MathML</a>, there is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M335">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M336">View MathML</a>. By integrating

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M337">View MathML</a>

from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M338">View MathML</a> to t, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M339">View MathML</a>

Similarly, we see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M340">View MathML</a> is bounded. Therefore, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M341">View MathML</a>

Since every solution of (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M297">View MathML</a>) is contained in Ω, the excision property implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M343">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M344">View MathML</a> on Ω, we finally get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M345">View MathML</a>

This completes the proof. □

We now prove three solutions theorem for (P).

Proof of Theorem 1.1

Define

and let us consider Then noting that every solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M142">View MathML</a> of (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M348">View MathML</a>) satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M349">View MathML</a>, we may choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M350">View MathML</a>, by (H) such that

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M352">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M353">View MathML</a> be the first eigenvalues of for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M3">View MathML</a> respectively and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M355">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M356">View MathML</a> be corresponding eigenfunctions with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M357">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M358">View MathML</a> are positive and concave [19], we may choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M359">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M360">View MathML</a><a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M361">View MathML</a> and for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M43">View MathML</a>

We show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M364">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M365">View MathML</a> are a strict upper solution and a strict lower solution of (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M348">View MathML</a>) respectively. Indeed,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M367">View MathML</a>

Similarly, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M368">View MathML</a>

Moreover,

Similarly, we also get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M370">View MathML</a>

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M289">View MathML</a>, large enough, define

Then by Theorem 2.2, there exist two solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M373">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M374">View MathML</a> of (P) satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M69">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M376">View MathML</a>. Therefore, by Lemma 2.6, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M377','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M377">View MathML</a>

and by the excision property, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M378">View MathML</a>

This completes the proof.

3 Application

In this section, we prove the existence, nonexistence, and multiplicity of positive solutions for (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M12">View MathML</a>) by using three solutions theorem in Section 2. Let us define a cone

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M380','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M380">View MathML</a>

and define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M381','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M381">View MathML</a> by taking

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M383">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M384','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M384">View MathML</a> are unique zeros of

respectively. And define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M386">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M387','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M387">View MathML</a>

Then it is known that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M388">View MathML</a> is completely continuous [25] and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M389','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M389">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M390','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M390">View MathML</a> is equivalent to the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M142">View MathML</a> is a positive solution of (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M12">View MathML</a>). We know from Theorem 2.5 that under assumptions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M393','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M393">View MathML</a><a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M3">View MathML</a> and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M84">View MathML</a>), any solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M142">View MathML</a> of problem (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M12">View MathML</a>) is in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M398">View MathML</a>.

Remark 3.1 If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M142">View MathML</a> is a solution of (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M12">View MathML</a>), then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M401','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M401">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M402','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M402">View MathML</a>.

For later use, we introduce the following well-known result. See [12] for proof and details.

Proposition 3.2LetXbe a Banach space, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M403">View MathML</a>an order cone inX. Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M404','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M404">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M405">View MathML</a>are bounded open subsets inXwith<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M406">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M407','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M407">View MathML</a>. Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M408','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M408">View MathML</a>be a completely continuous operator such that either

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M409">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M410','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M410">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M411','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M411">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M412">View MathML</a>

or

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M411','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M411">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M410','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M410">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M409">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M412">View MathML</a>.

ThenAhas a fixed point in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M417','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M417">View MathML</a>.

Lemma 3.3Assume<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M121">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M3">View MathML</a>, (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M87">View MathML</a>) and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M90">View MathML</a>). Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M422','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M422">View MathML</a>be a compact subset of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M423','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M423">View MathML</a>. Then there exists a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M424','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M424">View MathML</a>such that for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M425','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M425">View MathML</a>and all possible positive solutions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M142">View MathML</a>of (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M12">View MathML</a>), one has<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M428','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M428">View MathML</a>.

Proof If it is not true, then there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M429','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M429">View MathML</a> and solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M430','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M430">View MathML</a> of (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M431">View MathML</a>) such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M432','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M432">View MathML</a>. We note that

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M434','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M434">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M435','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M435">View MathML</a>

This implies both <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M436','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M436">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M437','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M437">View MathML</a>. Moreover, by the above estimation,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M438','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M438">View MathML</a>

Thus we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M439','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M439">View MathML</a>

as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M436','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M436">View MathML</a> and this contradiction completes the proof. □

Lemma 3.4Assume<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M121">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M3">View MathML</a>, (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M87">View MathML</a>) and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M90">View MathML</a>). If (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M12">View MathML</a>) has a lower solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M446','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M446">View MathML</a>for some<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M447','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M447">View MathML</a>, then (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M12">View MathML</a>) has a solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M142">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M450','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M450">View MathML</a>.

Proof It suffices to show the existence of an upper solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M451','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M451">View MathML</a> of (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M12">View MathML</a>) satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M453','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M453">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M454','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M454">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M3">View MathML</a> be positive solutions of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M456','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M456">View MathML</a>

(Case I) Both f and g are bounded.

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M454','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M454">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M3">View MathML</a>) are positive concave functions and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M459','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M459">View MathML</a>, we may choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M177">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M461','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M461">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M462','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M462">View MathML</a>. We now show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M463','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M463">View MathML</a> is an upper solution of (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M12">View MathML</a>). In fact,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M465','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M465">View MathML</a>

Similarly,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M466','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M466">View MathML</a>

(Case II) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M467','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M467">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M468','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M468">View MathML</a>.

Using (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M87">View MathML</a>), choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M177">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M471','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M471">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M472','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M472">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M473','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M473">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M474','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M474">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M475','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M475">View MathML</a>

And

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M476','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M476">View MathML</a>

Thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M165">View MathML</a> is an upper solution of (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M12">View MathML</a>).

(Case III) g is bounded and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M479','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M479">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M468','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M468">View MathML</a>.

Choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M177">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M482','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M482">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M483','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M483">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M484','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M484">View MathML</a> and let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M485','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M485">View MathML</a>

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M486','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M486">View MathML</a>

And

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M487','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M487">View MathML</a>

Consequently, by Theorem 2.2, (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M12">View MathML</a>) has a solution satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M489','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M489">View MathML</a>

 □

Lemma 3.5Assume<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M121">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M3">View MathML</a>, (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M84">View MathML</a>), (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M87">View MathML</a>) and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M90">View MathML</a>). Then there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M495','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M495">View MathML</a>such that if (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M12">View MathML</a>) has a positive solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M142">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M498','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M498">View MathML</a>.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M142">View MathML</a> be a positive solution of (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M12">View MathML</a>). Without loss of generality, we may assume <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M501','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M501">View MathML</a>. From (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M84">View MathML</a>), we know that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M503','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M503">View MathML</a>

(3.1)

From (3.1) and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M87">View MathML</a>), we can choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M505','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M505">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M506','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M506">View MathML</a>

(3.2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M507','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M507">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M3">View MathML</a>. Using (3.2) and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M90">View MathML</a>), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M510','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M510">View MathML</a>

Thus we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M511','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M511">View MathML</a>

 □

Lemma 3.6Assume<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M121">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M3">View MathML</a>, (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M84">View MathML</a>), (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M87">View MathML</a>) and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M90">View MathML</a>). Then for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M289">View MathML</a>, there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M518','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M518">View MathML</a>such that for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M519','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M519">View MathML</a>, (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M12">View MathML</a>) has a positive solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M142">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M522','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M522">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M523','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M523">View MathML</a>.

Proof We know that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M142">View MathML</a> satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M401','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M401">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M526','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M526">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M142">View MathML</a> is a solution of (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M12">View MathML</a>). Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M529','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M529">View MathML</a> are completely continuous, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M530','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M530">View MathML</a> is also completely continuous. Given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M289">View MathML</a>, choose

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M532','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M532">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M533','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M533">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M534','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M534">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M535','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M535">View MathML</a>, then for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M536','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M536">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M537','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M537">View MathML</a>. From the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M538','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M538">View MathML</a>, we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M539','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M539">View MathML</a> is the maximum value of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M538','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M538">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M302">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M542','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M542">View MathML</a>, then from the choice of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M543','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M543">View MathML</a>, we have

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M545','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M545">View MathML</a>, then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M546','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M546">View MathML</a>

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M547','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M547">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M548','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M548">View MathML</a>

By the concavity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M538','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M538">View MathML</a>, we get for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M550','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M550">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M551','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M551">View MathML</a>

(3.3)

By similar argument as the above, with (3.3), we may show that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M552','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M552">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M553','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M553">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M3">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M555','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M555">View MathML</a>, from (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M87">View MathML</a>), we may choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M557','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M557">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M558','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M558">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M559','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M559">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M560','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M560">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M561','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M561">View MathML</a> and for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M562','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M562">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M563','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M563">View MathML</a>

By Proposition 3.2, (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M564','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M564">View MathML</a>) has a positive solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M565','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M565">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M566','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M566">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M567','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M567">View MathML</a>. We know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M565','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M565">View MathML</a> is a lower solution of (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M12">View MathML</a>) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M570','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M570">View MathML</a> and by Lemma 3.4, the proof is complete. □

We now prove one of the main results for this paper.

Proof of Theorem 1.2

From Lemma 3.6 and Lemma 3.5, we know that the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M571','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M571">View MathML</a> is not empty and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M572','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M572">View MathML</a>. By Lemma 3.3 and complete continuity of T, there exist sequences <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M573','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M573">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M574','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M574">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M575','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M575">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M576','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M576">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M577','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M577">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M578','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M578">View MathML</a> a solution of (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M579','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M579">View MathML</a>). We claim that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M578','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M578">View MathML</a> is a nontrivial solution of (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M579','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M579">View MathML</a>). Suppose that it is not true, then there exists a sequence of solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M582','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M582">View MathML</a> for (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M431">View MathML</a>) such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M584','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M584">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M585','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M585">View MathML</a>. As in the proof of Lemma 3.3, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M586','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M586">View MathML</a>

But from (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M84">View MathML</a>), we have a contradiction to the fact that the right side of the above inequality converges to zero as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M588','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M588">View MathML</a>. Thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M578','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M578">View MathML</a> is a nontrivial solution of (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M590','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M590">View MathML</a>). According to Lemma 3.4 and the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M591','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M591">View MathML</a>, we know that (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M12">View MathML</a>) has at least one positive solution at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M593','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M593">View MathML</a> and no positive solution for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M128">View MathML</a>. To prove the existence of the second positive solution of (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M12">View MathML</a>) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M596','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M596">View MathML</a>, we will use Theorem 1.1. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M596','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M596">View MathML</a>. Then we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M598','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M598">View MathML</a> a lower solution of (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M12">View MathML</a>) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M600','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M600">View MathML</a> a strict lower solution of (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M12">View MathML</a>) in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M62">View MathML</a> satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M603','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M603">View MathML</a>. For upper solutions, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M352">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M353">View MathML</a> be the first eigenvalues of for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M3">View MathML</a> respectively and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M355">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M356">View MathML</a> be corresponding eigenfunctions with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M609','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M609">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M355">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M356">View MathML</a> are in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M39">View MathML</a> and positive [19], we may choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M613','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M613">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M614','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M614">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M615','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M615">View MathML</a>

Also by the fact <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M616','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M616">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M184">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M618','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M618">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M619','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M619">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M620','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M620">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M621','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M621">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M622','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M622">View MathML</a> and it is a strict upper solution of (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M12">View MathML</a>) in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M62">View MathML</a>. Indeed,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M625','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M625">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M626','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M626">View MathML</a>

Finally, from Lemma 3.6, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M627','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M627">View MathML</a> such that (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M628','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M628">View MathML</a>) has a positive solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M629','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M629">View MathML</a> satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M630','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M630">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M631','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M631">View MathML</a>. By using the concavity of solutions, it is easily verified that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M632','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M632">View MathML</a>

Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M633','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M633">View MathML</a> is an upper solution of (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M12">View MathML</a>) in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M635','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M635">View MathML</a>. Now by Theorem 1.1, (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M12">View MathML</a>) has at least two positive solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M66">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M67">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M639','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M639">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M640','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M640">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M641','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M641">View MathML</a><a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M642','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M642">View MathML</a>.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors have equally contributed in obtaining new results in this article and also read and approved the final manuscript.

Acknowledgements

The authors express their thanks to Professors Ryuji Kajikiya, Yuki Naito and Inbo Sim for valuable discussions related to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/63/mathml/M34">View MathML</a>-regularity of solutions and also thank to the referees for their careful reading and valuable remarks and suggestions. The first author was supported by Pusan National University Research Grant, 2011. The second author was supported by Mid-career Researcher Program (No. 2010-0000377) and Basic Science Research Program (No. 2012005767) through NRF grant funded by the MEST.

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