Open Access Research

Infinitely many solutions for a class of quasilinear elliptic equations with p-Laplacian in R N

Gao Jia*, Jie Chen and Long-jie Zhang

Author Affiliations

College of Science, University of Shanghai for Science and Technology, Shanghai, 200093, China

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Boundary Value Problems 2013, 2013:179  doi:10.1186/1687-2770-2013-179


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/179


Received:15 December 2012
Accepted:22 July 2013
Published:6 August 2013

© 2013 Jia et al.; licensee Springer

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

Abstract

In this paper, we study the multiplicity of solutions for a class of quasilinear elliptic equations with p-Laplacian in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M1">View MathML</a>. In this case, the functional J is not differentiable. Hence, it is difficult to work under the classical framework of the critical point theory. To overcome this difficulty, we use a nonsmooth critical point theory, which provides the existence of critical points for nondifferentiable functionals.

MSC: 35J20, 35J92, 58E05.

Keywords:
quasilinear elliptic equations; nondifferentiable functional; p-Laplacian; multiple solutions

1 Introduction and main results

Recently, the multiplicity of solutions for the quasilinear elliptic equations has been studied extensively, and many fruitful results have been obtained. For example, in [1], Shibo Liu considered the existence of multiple nonzero solutions of the Dirichlet boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M3">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M5">View MathML</a> denotes the p-Laplacian operator, Ω is a bounded domain in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M1">View MathML</a> with smooth boundary Ω.

Moreover, Aouaoui studied the following quasilinear elliptic equation in [2]:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M7">View MathML</a>

(1.2)

and proved the multiplicity of solutions of the problem (1.2) by using the nonsmooth critical point theory. One can refer to [3,4] and [5] for more results.

In this paper, we shall investigate the existence of infinitely many solutions of the following problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M8">View MathML</a>

(1.3)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M9">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M10">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M11">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M12">View MathML</a> is a given continuous function satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M13">View MathML</a>

In order to determine weak solutions of (1.3) in a suitable functional space E, we look for critical points of the functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M14">View MathML</a> defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M15">View MathML</a>

(1.4)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M16">View MathML</a>. Under reasonable assumptions, the functional J is continuous, but not even locally Lipschitz. However, one can see from [4,6] and [7] that the Gâteaux-derivative of J exists in the smooth directions, i.e., it is possible to evaluate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M17">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M18">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M19">View MathML</a>.

Definition 1.1 A critical point u of the functional J is defined as a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M18">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M21">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M22">View MathML</a>, i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M23">View MathML</a>

(1.5)

Our approach to study (1.3) is based on the nonsmooth critical point theory developed in [8] and [9]. Dealing with this class of problems, the main difficulty is that the associated functional is not differentiable in all directions.

The main goal here is to establish multiplicity of results for (1.3), when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M24">View MathML</a> is odd and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M25">View MathML</a> is even in s. Such solutions for (1.3) will follow from a version of the symmetric mountain pass theorem due to Ambrosetti and Rabinowitz [10,11]. Compared with problem (1.2) in [2], problem (1.3) is much more difficult, since the discreteness of the spectrum is not guaranteed. Therefore, we only consider the first eigenvalue <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M26">View MathML</a>.

To state and prove our main result, we consider the following assumptions.

Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M27">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M28">View MathML</a>.

(H1) Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M29">View MathML</a> be a function such that

• for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M30">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M25">View MathML</a> is measurable with respect to x;

• for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M32">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M25">View MathML</a> is a function of class <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M34">View MathML</a> with respect to s;

• there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M35">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M36">View MathML</a>

(1.6)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M37">View MathML</a>

(1.7)

(H2) There exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M38">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M39">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M40">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M41">View MathML</a>

(1.8)

(H3) Let a Carathéodory function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M42">View MathML</a> satisfy <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M43">View MathML</a>, a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M32">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M45">View MathML</a>

(1.9)

where θ is the same as that in (H2).

(H4) There exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M46">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M47">View MathML</a>

(1.10)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M48">View MathML</a> is a positive constant.

Example 1.1 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M49">View MathML</a>. The following function satisfies hypotheses (H1) and (H2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M50">View MathML</a>

and the corresponding constants are

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M51">View MathML</a>

Example 1.2 The following function satisfies hypotheses (H3) and (H4)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M52">View MathML</a>

On the other hand, we define the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M53">View MathML</a>. It follows from [12] that the discreteness of the spectrum is not guaranteed. Hence, we only consider the first eigenvalue <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M54">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M55">View MathML</a>

Next, we can state the main theorem of the paper.

Theorem 1.1Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M25">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M24">View MathML</a>satisfy (H1)-(H4). Moreover, let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M58">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M59">View MathML</a>, a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M32">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M61">View MathML</a>. If there exists a positive numberμsuch that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M62">View MathML</a>, then problem (1.3) has infinitely many distinct solutions in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M63">View MathML</a>, i.e., there exists a sequence<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M64">View MathML</a>, satisfying (1.3) and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M65">View MathML</a>, as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M66">View MathML</a>.

To explain our result, we introduce some functional spaces. We define the reflexive Banach space E of all functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M67">View MathML</a> with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M68">View MathML</a>.

Such a weighted Sobolev space has been used in many previous papers, see [13] and [14]. Now, we give an important property of the space E, which will play an essential role in proving our main results.

Remark 1.1 One can easily deduce <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M69">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M70">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M71">View MathML</a>. More details can be found in [2].

Throughout this paper, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M72">View MathML</a> denote the norm of E and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M73">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M74">View MathML</a>) means that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M75">View MathML</a> converges strongly (weakly) in corresponding spaces. ↪ stands for a continuous map, and ↪↪ means a compact embedding map. C denotes any universal positive constant unless specified.

The paper is organized as follows. In Section 2, we introduce the nonsmooth critical framework and preliminaries to our work. In Section 3, we give some lemmas to prove the main result. Finally, the proof of Theorem 1.1 is presented in Section 4.

2 Nonsmooth critical framework and preliminaries

Our results are based on the techniques of nonsmooth critical point theory. In this section, we recall some basic tools from [8] and [9].

Definition 2.1 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M76">View MathML</a> be a metric space, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M77">View MathML</a> be a continuous functional and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M78">View MathML</a>. We denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M79">View MathML</a> the supremum of the σ’s in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M80">View MathML</a> such that there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M81">View MathML</a> and a continuous map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M82">View MathML</a>, satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M83">View MathML</a>

The extended real number <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M79">View MathML</a> is called the weak slope of I at u.

Note that the notion above was independently introduced in [15], as well.

Definition 2.2 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M76">View MathML</a> be a metric space, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M77">View MathML</a> be a continuous functional and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M87">View MathML</a>. We say that I satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M88">View MathML</a>, i.e., the Palais-Smale condition at level c, if every sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M89">View MathML</a> in X with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M90">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M91">View MathML</a> admits a strongly convergent subsequence.

In order to treat the Palais-Smale condition, we need to introduce an auxiliary notion.

Definition 2.3 Let c be a real number. We say that functional I satisfies the concrete Palais-Smale condition at level c (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M92">View MathML</a> for short) if every sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M93">View MathML</a> satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M94">View MathML</a>

possesses a strongly convergent subsequence in E, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M95">View MathML</a> is some real number converging to zero.

Remark 2.1 Under assumptions (H1)-(H4), if the functional J satisfies (1.4), then J is continuous, and for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M96">View MathML</a> we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M97">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M98">View MathML</a> denotes the weak slope of J at u.

Remark 2.2 Let c be a real number. If J satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M92">View MathML</a>, then J satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M88">View MathML</a>.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M101">View MathML</a> be a sequence such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M102">View MathML</a>

Note that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M103">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M104">View MathML</a>

By Remark 2.1, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M105">View MathML</a>. Taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M106">View MathML</a>, the conclusion follows. □

3 Basic lemmas

To derive our main theorem, we need the following lemmas. The first lemma is the version of the Ambrosetti-Rabinowitz mountain pass lemma [10,11] and [16].

Lemma 3.1LetXbe an infinite-dimensional Banach space, and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M77">View MathML</a>be a continuous even functional satisfying<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M88">View MathML</a>for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M109">View MathML</a>. Assume that

(i) there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M110">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M111">View MathML</a>and a subspace<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M112">View MathML</a>of finite codimension such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M113">View MathML</a>

(ii) for every finite-dimensional subspace<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M114">View MathML</a>, there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M115">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M116">View MathML</a>

Then there exists a sequence<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M117">View MathML</a>of critical values ofIwith<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M118">View MathML</a>.

Lemma 3.2If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M18">View MathML</a>is a critical point ofJ, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M120">View MathML</a>.

Proof For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M121">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M122">View MathML</a>, consider the real functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M123">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M124">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M125">View MathML</a> defined in ℝ by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M126">View MathML</a>

(3.1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M127">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M128">View MathML</a>. Denoting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M129">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M130">View MathML</a>, we can take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M131">View MathML</a> as a test function in (1.5). Therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M132">View MathML</a>

Noting that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M133">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M134">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M135">View MathML</a>

From (1.10) and the fact <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M136">View MathML</a> we deduce

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M137">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M138">View MathML</a> a.e. in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M1">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M140">View MathML</a> in E as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M141">View MathML</a>. It follows from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M141">View MathML</a> that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M143">View MathML</a>

Denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M144">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M145">View MathML</a>, then the result is true. In the following discussion, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M146">View MathML</a> is assumed. By (1.6), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M147">View MathML</a>

(3.2)

Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M148">View MathML</a>, then we can get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M149">View MathML</a>

(3.3)

On the other hand, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M150">View MathML</a>

which implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M151">View MathML</a>

(3.4)

Eventually, one can deduce from (3.2)-(3.4) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M152">View MathML</a>

(3.5)

By Theorem 5.2 of [17], we get that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M153">View MathML</a>. Replacing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M154">View MathML</a> by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M155">View MathML</a>, we can similarly prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M156">View MathML</a>. We conclude that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M120">View MathML</a>, and the proof of Lemma 3.2 is completed. □

Lemma 3.3Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M158">View MathML</a>be a bounded sequence inEwith

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M159">View MathML</a>

(3.6)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M160">View MathML</a>is a sequence of real numbers converging to zero. Then there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M18">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M162">View MathML</a>a.e. in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M1">View MathML</a>and, up to a subsequence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M164">View MathML</a>is weakly convergent touinE. Moreover, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M165">View MathML</a>

(3.7)

i.e., uis a critical point ofJ.

Proof Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M89">View MathML</a> is bounded in E, and there is a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M18">View MathML</a> (see [18]) such that, up to a subsequence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M168">View MathML</a>

Moreover, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M89">View MathML</a> satisfies (3.6), by Theorem 2.1 of [19], we have, up to a further subsequence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M170">View MathML</a> a.e. in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M1">View MathML</a>.

We will use the device of [20]. We consider the test functions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M172">View MathML</a>

(3.8)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M173">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M174">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M122">View MathML</a>. According to (1.6) and (1.7), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M176">View MathML</a>

Since (3.6) holds by density for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M177">View MathML</a>, we can put <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M178">View MathML</a> in (3.6) and obtain that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M179">View MathML</a>

(3.9)

On the other hand, note that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M180">View MathML</a>

(3.10)

One can deduce from (3.10) and Fatou’s lemma that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M181">View MathML</a>

(3.11)

We consider the test functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M182">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M173">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M184">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M185">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M186">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M187">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M188">View MathML</a>

This together with (3.11) can prove that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M189">View MathML</a>

(3.12)

In a similar way, by considering the test functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M190">View MathML</a>, it is possible to prove that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M191">View MathML</a>

(3.13)

From (3.12) and (3.13), it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M192">View MathML</a>

(3.14)

Finally, we can deduce (3.7) from (3.14). □

Remark 3.1 (see [21])

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M89">View MathML</a> be a sequence in E satisfying (3.6). Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M194">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M195">View MathML</a>

(3.15)

In the following lemma, we will prove the boundedness of a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M92">View MathML</a> sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M101">View MathML</a> under (1.6), (1.8) and (1.9).

Lemma 3.4Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M109">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M89">View MathML</a>be a sequence inEsatisfying (3.6) and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M200">View MathML</a>

(3.16)

Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M89">View MathML</a>is bounded inE.

Proof Calculating <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M202">View MathML</a>, from (3.15) and (3.16), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M203">View MathML</a>

From (1.8) and (1.9), it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M204">View MathML</a>

(3.17)

Moreover, there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M122">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M206">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M207">View MathML</a>

Therefore, denoting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M208">View MathML</a>, we obtain from (3.17) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M209">View MathML</a>

(3.18)

By virtue of hypothesis (H3), we know that there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M210">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M211">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M212">View MathML</a>

(3.19)

From (3.18) and (3.19), it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M213">View MathML</a>

(3.20)

On the other hand, by Hölder’s inequality and Young’s inequality, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M214">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M215">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M216">View MathML</a>

(3.21)

Using (3.20) and (3.21), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M217">View MathML</a>

(3.22)

Choosing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M218">View MathML</a> in (3.22), we find that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M89">View MathML</a> is bounded in E. □

Lemma 3.5Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M89">View MathML</a>be the same as that in Lemma 3.3. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M89">View MathML</a>, up to a subsequence, converges strongly touinE.

Proof By Lemma 3.3, we know that u is a critical point of the functional J. Then, from Lemma 3.2, we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M120">View MathML</a>. Therefore, taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M223">View MathML</a> as a test function in (3.7), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M224">View MathML</a>

(3.23)

By virtue of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M89">View MathML</a> is bounded in E, we can assume that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M18">View MathML</a> satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M227">View MathML</a>

By Lemma 3.3, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M170">View MathML</a> a.e. in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M229">View MathML</a>. Then by Fatou’s lemma, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M230">View MathML</a>

(3.24)

Moreover, by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M231">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M232">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M233">View MathML</a>

(3.25)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M234">View MathML</a>

(3.26)

By using (3.23)-(3.26) and passing to limit in (3.15), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M235">View MathML</a>

(3.27)

On the other hand, by Lebesgue’s dominated convergence theorem and the weak convergence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M75">View MathML</a> to u in E, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M237">View MathML</a>

(3.28)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M238">View MathML</a>

(3.29)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M239">View MathML</a>

(3.30)

Moreover, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M240">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M241">View MathML</a> are bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M242">View MathML</a>, then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M243">View MathML</a>

Therefore, from the definition of weak convergence, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M244">View MathML</a>

(3.31)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M245">View MathML</a>

(3.32)

Combining (3.27)-(3.32), it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M246">View MathML</a>

It is well known that the following inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M247">View MathML</a>

(3.33)

holds for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M248">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M249">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M250">View MathML</a>. Therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M251">View MathML</a>

According to (1.6), we conclude that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M89">View MathML</a> converges strongly to u in E. □

Lemma 3.6For every real numberc, the functionalJsatisfies<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M92">View MathML</a>.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M89">View MathML</a> be a sequence in E satisfying (3.6) and (3.16). By Lemma 3.4, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M89">View MathML</a> is bounded in E. Therefore, the conclusion can be deduced from Lemma 3.5. □

4 Proof of Theorem 1.1

It is easy to check that the functional J is continuous and even. Moreover, by Remark 2.2 and Lemma 3.6, J satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M88">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M109">View MathML</a>.

On the other hand, from (1.4), (1.6), (1.9) and (1.10), for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M18">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M259">View MathML</a>

(4.1)

We discuss (4.1) in the following two cases:

In case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M260">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M261">View MathML</a>

In case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M262">View MathML</a>, by the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M54">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M264">View MathML</a>

i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M265">View MathML</a>. Therefore, if λ satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M266">View MathML</a>, there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M110">View MathML</a> small enough and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M81">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M269">View MathML</a>

Hence, condition (i) of Lemma 3.1 holds with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M270">View MathML</a>.

Now we consider a finite-dimensional subspace W of E. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M271">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M272">View MathML</a>. From (1.6), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M273">View MathML</a>

(4.2)

By virtue of (1.9) and (1.10), we know that there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M274">View MathML</a>, satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M275">View MathML</a> a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M276">View MathML</a> and a positive constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M277">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M278">View MathML</a>

(4.3)

Combining (4.2)-(4.3), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M279">View MathML</a>

(4.4)

Since W is finite-dimensional, then all norms of W are equivalent. From (4.4), there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M280">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M281">View MathML</a>

In view of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M38">View MathML</a> , we deduce that the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M283">View MathML</a> is bounded in E and condition (ii) of Lemma 3.1 holds. By Lemma 3.1, the conclusion follows.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

We declare that all authors collaborated and dedicated the same amount of time in order to perform this article.

Acknowledgements

The authors express their sincere thanks to the referees for their valuable criticism of the manuscript and for helpful suggestions. This work has been supported by the Natural Science Foundation of China (No. 11171220) and Shanghai Leading Academic Discipline Project (XTKX2012).

References

  1. Liu, SB: Multiplicity results for coercive p-Laplacian equations. J. Math. Anal. Appl.. 316, 229–236 (2006). Publisher Full Text OpenURL

  2. Aouaoui, S: Multiplicity of solutions for quasilinear elliptic equations in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M1">View MathML</a>. J. Math. Anal. Appl.. 370(2), 639–648 (2010). Publisher Full Text OpenURL

  3. Alves, CO, Carrião, PC, Miyagaki, OH: Existence and multiplicity results for a class of resonant quasilinear elliptic problems on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M1">View MathML</a>. Nonlinear Anal.. 39, 99–110 (2000). Publisher Full Text OpenURL

  4. Canino, A: Multiplicity of solutions for quasilinear elliptic equations. Topol. Methods Nonlinear Anal.. 6, 357–370 (1995)

  5. Squassina, M: Existence of multiple solutions for quasilinear diagonal elliptic systems. Electron. J. Differ. Equ.. 1999, 1–12 (1999)

  6. Arcoya, D, Boccardo, L: Critical points for multiple integrals of the calculus of variations. Arch. Ration. Mech. Anal.. 134, 249–274 (1996). Publisher Full Text OpenURL

  7. Arcoya, D, Boccardo, L: Some remarks on critical point theory for nondifferentiable functionals. NoDEA Nonlinear Differ. Equ. Appl.. 6, 79–100 (1999). Publisher Full Text OpenURL

  8. Corvellec, JN, Degiovanni, M, Marzocchi, M: Deformation properties of continuous functionals and critical point theory. Topol. Methods Nonlinear Anal.. 1, 151–171 (1993)

  9. Degiovanni, M, Marzocchi, M: A critical point theory for nonsmooth functionals. Ann. Mat. Pura Appl.. 167(4), 73–100 (1994)

  10. Ambrosetti, A, Rabinowitz, PH: Dual variational methods in critical point theory and applications. J. Funct. Anal.. 14, 349–381 (1973). Publisher Full Text OpenURL

  11. Silva, EAB: Critical point theorems and applications to differential equations. Ph.D. thesis, University of Wisconsin-Madison (1988)

  12. Brasco, L, Franzina, G: On the Hong-Krahn-Szego inequality for the p-Laplace operator. Manuscr. Math.. 141, 537–557 (2013). Publisher Full Text OpenURL

  13. Bartsh, T, Wang, ZQ: Existence and multiplicity results for some superlinear elliptic problems on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/179/mathml/M1">View MathML</a>. Commun. Partial Differ. Equ.. 20, 1725–1741 (1995). Publisher Full Text OpenURL

  14. Rabinowitz, PH: On a class of nonlinear Schrödinger equations. Z. Angew. Math. Phys.. 43, 270–291 (1992). Publisher Full Text OpenURL

  15. Katriel, G: Mountain pass theorems and global homeomorphism theorems. Ann. Inst. Henri Poincaré, Anal. Non Linéaire. 11, 189–209 (1994)

  16. Rabinowitz, PH: Minimax Methods in Critical Point Theory with Applications to Differential Equations, Am. Math. Soc., Providence (1986)

  17. Ladyzenskaya, OA, Uralceva, NN: Equations aux dérivées partielles de type elliptiques, Dunod, Paris (1968)

  18. Brezis, H, Lieb, E: A relation between pointwise convergence of functions and convergence of functionals. Proc. Am. Math. Soc.. 88, 486–490 (1983)

  19. Boccardo, L, Murat, F: Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations. Nonlinear Anal.. 19, 581–597 (1992). Publisher Full Text OpenURL

  20. Boccardo, L, Murat, F, Puel, JP: Existence de solutions non bornées pour certaines équations quasi-linéaires. Port. Math.. 41, 507–534 (1982)

  21. Brezis, H, Browder, FE: Sur une propriété des espaces de Sobolev. C. R. Math. Acad. Sci. Paris. 287, 113–115 (1978)