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This article is part of the series Proceedings of the International Congress in Honour of Professor Hari M. Srivastava.

Open Access Research

Positive solutions for Sturm-Liouville BVPs on time scales via sub-supersolution and variational methods

Quan-Guo Zhang1, Xi-Ping He2 and Hong-Rui Sun1*

Author Affiliations

1 School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu, 730000, People’s Republic of China

2 Center of Teaching Guidance, Gansu Radio and TV University, Lanzhou, Gansu, 730000, People’s Republic of China

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Boundary Value Problems 2013, 2013:123  doi:10.1186/1687-2770-2013-123


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/123


Received:20 January 2013
Accepted:26 April 2013
Published:13 May 2013

© 2013 Zhang et al.; 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

This paper is concerned with the existence of one and two positive solutions for the following Sturm-Liouville boundary value problem on time scales

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M1">View MathML</a>

Under a locally nonnegative assumption on the nonlinearity f and some other suitable hypotheses, positive solutions are sought by considering the corresponding truncated problem, constructing the variational framework and combining the sub-supersolution method with the mountain pass lemma.

MSC: 34B10, 34B18.

Keywords:
positive solution; Sturm-Liouville; time scales; sub-supersolution; variational methods

1 Introduction

The theory of dynamic equations on time scales has become a new important mathematical branch [1,2] since it was initiated by Hilger in 1988 [3]. Since then, boundary value problems (BVPs) for dynamic equations on time scales have received considerable attention, various fixed point theorems, sub-supersolution method and Leray-Schauder degree theory have been applied to get many interesting results about the existence of solutions; see [1,2,4-12] and the references therein. Variational method is also an important method for dealing with the existence of BVPs. Recently, some authors have used the theory to study the existence of solutions of some BVPs on time scales [4,5,13,14].

Especially, in [4,5], Agarwal et al. studied the following dynamic equation on time scales:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M2">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M3">View MathML</a>

with the Dirichlet boundary condition. They gave some sufficient conditions for the existence of single and multiple positive solutions by using the variational method and critical point theory.

In [14], we considered the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M4">View MathML</a>

(1.1)

and obtained the existence of many solutions depending on the value of the parameter λ which lie in some different intervals under some suitable hypotheses. The main approaches are also the variational method and some known critical point theorems and a three critical point theorem established in [15]. Erbe et al.[8] also established some existence criteria of positive solutions by a fixed point theorem in a cone with the globally nonnegative hypothesis of f.

Motivated by the papers [4,5,8,14], in this paper, we continue to study the problem (1.1) in the case of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M5">View MathML</a>, that is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M6">View MathML</a>

(1.2)

Here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M7">View MathML</a> is a time scale and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M8">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M9">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M10">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M11">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M12">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M13">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M14">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M15">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M16">View MathML</a>. The purpose of this paper is to discuss the existence and multiplicity of positive solutions to the problem (1.2) under the local non-negativity assumption of f and some other hypotheses. The main tools are the truncated method, the variational method, the sub-supersolution method and the mountain pass lemma. First, inspired by the method in [4], we convert the existence of a positive solution of (1.2) to the existence of a solution of an associated problem of (1.2). In contrast with the paper [4], the appearance of term <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M17">View MathML</a>, our problem is more complicated and the proof is also different from [4] (see Lemma 2.3 for details). Next, we construct a supersolution of (1.2) and give the existence of one positive solution. Finally, under our weaker assumption on f (see (H1) and (H2) for details), since we cannot verify that the corresponding functional for the associated problem satisfies the P.S. condition, we consider the corresponding truncated problem. To prove the existence of the second positive solution by the mountain pass lemma, we also give an estimate of a nonnegative solution of (1.2) and prove the solution of a truncated problem is also a solution of (1.2) for n large enough (see Theorem 3.3 for details). To the best of our knowledge, the results are new both in the continuous and in the discrete case.

The paper is organized as follows. In Section 2, we present some basic properties of some related Sobolev space on time scales, construct the variational framework, give some properties of this framework and some necessary lemmas. In Section 3, we firstly get the existence of a single positive solution of (1.2) by using the sub-supersolution method; then applying the truncated method, analytic technique and mountain pass lemma, we establish the existence of two positive solutions.

2 Preliminaries and variational formulation

In this section, we list the definition and properties of the Sobolev space on time scales [16], give some lemmas which we need for the proof of the main result and construct a variational framework.

For convenience, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M18">View MathML</a>[16,17], we denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M19">View MathML</a>. We let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M20">View MathML</a>[18] denote the class of absolutely continuous functions on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M21">View MathML</a> and the Sobolev space is defined as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M22">View MathML</a>

with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M23">View MathML</a>

We know the immersion <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M24">View MathML</a> is compact. Analogous to the proof in the real numbers situation, one can deduce the following result on time scales.

Lemma 2.1If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M25">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M26">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M27">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M28">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M29">View MathML</a>

Using Lemma 2.1, by a similar proof of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M30">View MathML</a>, one can derive the following.

Lemma 2.2If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M27">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M32">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M33">View MathML</a>, Δ-a.e. in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M21">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M35">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M36">View MathML</a>.

For convenience, we denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M37">View MathML</a>

and for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M38">View MathML</a>, we set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M39">View MathML</a>

In order to discuss the existence of a positive solution of (1.2), we consider the following problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M40">View MathML</a>

(2.1)

First, we give an important lemma.

Lemma 2.3If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M41">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M42">View MathML</a>, uis a solution of (2.1), thenuis nonnegative in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M21">View MathML</a>. Furthermore, if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M44">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M42">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M17">View MathML</a>is nondecreasing in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M47">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M48">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M49">View MathML</a>.

Proof Let u be a solution of (2.1). In view of Lemma 2.2, we know <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M32">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M33">View MathML</a>, Δ-a.e. in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M21">View MathML</a>. Multiplying (2.1) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M53">View MathML</a>, integrating over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M54">View MathML</a> and employing the integration by parts formula for an absolutely continuous function on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M7">View MathML</a>, we find that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M56">View MathML</a>

Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M57">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M58">View MathML</a>.

Next, we show that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M17">View MathML</a> is nondecreasing in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M47">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M44">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M42">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M48">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M64">View MathML</a>.

In fact, if the conclusion is false, we can suppose that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M65">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M66">View MathML</a> and one of the following two cases holds:

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M48">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M68">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M69">View MathML</a>,

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M70">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M71">View MathML</a> and there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M72">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M48">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M74">View MathML</a>.

For the case (i), if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M75">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M76">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M72">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M78">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M79">View MathML</a>. According to the nonnegativity of u on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M21">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M66">View MathML</a>, it is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M75">View MathML</a> is impossible. Thus we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M83">View MathML</a>

(2.2)

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M84">View MathML</a>, then we know <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M85">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M86">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M87">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M88">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M89">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M90">View MathML</a>. Hence, in this case, we always have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M91">View MathML</a>

(2.3)

Therefore

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M92">View MathML</a>

(2.4)

However, since u is a solution of (1.2), (2.2) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M66">View MathML</a>, we know that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M94">View MathML</a>

(2.5)

which contradicts (2.4).

For the case (ii), it can be divided into two cases to consider.

(1) If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M95">View MathML</a>, then one can deduce that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M89">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M97">View MathML</a>. From this and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M98">View MathML</a>, we have a contradiction.

(2) If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M99">View MathML</a>, then we always have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M100">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M101">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M76">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M90">View MathML</a>. Similar to (i), we get (2.2) and (2.3). But this is impossible from (2.4) and (2.5).

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M104">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M105">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M106">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M107">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M108">View MathML</a>. So, we get a contradiction to (2.4) and (2.5).

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M104">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M110">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M89">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M90">View MathML</a>. Hence, we get (2.2) and (2.3). But this contradicts (2.4) and (2.5).

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M104">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M114">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M89">View MathML</a>. By assumption, u is a solution of (1.2), so we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M116">View MathML</a>, which contradicts <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M89">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M48">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M119">View MathML</a>. □

Remark 2.4 From the proof of Lemma 2.3, we can easily find that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M30">View MathML</a>, then the monotonicity assumption of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M17">View MathML</a> can be omitted.

By Lemma 2.3, under the hypothesis

(H1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M17">View MathML</a> is nondecreasing in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M47">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M44">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M42">View MathML</a>,

in order to prove the existence of a positive solution of (1.2), it suffices to consider the existence of a solution of (2.1). Now we establish the corresponding variational formulations for (2.1). We set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M126">View MathML</a>

then E is a Banach space with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M127">View MathML</a>, and we can find that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M128">View MathML</a> can be taken as an equivalent norm on E. Define the functional I on E as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M129">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M130">View MathML</a>.

Note that the appearance of the term <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M131">View MathML</a> in the functional I guarantees that I is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M132">View MathML</a>, see next lemma. By the definition of Fréchet derivative and the fact that the immersion <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M133">View MathML</a> is compact, we have the following results.

Lemma 2.5The following statements are valid.

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M134">View MathML</a>, and for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M135">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M136">View MathML</a>

(ii) We define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M137">View MathML</a>

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M138">View MathML</a>

Jis weakly continuous inEand<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M139">View MathML</a>is compact.

(iii) The solutions of (2.1) match up to the critical points ofIinE.

For the eigenvalue problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M140">View MathML</a>

(2.6)

we have the following lemma.

Lemma 2.6 [[14], Lemma 3.1]

The eigenvalues of (2.6) may be arranged as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M141">View MathML</a> , and there exists a countable orthonormal basis ofEconsisting of eigenfunction associated eigenvalues of (2.6) and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M142">View MathML</a>

(2.7)

Remark 2.7 By (2.7) and Lemma 2.2, we know the eigenfunction <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M143">View MathML</a> corresponding to the eigenvalue <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M144">View MathML</a> satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M145">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M64">View MathML</a>. Furthermore, by the Krein-Rutman theorem [[19], Theorem 7.C], we know <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M147">View MathML</a> with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M148">View MathML</a>

Lemma 2.8 [[1], Theorem 4.73], [8]

The problem (1.2) has the Green function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M149">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M150">View MathML</a>, ψ, φare solutions of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M151">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M152">View MathML</a>

respectively, and satisfy<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M153">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M154">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M155">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M156">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M157">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M158">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M159">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M160">View MathML</a>.

Lemma 2.9The function defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M161">View MathML</a>

belongs to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M162">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M163">View MathML</a>is given in Remark 2.7.

Proof Clearly, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M164">View MathML</a> is well defined in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M165">View MathML</a>.

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M166">View MathML</a>, by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M147">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M168">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M169">View MathML</a>. Hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M170">View MathML</a>.

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M171">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M172">View MathML</a>. By Remark 2.7, we know <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M173">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M174">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M175">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M176">View MathML</a>, then there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M72">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M145">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M179">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M180">View MathML</a>, then by L’Hôspital rule [[1], Theorem 1.119] and Lemma 2.8, we know

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M181">View MathML</a>

Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M164">View MathML</a> is bounded for s close to 0.

Similarly, we can derive that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M164">View MathML</a> is bounded for s close to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M184">View MathML</a>. □

In order to derive the main result, we list the following well-known mountain pass lemma.

Lemma 2.10 [[20], Theorem 6.1]

Suppose<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M185">View MathML</a>satisfies the P.S. condition. Suppose<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M186">View MathML</a>and

(i) there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M187">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M188">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M189">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M190">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M191">View MathML</a>;

(ii) there is<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M192">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M193">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M194">View MathML</a>.

Define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M195">View MathML</a>

Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M196">View MathML</a>is a critical value.

3 Main results

In this section, we establish some existence criteria of a positive solution of (1.1) by employing the sub-supersolution method and critical point theory.

First, using a method analogous to that in [21], we construct a supersolution to employ the sub-supersolution method.

Theorem 3.1Assume that (H1) holds and there are constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M197">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M198">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M199">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M200">View MathML</a>is fixed, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M201">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M202">View MathML</a>represents the unique positive solution of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M203">View MathML</a>

Then the problem (1.2) has at least one positive solution.

Proof For fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M200">View MathML</a>, let v be the unique positive solution of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M205">View MathML</a>

(3.1)

Then, from the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M202">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M207">View MathML</a>. Then, by the assumptions, it is easy to see that v is a supersolution of (1.2). In addition, condition (H1) guarantees that the constant function 0 is a strict subsolution of (1.2). Therefore, the sub-supersolution method implies (1.2) has a positive solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M208">View MathML</a>. □

Remark 3.2 Furthermore, by Lemma 2.8, we know

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M209">View MathML</a>

Theorem 3.3Under the hypothesis of Theorem 3.1 and suppose the condition

(H2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M210">View MathML</a>uniformly for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M42">View MathML</a>

holds, then the problem (1.2) has at least two positive solutions.

In order to prove this theorem, we first present some necessary lemmas.

Lemma 3.4Letv, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M208">View MathML</a>be given in the proof of Theorem 3.1, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M208">View MathML</a>is a local minimizer ofIinE.

Proof Denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M214">View MathML</a>

By the assumptions and Lemma 2.8, we easily find <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M215">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M216">View MathML</a>. Hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M208">View MathML</a> is a local minimizer of I in W.

Next, by a similar argument to that in [22], we assert that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M208">View MathML</a> is also a local minimizer of I in E.

In fact, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M208">View MathML</a> is not a local minimizer of I in E, then for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M220">View MathML</a> there is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M221">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M222">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M223">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M224">View MathML</a>. By the Lagrange multiplier rule, we know there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M225">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M226">View MathML</a>

(3.2)

Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M208">View MathML</a> is a solution of (1.2), so

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M228">View MathML</a>

Thus, from (3.2), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M229">View MathML</a>

Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M230">View MathML</a> is a solution of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M231">View MathML</a>

(3.3)

It is easy to show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M232">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M233">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M234">View MathML</a>. But this contradicts the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M208">View MathML</a> is a local minimizer of I in W. □

Next, under hypothesis (H2), in order to show the existence of the second positive solution of (1.2) by employing the mountain pass lemma, we need to show that I satisfies the P.S. condition. However, by (H2), we cannot justify this; therefore, we consider the truncation function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M236">View MathML</a> and the truncation functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M237">View MathML</a> defined as follows.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M238">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M239">View MathML</a> be an increasing positive sequence with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M240">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M241">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M242">View MathML</a> , define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M243">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M244">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M245">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M246">View MathML</a>

Lemma 3.5Assume that (H1) and (H2) hold, then there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M247">View MathML</a>such that the functional<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M237">View MathML</a>satisfies the P.S. condition inEfor<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M249">View MathML</a>.

Proof For given n, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M250">View MathML</a> be the P.S. sequence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M237">View MathML</a>, that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M252">View MathML</a> is bounded and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M253">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M254">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M255">View MathML</a> is bounded, one can deduce that I satisfies the P.S. condition by a similar proof to Proposition B.35 in [23].

Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M255">View MathML</a> is unbounded. Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M257">View MathML</a>

we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M258">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M254">View MathML</a>. Denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M260">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M261">View MathML</a>. So, without loss of generality, we can assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M262">View MathML</a> in E, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M263">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M264">View MathML</a>. Note that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M265">View MathML</a>

(3.4)

by the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M236">View MathML</a> and passing to limit in (3.4), one can derive that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M267">View MathML</a>.

In view of (H2) and the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M236">View MathML</a>, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M220">View MathML</a> small enough, there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M270">View MathML</a> independent of n, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M247">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M272">View MathML</a>

(3.5)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M273">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M274">View MathML</a>

(3.6)

Passing to the limit in (3.6), we know <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M275">View MathML</a>. Hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M276">View MathML</a>, which contradicts <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M267">View MathML</a>. Therefore <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M255">View MathML</a> is bounded. So, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M237">View MathML</a> satisfies the P.S. condition for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M249">View MathML</a>. □

By Lemmas 3.4 and 3.5, we deduce that for n large enough, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M237">View MathML</a> has a nontrivial critical point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M282">View MathML</a> by using the mountain pass lemma and Theorem 1 in [24]. In order to obtain a solution of (1.2), we need to get an estimate of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M282">View MathML</a>. Therefore, we first give an estimate of a nonnegative solution of (1.2) employing a method similar to that in [25].

Lemma 3.6Suppose (H2) holds, then there is<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M284">View MathML</a>such that for any nonnegative solutionuof (1.2), we have<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M285">View MathML</a>.

Proof If u is a nonnegative solution of (1.2), then by the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M163">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M287">View MathML</a>

(3.7)

Condition (H2) implies there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M288">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M289">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M290">View MathML</a>

(3.8)

Hence, from (3.7) and (3.8), we derive that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M291">View MathML</a>

So, using (3.7), we know

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M292">View MathML</a>

Thus, by (H2), we know <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M293">View MathML</a> is uniformly bounded. Note that by Lemma 2.9, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M294">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M295">View MathML</a>

Hence, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M284">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M285">View MathML</a>. □

Remark 3.7 Note that we only need (H2) to derive (3.8). Hence, (3.5) implies Lemma 3.6 is also valid for the truncation problem.

Proof of Theorem 3.3 Since the positive solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M208">View MathML</a> derived from Theorem 3.1 is a local minimizer of I and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M299">View MathML</a>, we can choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M300">View MathML</a> large enough such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M301">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M249">View MathML</a>. Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M208">View MathML</a> is also a local minimizer of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M237">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M249">View MathML</a>. Then, from the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M236">View MathML</a> and Lemma 3.5, we know the mountain pass lemma and Theorem 1 in [24] imply that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M237">View MathML</a> has the second critical point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M308">View MathML</a>. Furthermore, by Remark 3.7, we know <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M308">View MathML</a> is also a critical point of I. Thus the problem (1.2) has the second positive solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/123/mathml/M308">View MathML</a>. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All the authors typed, read, and approved the final manuscript.

Acknowledgements

Dedicated to Professor Hari M Srivastava.

The research of HR Sun has been supported by the program for New Century Excellent Talents in University (NECT-12-0246) and FRFCU (lzujbky-2013-k02).

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