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Multiplicity of solutions for nonlinear impulsive differential equations with Dirichlet boundary conditions

Chenxing Zhou1, Fenghua Miao1 and Sihua Liang12*

Author Affiliations

1 College of Mathematics, Changchun Normal University, Changchun, Jilin, 130032, P.R. China

2 Key Laboratory of Symbolic Computation and Knowledge Engineering of Ministry of Education, Jilin University, Changchun, 130012, P.R. China

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Boundary Value Problems 2013, 2013:200  doi:10.1186/1687-2770-2013-200


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/200


Received:17 June 2013
Accepted:20 August 2013
Published:5 September 2013

© 2013 Zhou 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 consider the existence of solutions for nonlinear impulsive differential equations with Dirichlet boundary conditions. Infinitely many solutions are obtained by using a version of the symmetric mountain-pass theorem, and this sequence of solutions converge to zero. Some recent results are extended.

MSC: 34B37, 35B38.

Keywords:
impulsive effects; variational methods; Dirichlet boundary value problem; critical points

1 Introduction

In this paper, we study the following nonlinear impulsive differential equations with Dirichlet boundary conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M1">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M5">View MathML</a> is continuous, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M6">View MathML</a>, N is a positive integer, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M7">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M8">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M9">View MathML</a> are continuous. With the help of the symmetric mountain-pass lemma due to Kajikiya [1], we prove that there are infinitely many small weak solutions for equations (1.1) with the general nonlinearities <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M10">View MathML</a>.

In recent years, a great deal of works have been done in the study of the existence of solutions for impulsive boundary value problems, by which a number of chemotherapy, population dynamics, optimal control, ecology, industrial robotics and physics phenomena are described. For the general aspects of impulsive differential equations, we refer the reader to the classical monograph [2]. For some general and recent works on the theory of impulsive differential equations, we refer the reader to [3-13]. Some classical tools or techniques have been used to study such problems in the literature. These classical techniques include the coincidence degree theory [14], the method of upper and lower solutions with a monotone iterative technique [15], and some fixed point theorems in cones [16,17].

On the other hand, in the last few years, many authors have used a variational method to study the existence and multiplicity of solutions for boundary value problems without impulsive effects [18-21]. For related basic information, we refer the reader to [22,23].

For a second order differential equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M11">View MathML</a>, one usually considers impulses in the position u and the velocity <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M12">View MathML</a>. However, in the motion of spacecraft, one has to consider instantaneous impulses depending on the position that result in jump discontinuities in velocity, but with no change in the position [24-27].

A new approach via critical point and variational methods is proved to be very effective in studying the boundary problem for differential equations. For some general and recent works on the theory of critical point theory and variational methods, we refer the reader to [28-37].

More precisely, in [28] the authors studied the following equations with Dirichlet boundary conditions:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M13">View MathML</a>

(1.2)

They obtained the existence of solutions for problems by using the variational method. Zhang and Yuan [30] extended the results in [28]. They obtained the existence of solutions for problem (1.2) with a perturbation term. Also, they obtained infinitely many solutions for problem (1.2) under the assumption that the nonlinearity f is a superlinear case. Soon after that, Zhou and Li [29] extended problem (1.2). In all the above-mentioned works, the information on the sequence of solutions was not given.

Motivated by the fact above, the aim of this paper is to show the existence of infinitely many solutions for problem (1.1), and that there exists a sequence of infinitely many arbitrarily small solutions, converging to zero, by using a new version of the symmetric mountain-pass lemma due to Kajikiya [1]. Our main results extend the existing study.

Throughout this paper, we assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M14">View MathML</a> is continuous, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M15">View MathML</a> satisfies the following conditions:

(I1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M16">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M17">View MathML</a>) are odd and satisfy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M18">View MathML</a>

(I2) There exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M19">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M17">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M21">View MathML</a>

(H1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M22">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M23">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M24">View MathML</a>;

(H2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M25">View MathML</a> uniformly for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M26">View MathML</a>;

(H3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M27">View MathML</a> uniformly for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M28">View MathML</a>.

The main result of this paper is as follows.

Theorem 1.1Suppose that (I1)-(I2) and (H1)-(H3) hold. Then problem (1.1) has a sequence of nontrivial solutions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M29">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M30">View MathML</a>as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M31">View MathML</a>.

Remark 1.1 Without the symmetry condition (i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M32">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M33">View MathML</a>), we can obtain at least one nontrivial solution by the same method in this paper.

Remark 1.2 We should point out that Theorem 1.1 is different from the previous results of [28-37] in three main directions:

(1) We do not make the nonlinearity f satisfy the well-known Ambrosetti-Rabinowitz condition [23];

(2) We try to use Lusternik-Schnirelman’s theory for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M34">View MathML</a>-invariant functional. But since the functional is not bounded from below, we could not use the theory directly. So, we follow [38] to consider a truncated functional.

(3) We can obtain a sequence of nontrivial solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M29">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M30">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M31">View MathML</a>.

Remark 1.3 There exist many functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M16">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M10">View MathML</a> satisfying conditions (I1)-(I2) and (H1)-(H3), respectively. For example, when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M40">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M41">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M42">View MathML</a>.

2 Preliminary lemmas

In this section, we first introduce some notations and some necessary definitions.

Definition 2.1 Let E be a Banach space and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M43">View MathML</a>. J is said to be sequentially weakly lower semi-continuous if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M44">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M45">View MathML</a> in E.

Definition 2.2 Let E be a real Banach space. For any sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M46">View MathML</a>, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M47">View MathML</a> is bounded and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M48">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M49">View MathML</a> possesses a convergent subsequence, then we say J satisfies the Palais-Smale condition (denoted by (PS) condition for short).

In the Sobolev space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M50">View MathML</a>, consider the inner product

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M51">View MathML</a>

which induces the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M52">View MathML</a>

It is a consequence of Poincaré’s inequality that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M53">View MathML</a>

Here, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M54">View MathML</a> is the first eigenvalue of the Dirichlet problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M55">View MathML</a>

(2.1)

In this paper, we will assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M56">View MathML</a>. We can also define the inner product

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M57">View MathML</a>

which induces the equivalent norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M58">View MathML</a>

Lemma 2.1[29]

If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M59">View MathML</a>, then the norm<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M60">View MathML</a>and the norm<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M61">View MathML</a>are equivalent.

Lemma 2.2[29]

There exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M62">View MathML</a>such that if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M63">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M64">View MathML</a>

(2.2)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M65">View MathML</a>.

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M66">View MathML</a>, we have that u and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M12">View MathML</a> are both absolutely continuous, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M68">View MathML</a>, hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M69">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M26">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M63">View MathML</a>, then u is absolutely continuous and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M72">View MathML</a>. In this case, the one-side derivatives <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M73">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M74">View MathML</a> may not exist. As a consequence, we need to introduce a different concept of solution. Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M75">View MathML</a> satisfies the Dirichlet condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M76">View MathML</a>. Assume that, for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M17">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M78">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M79">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M80">View MathML</a>.

Taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M81">View MathML</a> and multiplying the two sides of the equality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M82">View MathML</a>

by v and integrating between 0 and T, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M83">View MathML</a>

(2.3)

Moreover, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M76">View MathML</a>, one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M85">View MathML</a>

Combining (2.3), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M86">View MathML</a>

Lemma 2.3A weak solution of (1.1) is a function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M87">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M88">View MathML</a>

(2.4)

for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M81">View MathML</a>.

Consider <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M90">View MathML</a> defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M91">View MathML</a>

(2.5)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M92">View MathML</a>. Using the continuity of f and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M16">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M94">View MathML</a>, we obtain the continuity and differentiability of J and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M95">View MathML</a>. For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M81">View MathML</a>, one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M97">View MathML</a>

(2.6)

Thus, the solutions of problem (1.1) are the corresponding critical points of J.

Lemma 2.4If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M63">View MathML</a>is a weak solution of problem (1.1), thenuis a classical solution of problem (1.1).

Proof Obviously, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M76">View MathML</a> since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M63">View MathML</a>. By the definition of weak solution, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M81">View MathML</a>, one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M102">View MathML</a>

(2.7)

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M103">View MathML</a>, choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M81">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M105">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M106">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M107">View MathML</a>

By the definition of weak derivative, the equality above implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M108">View MathML</a>

(2.8)

Hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M79">View MathML</a> and u satisfies the equation in (1.1) a.e. on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M110">View MathML</a>. By integrating (2.7), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M111">View MathML</a>

Combining this fact with (2.8), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M112">View MathML</a>

Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M113">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M94">View MathML</a>, and the impulsive condition in (1.1) is satisfied. This completes the proof. □

Lemma 2.5If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M115">View MathML</a>, then the functionalJis sequentially weakly lower semi-continuous.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M29">View MathML</a> be a weakly convergent sequence to u in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M117">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M118">View MathML</a>

We have that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M29">View MathML</a> converges uniformly to u on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M120">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M121">View MathML</a>

This completes the proof. □

Under assumptions (H1) and (H2), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M122">View MathML</a>

which means that for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M123">View MathML</a>, there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M124">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M125">View MathML</a>

(2.9)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M126">View MathML</a>

(2.10)

Hence, for every positive constant k, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M127">View MathML</a>

(2.11)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M128">View MathML</a>.

Lemma 2.6Suppose that (I1)-(I2) and (H1)-(H3) hold, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M129">View MathML</a>satisfies the (PS) condition.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M29">View MathML</a> be a sequence in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M50">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M47">View MathML</a> is bounded and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M48">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M31">View MathML</a>. First, we prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M29">View MathML</a> is bounded. By (2.5), (2.6) and (2.11), one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M136">View MathML</a>

By condition (I1), we can deduce that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M137">View MathML</a>

Setting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M138">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M139">View MathML</a>

(2.12)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M140">View MathML</a> and M is a positive constant. On the other hand, by (I1), (2.5) and (2.10), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M141">View MathML</a>

(2.13)

Thus, (2.12) and (2.13) imply that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M29">View MathML</a> is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M50">View MathML</a>. Going if necessary to a subsequence, we can assume that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M63">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M145">View MathML</a>

as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M146">View MathML</a>. Hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M147">View MathML</a>

as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M146">View MathML</a>. Moreover, one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M149">View MathML</a>

Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M150">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M151">View MathML</a>. That is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M29">View MathML</a> converges strongly to u in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M50">View MathML</a>. That is J satisfies the (PS) condition. □

3 Existence of a sequence of arbitrarily small solutions

In this section, we prove the existence of infinitely many solutions of (1.1), which tend to zero. Let X be a Banach space and denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M154">View MathML</a>

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M155">View MathML</a>, we define genus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M156">View MathML</a> as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M157">View MathML</a>

If there is no mapping φ as above for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M158">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M159">View MathML</a>. We list some properties of the genus (see [1]).

Proposition 3.1LetAandBbe closed symmetric subsets ofX, which do not contain the origin. Then the following hold.

(1) If there exists an odd continuous mapping fromAtoB, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M160">View MathML</a>;

(2) If there is an odd homeomorphism fromAtoB, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M161">View MathML</a>;

(3) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M162">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M163">View MathML</a>;

(4) Thenn-dimensional sphere<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M164">View MathML</a>has a genus of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M165">View MathML</a>by the Borsuk-Ulam theorem;

(5) IfAis compact, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M166">View MathML</a>and there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M167">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M168">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M169">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M170">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M171">View MathML</a> denote the family of closed symmetric subsets A of X such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M172">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M173">View MathML</a>. The following version of the symmetric mountain-pass lemma is due to Kajikiya [1].

Lemma 3.1LetEbe an infinite-dimensional space and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M174">View MathML</a>, and suppose the following conditions hold.

(C1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M175">View MathML</a>is even, bounded from below, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M176">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M175">View MathML</a>satisfies the Palais-Smale condition;

(C2) For each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M178">View MathML</a>, there exists an<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M179">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M180">View MathML</a>.

Then either (R1) or (R2) below holds.

(R1) There exists a sequence<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M181">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M182">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M183">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M181">View MathML</a>converges to zero;

(R2) There exist two sequences<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M181">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M186">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M182">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M183">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M189">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M190">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M191">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M192">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M193">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M194">View MathML</a>converges to a nonzero limit.

Remark 3.1 From Lemma 3.1, we have a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M181">View MathML</a> of critical points such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M196">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M189">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M198">View MathML</a>.

In order to get infinitely many solutions, we need some lemmas. Under the assumptions of Theorem 1.1, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M199">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M200">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M201">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M202">View MathML</a>. As <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M203">View MathML</a> attains a local but not a global minimum (P is not bounded below), we have to perform some sort of truncation. To this end, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M204">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M205">View MathML</a> be such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M206">View MathML</a>, where m is the local minimum of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M203">View MathML</a>, and M is the local maximum and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M208">View MathML</a>. For these values <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M205">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M204">View MathML</a>, we can choose a smooth function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M211">View MathML</a> defined as follows

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M212">View MathML</a>

Then it is easy to see <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M213">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M211">View MathML</a> is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M215">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M216">View MathML</a> and consider the perturbation of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M129">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M218">View MathML</a>

(3.1)

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M219">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M220">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M221">View MathML</a>

From the arguments above, we have the following.

Lemma 3.2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M222">View MathML</a>is defined as in (3.1). Then

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M223">View MathML</a>andGis even and bounded from below;

(ii) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M224">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M225">View MathML</a>, consequently, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M226">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M227">View MathML</a>;

(iii) Suppose that (I1)-(I2) and (H1)-(H3) hold, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M222">View MathML</a>satisfies the (PS) condition.

Proof It is easy to see (i) and (ii). (iii) are consequences of (ii) and Lemma 2.6. □

Lemma 3.3Assume that (I2) and (H3) hold. Then for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M178">View MathML</a>, there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M230">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M231">View MathML</a>.

Proof Firstly, by (H3) of Theorem 1.1, for any fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M232">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M233">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M234">View MathML</a>

Secondly, from Lemma 5 of [33], we have that for any finite dimensional subspace <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M235">View MathML</a> of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M50">View MathML</a> and any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M237">View MathML</a>, there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M238">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M239">View MathML</a>

Therefore, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M237">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M241">View MathML</a> and ρ small enough, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M242">View MathML</a>

since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M243">View MathML</a>. That is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M244">View MathML</a>

This completes the proof. □

Now, we give the proof of Theorem 1.1 as following.

Proof of Theorem 1.1 Recall that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M245">View MathML</a>

and define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M246">View MathML</a>

By Lemma 3.2(i) and Lemma 3.3, we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M247">View MathML</a>. Therefore, assumptions (C1) and (C2) of Lemma 3.1 are satisfied. This means that G has a sequence of solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/200/mathml/M29">View MathML</a> converging to zero. Hence, Theorem 1.1 follows by Lemma 3.2(ii). □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

CZ carried out the theoretical studies, and participated in the sequence alignment and drafted the manuscript. FM participated in the design of the study and performed the statistical analysis. SL conceived of the study, and participated in its design and coordination. All authors read and approved the final manuscript.

Acknowledgements

The authors are supported by the Research Foundation during the 12th Five-Year Plan Period of Department of Education of Jilin Province, China (Grant [2013] No. 252), the China Postdoctoral Science Foundation (Grant No. 2012M520665), the Youth Foundation for Science and Technology Department of Jilin Province (20130522100JH), the open project program of Key Laboratory of Symbolic Computation and Knowledge Engineering of Ministry of Education, Jilin University (Grant No. 93K172013K03), the Natural Science Foundation of Changchun Normal University.

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