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Multiple solutions of semilinear elliptic systems on the Heisenberg group

Gao Jia*, Long-jie Zhang and Jie Chen

Author Affiliations

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

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Boundary Value Problems 2013, 2013:157  doi:10.1186/1687-2770-2013-157


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/157


Received:17 November 2012
Accepted:10 June 2013
Published:1 July 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, a class of semilinear elliptic systems which have a strong resonance at the first eigenvalue on the Heisenberg group is considered. Under certain assumptions, by virtue of the variational methods, the multiple weak solutions of the systems are obtained.

MSC: 35J20, 35J25, 65J67.

Keywords:
semilinear elliptic system; strong resonance; variational method; Heisenberg group

1 Introduction

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M1">View MathML</a> be the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M2">View MathML</a> equipped with the following group operation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M3">View MathML</a>

where ‘⋅’ denotes the usual inner-product in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M4">View MathML</a>. This operation endows <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M1">View MathML</a> with the structure of a Lie group. The vector fields <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M6">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M7">View MathML</a>, T, given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M8">View MathML</a>

form a basis for the tangent space at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M9">View MathML</a>.

Definition 1.1 The Heisenberg Laplacian is by definition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M10">View MathML</a>

and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M11">View MathML</a> denote the 2N-vector <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M12">View MathML</a>.

Definition 1.2 The space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M13">View MathML</a> is defined as the completion of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M14">View MathML</a> in the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M15">View MathML</a>

Some existence and nonexistence for the semilinear equations or systems on the Heisenberg group have been studied by Garofalo, Lanconelli and Niu, see [1,2], etc.

In this paper, we study the problems on the existence and multiplicity of solutions for the system

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M16">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M17">View MathML</a> is a bounded smooth domain, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M18">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M19">View MathML</a>. Moreover, we assume that there is some function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M20">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M21">View MathML</a>. Here ∇F denotes the gradient in the variable u and v, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M22">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M23">View MathML</a>.

In fact, the condition in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M4">View MathML</a> was studied by da Silva; we can see [3]. In this paper we study the problem on the Heisenberg group <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M1">View MathML</a>. The elliptic problems at resonance have been studied by many authors; see [4-7].

We use the variation methods to solve problem (1.1). Finding weak solutions of (1.1) in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M26">View MathML</a> is equivalent to finding critical points of the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M27">View MathML</a> functional given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M28">View MathML</a>

(1.2)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M29">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M30">View MathML</a> denotes the usual inner product in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M31">View MathML</a>.

We introduce the eigenvalue problem with weights. Let us denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M32">View MathML</a> the set of all continuous, cooperative and symmetric matrices A of order 2, given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M33">View MathML</a>

where the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M34">View MathML</a> satisfy the following conditions:

(A1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M35">View MathML</a> is cooperative, that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M36">View MathML</a>.

(A2) There is an <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M37">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M38">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M39">View MathML</a>.

Given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M40">View MathML</a>, consider the weighted eigenvalue problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M41">View MathML</a>

if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M40">View MathML</a>. By virtue of the spectral theory for compact operators, we obtain the sequence of eigenvalues

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M43">View MathML</a>

such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M44">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M45">View MathML</a>; see [6,8,9]. Here, each eigenvalue <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M46">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M47">View MathML</a> has finite multiplicity, and we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M48">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M49">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M47">View MathML</a>.

Remark 1.1

(1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M51">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M47">View MathML</a>.

(2) The following variational inequalities hold:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M53">View MathML</a>

(1.3)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M54">View MathML</a>

(1.4)

The variational inequalities will be used in the next section. We would like to mention that the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M55">View MathML</a> is positive in Ω. In the paper, without loss of generality, we assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M56">View MathML</a>.

We now state the assumptions and the main results in this paper. Firstly, we define the following functions:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M57">View MathML</a>

(1.5)

The above functions belong to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M58">View MathML</a> and the limits are taken a.e. and uniformly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M59">View MathML</a>.

Now we make the following basic hypotheses:

(E0) There exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M60">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M61">View MathML</a>

(E1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M62">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M63">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M64">View MathML</a>.

(E2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M65">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M64">View MathML</a>.

(E3) There exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M67">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M68">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M69">View MathML</a>

(E4) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M70">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M71">View MathML</a>.

(E5) There exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M72">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M73">View MathML</a>

(E6) There are <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M74">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M75">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M76">View MathML</a>

We can prove that the associated functional J has the saddle geometry. Actually, we have the following results.

Theorem 1.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M17">View MathML</a>be a bounded smooth domain, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M78">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M79">View MathML</a>. Assume that there is some function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M80">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M22">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M82">View MathML</a>. Furthermore, if the conditions (E0), (E1), (E2) are satisfied, problem (1.1) has at least one solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M83">View MathML</a>.

Remark 1.2 For the hypotheses <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M84">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M85">View MathML</a>, problem (1.1) admits the trivial solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M86">View MathML</a>. In this case, the main point is to assure the existence of nontrivial solutions.

Theorem 1.2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M17">View MathML</a>be a bounded smooth domain, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M78">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M79">View MathML</a>. Assume that there is some function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M80">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M22">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M82">View MathML</a>. Furthermore, if the conditions (E0), (E2), (E3), (E4) and (E5) are satisfied, then problem (1.1) has at least two nontrivial solutions.

Theorem 1.3Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M17">View MathML</a>be a bounded smooth domain, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M78">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M95">View MathML</a>. Assume that there is some function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M80">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M22">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M82">View MathML</a>. Furthermore, if the conditions (E0), (E1), (E2), (E3), (E4) and (E6) are satisfied, then problem (1.1) has at least three nontrivial solutions.

2 Preliminaries and fundamental lemmas

In this section, we prove some lemmas needed in the proof of our main theorems.

We first introduce the Folland-Stein embedding theorem (see [10]) as follows.

Lemma 2.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M17">View MathML</a>be a bounded domain and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M100">View MathML</a>. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M13">View MathML</a>compactly embedding in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M102">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M103">View MathML</a>.

To establish Lemmas 2.7 and 2.8, we introduce the following corollary of the Ekeland variation principle (see [11]).

Lemma 2.2Xis a metric space, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M104">View MathML</a>is bounded from below, which satisfies the<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M105">View MathML</a>condition, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M106">View MathML</a>is a critical value ofE.

Next, we describe some results under the geometry for the functional I.

Lemma 2.3Under hypotheses (E0) and (E1), the functionalIhas the following saddle geometry:

(L3-1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M107">View MathML</a>if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M108">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M109">View MathML</a>.

(L3-2) There is<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M110">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M111">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M112">View MathML</a>.

(L3-3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M113">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M114">View MathML</a>.

Proof (L3-1). From (1.2), (1.4) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M115">View MathML</a>

Using (E0), we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M116">View MathML</a>, as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M108">View MathML</a>.

(L3-2). By simple calculation, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M118">View MathML</a>

By using (E0), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M119">View MathML</a>

So, we choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M120">View MathML</a>.

(L3-3). By (E1) and the variational inequality (1.4), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M121">View MathML</a>

the proof of this lemma is completed. □

Next, we prove the Palais-Smale conditions at some levels for the functional I. We recall that I: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M122">View MathML</a> is said to satisfy the Palais-Smale conditions at the level <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M123">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M105">View MathML</a> in short) if any sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M125">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M126">View MathML</a>

as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M127">View MathML</a>, possesses a convergent subsequence in E. Moreover, we say that I satisfies the (PS) conditions when we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M105">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M123">View MathML</a>.

Lemma 2.4Assume that the condition (E0) holds. Then the functionalIhas the<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M105">View MathML</a>conditions whenever<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M131">View MathML</a>or<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M132">View MathML</a>.

Proof We only prove the condition for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M133">View MathML</a>. For the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M134">View MathML</a>, we can use similar methods.

1. Boundedness of the (PS) sequence.

The proof is by contradiction. Suppose that there exists a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M105">View MathML</a> unbounded sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M136">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M137">View MathML</a>. For the ease of notation and without loss of generality, we assume that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M138">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M139">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M140">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M127">View MathML</a>.

We define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M142">View MathML</a>, hence there is an <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M143">View MathML</a> with the following properties:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M144">View MathML</a> in E,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M145">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M146">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M147">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M148">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M145">View MathML</a> a.e. in Ω.

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M150">View MathML</a>, obviously <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M151">View MathML</a>. By simple calculation, it is easy to obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M152">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M153">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M154">View MathML</a>. We have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M155">View MathML</a>

From the convergence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M156">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M157">View MathML</a>

We see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M56">View MathML</a>, and by the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M159">View MathML</a>, we obtain that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M160">View MathML</a>. So, we suppose initially that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M161">View MathML</a>. Because <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M55">View MathML</a> is positive, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M163">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M164">View MathML</a>, it is obvious that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M165">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M166">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M167">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M127">View MathML</a>.

Hence, we can take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M169">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M170">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M171">View MathML</a>, and we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M172">View MathML</a>

Using (1.4), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M173">View MathML</a>

(2.1)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M139">View MathML</a>, it is easy to obtain that the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M175">View MathML</a> is bounded. On the other hand, because of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M138">View MathML</a>, on a subsequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M177">View MathML</a>, without loss of generality, we assume <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M178">View MathML</a>.

Now, using Hölder’s inequality and (E0), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M179">View MathML</a>

Thus, applying the dominated convergence theorem, we conclude that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M180">View MathML</a>

(2.2)

On the other hand,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M181">View MathML</a>

Using (2.2), (1.4), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M182">View MathML</a>

as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M127">View MathML</a>. Therefore, by variational inequalities (1.3) and (1.4), we obtain that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M184">View MathML</a>

Consequently, by virtue of Fatou’s lemma and (E0), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M185">View MathML</a>

which contradicts the condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M186">View MathML</a>. Hence, the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M105">View MathML</a> sequence is bounded.

2. Various convergence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M188">View MathML</a>.

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M188">View MathML</a> is a bounded sequence, there is an <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M190">View MathML</a> with the following properties:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M191">View MathML</a> in E,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M192">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M193">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M147">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M148">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M192">View MathML</a> a.e. in Ω.

3. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M188">View MathML</a> convergence to h in E.

From the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M105">View MathML</a> sequence, we have, as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M199">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M200">View MathML</a>

By Fatou’s lemma and the above convergence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M188">View MathML</a>, it is easy to show that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M202">View MathML</a>

as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M127">View MathML</a>. Hence, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M204">View MathML</a>

(2.3)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M205">View MathML</a>

(2.4)

By weak convergence, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M206">View MathML</a>

(2.5)

Using (2.3), (2.4) and (2.5), by simple calculation, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M207">View MathML</a>

The proof is completed. □

Lemma 2.5Suppose that (E0) and (E3) are satisfied. Then the origin is a local minimum for the functional I.

Proof Using (E3), we can choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M208">View MathML</a> and a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M209">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M210">View MathML</a>

Consequently, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M211">View MathML</a>

where ρ is small enough and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M212">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M213">View MathML</a> is provided by (E5). Therefore the proof has been completed. □

To complete the mountain pass geometry, we prove the following result.

Lemma 2.6Let the hypotheses (E0), (E4) and (E5) hold. Then there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M214">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M215">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M216">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M217">View MathML</a>.

Proof Using (E2) and (E5), we take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M218">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M213">View MathML</a> is provided by (E5). Thus, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M220">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M221">View MathML</a>. If we take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M222">View MathML</a>, then the conclusion follows. □

Lemma 2.7Under hypotheses (E0), (E4) and (E5), problem (1.1) has at least one nontrivial solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M214">View MathML</a>. Moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M224">View MathML</a>has negative energy, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M225">View MathML</a>.

Proof By (E0) and (1.4), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M226">View MathML</a>

Therefore, the functional I is bounded below. In this case, we would like to mention that I has the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M105">View MathML</a> conditions with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M228">View MathML</a>. For seeing this, by Lemma 2.4, we take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M229">View MathML</a> provided by (E5) we can obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M230">View MathML</a>

Consequently, applying Lemma 2.2, we have one critical point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M214">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M232">View MathML</a>. The proof of this lemma is completed. □

To prove Theorem 1.3, we establish the following lemma.

Lemma 2.8Assume that the conditions (E0), (E1), (E4) and (E6) hold. Then problem (1.1) has at least two nontrivial solutions with negative energy.

Proof Define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M233">View MathML</a>

We have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M234">View MathML</a>. Hence, we minimize the functional I restricted to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M235">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M236">View MathML</a>.

Firstly, we consider the functionals <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M237">View MathML</a>. Using Lemma 2.4, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M238">View MathML</a> possesses the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M105">View MathML</a> conditions whenever <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M186">View MathML</a>. Therefore, we obtain that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M238">View MathML</a> satisfies the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M105">View MathML</a> conditions with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M243">View MathML</a>.

In this way, by using Lemma 2.2 for the functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M238">View MathML</a>, we obtain two critical points which we denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M245">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M246">View MathML</a>, respectively. Thus, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M247">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M248">View MathML</a>.

Moreover, we affirm that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M245">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M246">View MathML</a> are nonzero critical points. To see this, from (E4) and (E6), we obtain that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M251">View MathML</a>

and I restricted to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M252">View MathML</a> is nonnegative. More specifically, given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M253">View MathML</a>, using (L3-3) in Lemma 2.3, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M254">View MathML</a>

(2.6)

Next, we prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M245">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M246">View MathML</a> are distinct. The proof of this affirmation is by contradiction. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M257">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M258">View MathML</a>. Using (2.6), we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M259">View MathML</a>. Therefore, we have a contradiction. Consequently, we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M260">View MathML</a>. Thus problem (1.1) has at least two nontrivial solutions. Moreover, these solutions have negative energy. □

3 Proof of main theorems

In this section, we prove Theorem 1.1, Theorem 1.2 and Theorem 1.3.

Proof of Theorem 1.1 From Lemma 2.4, the functional I satisfies the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M105">View MathML</a> conditions for some levels <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M123">View MathML</a>. Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M263">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M264">View MathML</a>. Using Lemma 2.3, we get that the functional I satisfies the saddle point geometry (see [12], Theorem 1.11). This implies that I has one critical point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M265">View MathML</a>. Theorem 1.1 is proved. □

Proof of Theorem 1.2 From Lemma 2.5 and Lemma 2.6, we know that the functional I satisfies the geometric conditions of the mountain pass theorem. Moreover, the functional I satisfies the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M105">View MathML</a> conditions for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M267">View MathML</a>. Thus, we have a solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M268">View MathML</a> given by the mountain pass theorem. Obviously, the solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M269">View MathML</a> satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M270">View MathML</a>.

On the other hand, by Lemma 2.7, we get another solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M224">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M216">View MathML</a>. It follows that problem (1.1) has at least two nontrivial solutions. The proof is completed. □

Proof of Theorem 1.3 Since the conditions (E0), (E3), (E4) and (E5) imply that Lemma 2.5 and Lemma 2.6 hold. Thus, we have one solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M269">View MathML</a> which satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M270">View MathML</a>.

On the other hand, using Lemma 2.8, we obtain two distinct critical points <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M275">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/157/mathml/M276">View MathML</a>. Therefore, we obtain that problem (1.1) has at least three nontrivial solutions. The proof is completed. □

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 was supported by the National Natural Science Foundation of China (11171220) and Shanghai Leading Academic Discipline Project (XTKX2012).

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