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Multiplicity of positive solutions for Kirchhoff type problem involving critical exponent and sign-changing weight functions

Chang-Mu Chu

Author Affiliations

College of Science, Guizhou Minzu University, Guiyang, Guizhou, 550025, People’s Republic of China

Boundary Value Problems 2014, 2014:19  doi:10.1186/1687-2770-2014-19

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/19


Received:29 August 2013
Accepted:25 December 2013
Published:16 January 2014

© 2014 Chu; 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 devoted to the study of a class of Kirchhoff type problems with critical exponent, concave nonlinearity, and sign-changing weight functions. By means of variational methods, the multiplicity of the positive solutions to this problem is obtained.

MSC: 35J20, 35J60, 47J30, 58E50.

Keywords:
Kirchhoff type problem; critical exponent; concave nonlinearity; sign-changing weight functions; variational methods

1 Introduction and main results

This paper is concerned with the existence and multiplicity of positive solutions for the following problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M1">View MathML</a>

(1)

where Ω is a smooth bounded domain in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M2">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M3">View MathML</a> and the parameters <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M4">View MathML</a>. The weight functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M6">View MathML</a> satisfy the following conditions:

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M7">View MathML</a>) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M8">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M9">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M10">View MathML</a>;

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M11">View MathML</a>) there exist positive constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M12">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M13">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M14">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M15">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M16">View MathML</a>;

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M17">View MathML</a>) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M18">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M19">View MathML</a>;

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M20">View MathML</a>) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M21">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M22">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M23">View MathML</a>;

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M24">View MathML</a>) there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M25">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M26">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M27">View MathML</a>.

In (1), if we replace <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M28">View MathML</a> by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M29">View MathML</a>, it reduces to the following Dirichlet problem of Kirchhoff type:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M30">View MathML</a>

(2)

Problem (2) is related to the stationary analogue of the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M31">View MathML</a>

proposed by Kirchhoff in [1] as an extension of the classical d’Alembert wave equation for free vibrations of elastic strings. Kirchhoff’s model takes into account the changes in the length of the string produced by transverse vibrations. It received great attention only after Lions [2] proposed an abstract framework for the problem. The solvability of the Kirchhoff type problem (2) has been paid much attention to by various authors. The positive solutions of such a problem are obtained by using variational methods [3-5]. Perera and Zhang [6] obtained a nontrivial solution of problem (2) via the Yang index and the critical group. He and Zou [7] obtained infinitely many solutions by using the local minimum methods and the fountain theorems. Recently, when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M29">View MathML</a> is a continuous superlinear nonlinearity with critical growth, the existence of positive solutions of the Kirchhoff type problem has been studied [8-13]. Moreover, the paper [14] considered problem (2) with concave and convex nonlinearities by using a Nehari manifold and fibering map methods, and one obtained the existence of multiple positive solutions. In addition, the corresponding results of the Kirchhoff type problem can be found in [15-25], and the references therein.

In the present paper, we deal with problem (1) and consider the existence and multiplicity of positive solutions of problem (1). About the critical growth situation, the aforementioned papers only showed the existence of positive solutions of the Kirchhoff type problem. Moreover, involving the concave and convex nonlinearities, [14] only considered the subcritical growth case. Therefore, our purpose is to extend the result of [14] to critical growth. The main results of this paper extend the corresponding results in [11] and [14].

Before stating our results, we give some notations and assumptions. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M33">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M34">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M35">View MathML</a>), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M36">View MathML</a>. In addition, we denote positive constants by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M37">View MathML</a> . The main results of this paper are as follows.

Theorem 1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M38">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M39">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M3">View MathML</a>. Suppose that (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M7">View MathML</a>) and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M17">View MathML</a>) hold, then there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M43">View MathML</a>such that problem (1) for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M44">View MathML</a>has at least one positive solution.

Theorem 2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M38">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M39">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M3">View MathML</a>. Suppose that (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M7">View MathML</a>), (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M11">View MathML</a>), (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M17">View MathML</a>), (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M20">View MathML</a>) and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M24">View MathML</a>) hold, then there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M53">View MathML</a>such that problem (1) for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M54">View MathML</a>has at least two positive solutions.

Remark 1 Our Theorem 2 extends the results for the critical case of Theorem 1.1 in [11]. Our Theorem 2 shows that we have at least two positive solutions of problem (1), but the authors of the reference only obtain at least one positive solution of problem (1). In addition, the results of Theorem 2.1 in [14] are extended to critical growth.

This paper is organized as follows. In Section 2, we give the local Palais-Smale condition. The proof of Theorems 1 and 2 is provided in Section 3.

2 The local Palais-Smale condition

In this section, we show that the corresponding functional of problem (1) satisfies the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M55">View MathML</a> condition. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M56">View MathML</a>, the corresponding functional of problem (1) is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M57">View MathML</a>

It is well known that the critical points of the functional I in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M58">View MathML</a> are weak solutions of problem (1). By the definition of weak solution u of problem (1), it means that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M59">View MathML</a> satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M60">View MathML</a>

for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M61">View MathML</a>.

Define the best Sobolev constant,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M62">View MathML</a>

(3)

From [26], we know that S is attained when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M63">View MathML</a> by functions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M64">View MathML</a>

Definition A sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M65">View MathML</a> is called a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M55">View MathML</a> sequence of I if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M67">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M68">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M69">View MathML</a>. We say that I satisfies the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M55">View MathML</a> condition if any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M55">View MathML</a> sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M65">View MathML</a> of I has a convergent subsequence.

Lemma 1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M38">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M39">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M3">View MathML</a>. Assume that (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M7">View MathML</a>) and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M17">View MathML</a>) hold. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M65">View MathML</a>is a<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M55">View MathML</a>sequence ofI, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M80">View MathML</a>is bounded in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M58">View MathML</a>.

Proof By the Hölder inequality and the Young inequality, it follows from (3) and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M7">View MathML</a>) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M83">View MathML</a>

(4)

for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M59">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M85">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M80">View MathML</a> be a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M55">View MathML</a> sequence of I. It follows from (4) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M88">View MathML</a>

which implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M89">View MathML</a>

Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M90">View MathML</a>, we see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M80">View MathML</a> is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M58">View MathML</a>. □

Lemma 2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M38">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M39">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M3">View MathML</a>. Assume that (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M7">View MathML</a>) and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M17">View MathML</a>) hold. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M80">View MathML</a>is a<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M55">View MathML</a>sequence ofI, then there exists a positive constantAdepending ona, q, S, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M100">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M101">View MathML</a>

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M80">View MathML</a> be a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M55">View MathML</a> sequence of I. By Lemma 1, we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M80">View MathML</a> is bounded. Therefore, up to a subsequence, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M59">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M106">View MathML</a> converges weakly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M58">View MathML</a>, strongly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M108">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M109">View MathML</a> and a.e. in Ω. By (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M7">View MathML</a>), (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M11">View MathML</a>), and the Dominated Convergence Theorem, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M112">View MathML</a>

Thus, by using also the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M113">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M114">View MathML</a>

from which it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M115">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M58">View MathML</a>. Since I is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M117">View MathML</a>, we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M118">View MathML</a>. In particular, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M119">View MathML</a>, which implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M120">View MathML</a>

It follows from (4) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M121">View MathML</a>

Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M122">View MathML</a>, and we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M123">View MathML</a>. □

Lemma 3Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M38">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M39">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M3">View MathML</a>. Assume that (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M7">View MathML</a>) and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M17">View MathML</a>) hold, thenIsatisfies the<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M55">View MathML</a>condition with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M130">View MathML</a> = <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M131">View MathML</a> + <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M132">View MathML</a> + <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M133">View MathML</a> + <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M134">View MathML</a><a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M135">View MathML</a>, whereAis the positive constant given in Lemma 2.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M65">View MathML</a> be a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M55">View MathML</a> sequence of I with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M130">View MathML</a>. By Lemma 1, we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M80">View MathML</a> is bounded. Up to a subsequence, we may assume that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M140">View MathML</a>

From Lemma 2, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M118">View MathML</a>. By (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M7">View MathML</a>) and the Dominated Convergence Theorem, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M143">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M144">View MathML</a>; by the Brezis-Lieb lemma [27], one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M145">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M146">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M147">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M148">View MathML</a>

(5)

According to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M149">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M119">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M151">View MathML</a>

(6)

Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M152">View MathML</a>, it follows from (6) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M153">View MathML</a>

From (3), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M154">View MathML</a>

As <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M155">View MathML</a>, we deduce that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M156">View MathML</a>

It follows from (5), (6), and Lemma 2 that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M157">View MathML</a>

which contradicts the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M130">View MathML</a>. Therefore, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M159">View MathML</a>, which implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M115">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M58">View MathML</a>. Hence I satisfies the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M55">View MathML</a> condition with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M130">View MathML</a>. □

3 The proof of the main results

In this section, we show the proofs of our Theorems 1 and 2. Before we come to the proof of Theorem 1, we first recall the following lemma in [28].

Lemma 4Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M164">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M165">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M166">View MathML</a>. Then there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M167">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M168">View MathML</a>.

Proof of Theorem 1 Using the hypotheses (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M7">View MathML</a>) and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M17">View MathML</a>), it follows from (3) and (4) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M171">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M172">View MathML</a>, we can find <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M173">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M174">View MathML</a> such that for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M175">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M176">View MathML</a>

(7)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M177">View MathML</a>.

From Lemma 4, we obtain the result that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M178">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M179">View MathML</a>

(8)

Therefore, one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M180">View MathML</a>

Fix <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M175">View MathML</a>; noticing that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M3">View MathML</a>, it implies from (8) that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M183">View MathML</a> small enough such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M184">View MathML</a>. Thus we deduce that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M185">View MathML</a>

By applying the Ekeland’s variational principle in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M186">View MathML</a>[29], we obtain the result that there exists a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M187">View MathML</a> sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M188">View MathML</a> of I.

By the expression of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M189">View MathML</a>, we can choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M190">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M191">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M44">View MathML</a>. It follows from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M193">View MathML</a> and Lemma 3 that I satisfies the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M187">View MathML</a> condition. Therefore, one has a subsequence still denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M80">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M196">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M197">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M58">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M199">View MathML</a>

which implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M200">View MathML</a> is a solution of problem (1). After a direct calculation, we derive <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M201">View MathML</a>, which implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M202">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M203">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M204">View MathML</a>. Applying the Harnack inequality [30], we see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M200">View MathML</a> is a positive solution of problem (1). The proof of Theorem 1 is completed. □

Lemma 5Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M38">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M39">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M3">View MathML</a>. Assume that (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M7">View MathML</a>), (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M11">View MathML</a>), (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M17">View MathML</a>), (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M20">View MathML</a>), and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M24">View MathML</a>) hold, then there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M214">View MathML</a>, such that for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M215">View MathML</a>, we can find<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M216">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M217">View MathML</a>.

Proof For convenience, we consider the functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M218">View MathML</a> defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M219">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M59">View MathML</a>. According to (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M20">View MathML</a>) and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M24">View MathML</a>), we can choose such a cut-off function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M223">View MathML</a> that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M224">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M225">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M226">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M227">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M228">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M229">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M230">View MathML</a> is a positive constant. Define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M231">View MathML</a>

According to (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M20">View MathML</a>) and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M24">View MathML</a>), similar to the calculation of [31], we have the following estimate (as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M234">View MathML</a>)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M235">View MathML</a>

(9)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M236">View MathML</a>. Define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M237">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M238">View MathML</a>. From (9), we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M239">View MathML</a>. Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M240">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M241">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M242">View MathML</a>, so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M243">View MathML</a> is attained for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M244">View MathML</a>. By

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M245">View MathML</a>

one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M246">View MathML</a>

Therefore, we deduce from (9) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M247">View MathML</a>

(10)

By the expression of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M189">View MathML</a>, we can choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M249">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M191">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M251">View MathML</a>. Using the definitions of I and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M252">View MathML</a>, from (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M11">View MathML</a>) and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M20">View MathML</a>), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M255">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M238">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M257">View MathML</a>. It follows that there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M258">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M259">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M260">View MathML</a>

(11)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M261">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M262">View MathML</a>. Moreover, using the definitions of I and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M252">View MathML</a>, it follows from (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M11">View MathML</a>) and (10) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M265">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M266">View MathML</a>, it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M267">View MathML</a>

By the above two inequalities, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M268">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M269">View MathML</a>

(12)

Hence, we can choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M270">View MathML</a> such that for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M271">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M272">View MathML</a>

Therefore, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M271">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M274">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M275">View MathML</a>

(13)

Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M276">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M277">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M274">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M279">View MathML</a>, we deduce from (11) and (13) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M280">View MathML</a>

 □

Proof of Theorem 2 Choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M281">View MathML</a>, from the proof of Theorem 1, we have already seen that problem (1) for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M54">View MathML</a> has a positive solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M200">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M284">View MathML</a>. Now we only need to find the second positive solution of problem (1). According to (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M7">View MathML</a>), we can see that (4) and (7) hold. It follows from (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M17">View MathML</a>) and Lemma 4 that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M287">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M288">View MathML</a>

According to (4), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M289">View MathML</a>

which implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M290">View MathML</a>

Hence, there exists a positive number <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M291">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M292">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M293">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M54">View MathML</a>. It implies from (7) that the functional I has the mountain pass geometry. Define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M295">View MathML</a>

From Lemma 5, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M296">View MathML</a>. Applying Lemma 3, we know that I satisfies the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M297">View MathML</a> condition. By the Mountain Pass Theorem [32], we obtain the result that problem (1) has the second solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M298">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M299">View MathML</a>. After a direct calculation, we derive

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M300">View MathML</a>

which implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M301">View MathML</a>. Hence we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M302">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M303">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M304">View MathML</a>. By the Harnack inequality, we obtain the result that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/19/mathml/M305">View MathML</a> is the second positive solution of problem (1). The proof of Theorem 2 is completed. □

Competing interests

The author declares that they have no competing interests.

Acknowledgements

The author would like to thank the referees for valuable comments and suggestions on improving this paper. This paper was supported by Science and Education Youth culture project in Guizhou Province (Contract Number: Guizhou Provincial Institute of Zi (2012) No. 157). This paper also was supported by Science and Technology Foundation of Guizhou Province (No. J[2013]2141; No. LKM[2011]31).

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