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This article is part of the series Jean Mawhin’s Achievements in Nonlinear Analysis.

Open Access Research

A note on the existence of solutions for a class of quasilinear elliptic equations: an Orlicz-Sobolev space setting

Yang Yang1* and Jihui Zhang2

Author Affiliations

1 School of Science, Jiangnan University, Wuxi, 214122, People’s Republic of China

2 School of Mathematics Science, Nanjing Normal University, Nanjing, 210097, People’s Republic of China

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Boundary Value Problems 2012, 2012:136  doi:10.1186/1687-2770-2012-136


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/136


Received:13 April 2012
Accepted:8 October 2012
Published:22 November 2012

© 2012 Yang and Zhang; licensee Springer

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

Abstract

In this note, we study the existence and multiplicity of solutions for the quasilinear elliptic problem as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M1">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M2">View MathML</a> is a bounded domain with a smooth boundary. The existence and multiplicity of solutions are obtained by a version of the symmetric mountain pass theorem.

Keywords:
Orlicz-Sobolev spaces; symmetric mountain pass theorem; quasilinear elliptic equations

1 Introduction

In this note, we discuss the existence and multiplicity of solutions of the following boundary value problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M3">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M2">View MathML</a> is a bounded domain with a smooth boundary Ω. The function a is such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M5">View MathML</a> defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M6">View MathML</a>

is an increasing homeomorphism from R onto itself and the continuous function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M7">View MathML</a> satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M8">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M9">View MathML</a>. Especially, when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M10">View MathML</a>, the problem (1.1) is the well-known p-Laplacian equation. There is a large number of papers on the existence of solutions for the p-Laplacian equation. But the problem (1.1) possesses more complicated nonlinearities. For example, it is inhomogeneous and has an important physical background, e.g.,

(a) nonlinear elasticity: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M11">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M12">View MathML</a>;

(b) plasticity: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M13">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M14">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M15">View MathML</a>;

(c) generalized Newtonian fluids: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M16">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M17">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M15">View MathML</a>.

So, in the discussions, some special techniques are needed, and the problem (1.1) has been studied in an Orlicz-Sobolev space and received considerable attention in recent years; see, for instance, the papers [1-9]. In paper [9], Fang and Tan discussed the problem (1.1) under the conditions that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M19">View MathML</a> was odd in t. They got the first result that when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M20">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M21">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M22">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M23">View MathML</a>, the problem (1.1) had a sequence of solutions by genus theory. The second result is that when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M19">View MathML</a> satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M25">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M26">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M27">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M28">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M29">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M30">View MathML</a>, the problem (1.1) has infinitely many pairs of solutions which correspond to the positive critical values by the symmetric mountain pass theorem.

Motivated by their results, in this note, we discuss the problem (1.1) when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M19">View MathML</a> is still odd in t but it satisfies weaker conditions than [9]; and furthermore, we need not know the behaviors of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M19">View MathML</a> near the zero. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M33">View MathML</a>, we can get multiplicity of solutions by a version of the symmetric mountain pass theorem.

The paper is organized as follows. In Section 2, we present some preliminary knowledge on the Orlicz-Sobolev spaces and give the main result. In Section 3, we make the proof.

2 Preliminaries

Obviously, the problem (1.1) allows a nonhomogeneous function p in the differential operator defining the problem (1.1). To deal with this situation, we introduce an Orlicz-Sobolev space setting for the problem (1.1) as follows.

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M34">View MathML</a>

then P and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M35">View MathML</a> are complementary N-functions (see [10]), which define the Orlicz spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M36">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M37">View MathML</a> respectively.

Throughout this paper, we assume the following condition on P:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M38">View MathML</a>

Under the condition (p), the Orlicz space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M39">View MathML</a> coincides with the set (equivalence classes) of measurable functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M40">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M41">View MathML</a>

and is equipped with the (Luxemburg) norm, i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M42">View MathML</a>

We will denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M43">View MathML</a> the corresponding Orlicz-Sobolev space with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M44">View MathML</a>

and define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M45">View MathML</a> as the closure of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M46">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M43">View MathML</a>. In this note, we will use the following equivalent norm on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M45">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M49">View MathML</a>

Now, we introduce the Orlicz-Sobolev conjugate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M50">View MathML</a> of P, which is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M51">View MathML</a>

where we suppose that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M52">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M53">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M54">View MathML</a>. Throughout this paper, we assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M55">View MathML</a>. Now, we will make the following assumptions on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M19">View MathML</a>.

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M57">View MathML</a>) There exists an odd increasing homeomorphism h from R to R, and nonnegative constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M58">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M59">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M60">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M61">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M62">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M63">View MathML</a>

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M64">View MathML</a>

then we can obtain complementary N-functions which define corresponding Orlicz spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M65">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M66">View MathML</a>.

Similar to the condition (p), we also assume the following condition on H:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M67">View MathML</a>

In order to prove our results, we now state some useful lemmas.

Lemma 2.1[10]

Under the condition (p), the spaces<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M68">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M45">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M43">View MathML</a>are separable and reflexive Banach spaces.

Lemma 2.2[10]

Under the condition (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M57">View MathML</a>), the embedding<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M72">View MathML</a>is compact.

Lemma 2.3[2]

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M73">View MathML</a>, we have

(1) if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M74">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M75">View MathML</a>;

(2) if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M76">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M77">View MathML</a>;

(3) if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M78">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M79">View MathML</a>;

(4) if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M80">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M81">View MathML</a>.

Lemma 2.4[11-13]

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M82">View MathML</a>, whereEis a real Banach space andVis finite dimensional. Suppose<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M83">View MathML</a>is an even functional satisfying<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M84">View MathML</a>and

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M85">View MathML</a>) there is a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M86">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M87">View MathML</a>;

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M88">View MathML</a>) there is a subspaceWofEwith<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M89">View MathML</a>and there is<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M90">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M91">View MathML</a>;

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M92">View MathML</a>) considering<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M90">View MathML</a>given by (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M88">View MathML</a>), Isatisfies (PS)cfor<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M95">View MathML</a>.

ThenIpossesses at least<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M96">View MathML</a>pairs of nontrivial critical points.

Using the version of the symmetric mountain pass theorem mentioned above, we can state our result as follows.

Theorem 2.1Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M19">View MathML</a>is odd int, satisfies (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M57">View MathML</a>) with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M99">View MathML</a>and the following assumptions:

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M100">View MathML</a>) there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M101">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M102">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M103">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M104">View MathML</a>, such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M105">View MathML</a>for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M106">View MathML</a>, a.e. in Ω.

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M107">View MathML</a>) there is<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M108">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M109">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M110">View MathML</a>uniformly a.e. in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M111">View MathML</a>.

Then for any given<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M112">View MathML</a>, the problem (1.1) possesses at leastkpairs of nontrivial solutions.

3 Main results and proofs

In this section, we assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M113">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M114">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M115">View MathML</a> is called a weak solution of the problem (1.1) if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M116">View MathML</a>

Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M117">View MathML</a>

and we know that the critical points of I are just the weak solutions of the problem (1.1).

For E is a separable and reflexive Banach space, then there exist (see [9]) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M118">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M119">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M120">View MathML</a>

Now, we set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M121">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M122">View MathML</a>, so

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M123">View MathML</a>

(3.1)

Lemma 3.1Given<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M124">View MathML</a>, there is<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M125">View MathML</a>such that for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M126">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M127">View MathML</a>.

Proof We prove the lemma by contradiction. Suppose that there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M124">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M129">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M125">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M131">View MathML</a>. Taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M132">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M133">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M125">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M135">View MathML</a>. Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M136">View MathML</a> is a bounded sequence, and we may suppose, without loss of generality, that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M137">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M45">View MathML</a>. Furthermore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M139">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M140">View MathML</a> since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M141">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M142">View MathML</a>. This shows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M143">View MathML</a>. On the other hand, by the compactness of embedding <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M72">View MathML</a>, we conclude that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M145">View MathML</a>. This proves the lemma. □

Lemma 3.2Supposefsatisfies (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M57">View MathML</a>), then there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M125">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M148">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M149">View MathML</a>

Proof Now suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M150">View MathML</a>. From (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M57">View MathML</a>), we know that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M152">View MathML</a>

Consequently, considering <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M124">View MathML</a> to be chosen posteriorly by Lemma 3.1, we have for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M126">View MathML</a> and j sufficiently large,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M155">View MathML</a>

Now, taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M156">View MathML</a> and noting that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M157">View MathML</a>, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M158">View MathML</a>, we can choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M124">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M160">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M161">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M162">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M126">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M164">View MathML</a>, the proof is complete. □

Lemma 3.3Supposefsatisfies (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M107">View MathML</a>). Then given<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M166">View MathML</a>, there exist a subspaceWof<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M45">View MathML</a>and a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M168">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M169">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M170">View MathML</a>.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M171">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M172">View MathML</a> be such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M173">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M174">View MathML</a>. First, we take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M175">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M176">View MathML</a>. Considering <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M177">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M178">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M179">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M180">View MathML</a> be such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M181">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M182">View MathML</a>. Next, we take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M183">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M184">View MathML</a>. After a finite number of steps, we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M185">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M186">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M187">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M188">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M189">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M190">View MathML</a>, by construction, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M169">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M192">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M193">View MathML</a>.

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M194">View MathML</a>, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M80">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M196">View MathML</a>. Now, it suffices to verify that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M197">View MathML</a>

From the condition (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M107">View MathML</a>), given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M199">View MathML</a>, there is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M200">View MathML</a> such that for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M201">View MathML</a>, a.e. x in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M111">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M203">View MathML</a>

Consequently, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M204">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M80">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M206">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M207">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M208">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M209">View MathML</a>. Observing that W is finite dimensional and we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M210">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M211">View MathML</a>, the inequality is obtained by taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M212">View MathML</a>; the proof is complete. □

Lemma 3.4Supposefsatisfies (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M100">View MathML</a>), thenIsatisfies the (PS) condition.

Proof We suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M214">View MathML</a>,

Noting that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M102">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M101">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M218">View MathML</a> is bounded. By [9], Lemma 3.1, we know that I satisfies the (PS) condition. □

Proof of Theorem 2.1 First, we recall that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M219">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M220">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M221">View MathML</a> are defined in (3.1). Invoking Lemma 3.2, we find <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M125">View MathML</a>, and I satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M85">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M224">View MathML</a>. Now, by Lemma 3.3, there is a subspace W of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M45">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M226">View MathML</a> and such that I satisfies (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M88">View MathML</a>). Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/136/mathml/M84">View MathML</a> and I is even, we may apply Lemma 2.4 to conclude that I possesses at least k pairs of nontrivial critical points. The proof is complete. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors read and approved the final manuscript.

Acknowledgements

Project supported by Natural Science Foundation of China, Tian Yuan Special Foundation (No. 11226116), Natural Science Foundation of Jiangsu Province of China for Young Scholar (No. BK201209), the China Scholarship Council, the Fundamental Research Funds for the Central Universities (No. JUSRP11118) and Foundation for young teachers of Jiangnan University (No. 2008LQN008).

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