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Existence of solutions for a general quasilinear elliptic system via perturbation method

Yujuan Jiao1*, Shengmao Fu2 and Yanli Wang3

Author Affiliations

1 College of Mathematics and Computer Science, Northwest University for Nationalities, Lanzhou, 730124, P.R. China

2 College of Mathematics and Statistics, Northwest Normal University, Lanzhou, 730070, P.R. China

3 School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, P.R. China

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Boundary Value Problems 2013, 2013:219  doi:10.1186/1687-2770-2013-219

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/219


Received:28 May 2013
Accepted:28 August 2013
Published:7 November 2013

© 2013 Jiao 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 following quasilinear elliptic system:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M1">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M6">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M7">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M8">View MathML</a> is the critical Sobolev exponent and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M9">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M10">View MathML</a>) is a bounded smooth domain. By using the perturbation method, we establish the existence of both positive and negative solutions for this system.

MSC: 35J60, 35B33.

Keywords:
quasilinear elliptic system; positive solution; negative solution; perturbation method

1 Introduction

Let us consider the following quasilinear elliptic system:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M11">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M13">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M6">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M7">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M8">View MathML</a> is the critical Sobolev exponent and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M9">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M10">View MathML</a>) is a bounded smooth domain. This system includes the following special class of system with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M21">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M22">View MathML</a>, i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M23">View MathML</a>

which is referred to as the so-called modified nonlinear Schrödinger system.

Our assumptions on the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M24">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M25">View MathML</a> are as follows.

(A1) The functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M26">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M27">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M28">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M29">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M30">View MathML</a>.

(A2) There exist constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M31">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M32">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M33">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M34">View MathML</a> satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M35">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M36">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M37">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M38">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M39">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M40">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M41">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M42">View MathML</a>.

(A3)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M43">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M40">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M41">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M42">View MathML</a>.

In recent years, much attention has been devoted to the quasilinear Schrödinger equation of the following form:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M47">View MathML</a>

(1.2)

See, for example, [1] where Poppenberg et al. proved the existence of a positive ground state solution by using a constrained minimization argument. Using a change of variables, Liu et al.[2] used an Orlicz space to prove the existence of a soliton solution for equation (1.2) via the mountain pass theorem. Colin and Jeanjean [3] also made use of a change of variables but worked in the Sobolev space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M48">View MathML</a>. They proved the existence of a positive solution for equation (1.2) from the classical results given by Berestycki and Lions [4]. Liu et al.[5] established the existence of both one-sign and nodal ground states of soliton-type solutions for equation (1.2) by the Nehari method. By using the Nehari manifold method and the concentration compactness principle (see [6]) in the Orlicz space, Guo and Tang [7] considered the following quasilinear Schrödinger system:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M49">View MathML</a>

(1.3)

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M50">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M51">View MathML</a> having a potential well and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M6">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M7">View MathML</a>, and they proved the existence of a ground state solution for system (1.3) which localizes near the potential well <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M55">View MathML</a> for λ large enough. Guo and Tang [8] considered also ground state solutions of the single quasilinear Schrödinger equation corresponding to system (1.3) by the same methods and obtained similar results. In particular, by the perturbation method, Liu et al.[9] considered the existence and multiplicity of solutions for the following quasilinear equation of the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M56">View MathML</a>

(1.4)

under suitable assumptions.

It is worth pointing out that the existence of one-bump or multi-bump bound state solutions for the related semilinear Schrödinger equation (1.2) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M57">View MathML</a> has been extensively studied. One can see Bartsch and Wang [10], Ambrosetti et al.[11], Ambrosetti et al.[12], Byeon and Wang [13], Cingolani and Lazzo [14], Cingolani and Nolasco [15], Del Pino and Felmer [16,17], Floer and Weinstein [18], Oh [19,20] and the references therein.

Motivated by the single equation (1.4), the purpose of this paper is to study the existence of both positive and negative solutions for the coupled quasilinear system (1.1). We mainly follow the idea of Liu et al.[9] to perturb the functional and obtain our main results. We point out that the procedure to system (1.1) is not trivial at all. Since the appearance of the quasilinear terms <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M58">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M59">View MathML</a>, we need more delicate estimates.

The paper is organized as follows. In Section 2, we introduce a perturbation of the functional and give our main results (Theorem 2.1 and Theorem 2.2). In Section 3, we verify the Palais-Smale condition for the perturbed functional. Section 4 is devoted to some asymptotic behavior of the sequences <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M60">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M61">View MathML</a> satisfying some conditions. Finally, our main results will be proved in Section 5.

Throughout this paper, we will use the same C to denote various generic positive constants, and we will use <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M62">View MathML</a> to denote quantities that tend to 0.

2 Perturbation of the functional and main results

In order to obtain the desired existence of solutions for system (1.1), in this section, we introduce a perturbation of the functional and give our main results.

The weak form of system (1.1) is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M63">View MathML</a>

(2.1)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M64">View MathML</a>, which is formally the variational formulation of the following functional:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M65">View MathML</a>

(2.2)

We may define the derivative of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M66">View MathML</a> at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M67">View MathML</a> in the direction of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M64">View MathML</a> as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M69">View MathML</a>

(2.3)

We call <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M67">View MathML</a> a critical point of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M66">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M72">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M73">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M74">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M75">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M76">View MathML</a>. That is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M67">View MathML</a> is a weak solution for system (1.1).

When we consider system (1.1) by using the classical critical point theory, we encounter the difficulties due to the lack of an appropriate working space. In general, it seems that there is no suitable space in which the variational functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M66">View MathML</a> possesses both smoothness and compactness properties. For smoothness, one would need to work in a space smaller than <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M79">View MathML</a> to control the term involving the quasilinear term in system (1.1), but it seems impossible to obtain bounds for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M80">View MathML</a> sequence in this setting. Several ideas and approaches, such as minimizations [1,21], the Nehari method [5] and change of variables [2,3], have been used in recent years to overcome the difficulties. In this paper, we consider the perturbed functional

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M81">View MathML</a>

(2.4)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M82">View MathML</a> is a parameter. Then it is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M83">View MathML</a> is a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M84">View MathML</a>-functional on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M85">View MathML</a>. We can define also the derivative of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M86">View MathML</a> at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M67">View MathML</a> in the direction of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M88">View MathML</a> as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M89">View MathML</a>

(2.5)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M64">View MathML</a>. The idea of this paper is to obtain the existence of the critical points of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M86">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M92">View MathML</a> small and establish suitable estimates for the critical points as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M93">View MathML</a> so that we may pass to the limit to get the solutions for the original system (1.1).

Our main results are as follows.

Theorem 2.1Assume that (A1)-(A3) hold, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M6">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M7">View MathML</a>. Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M97">View MathML</a>and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M98">View MathML</a>be a sequence of critical points of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M99">View MathML</a>satisfying<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M100">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M101">View MathML</a>for someCindependent ofn. Then, up to a subsequence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M102">View MathML</a>

as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M103">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M67">View MathML</a>is a critical point of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M66">View MathML</a>.

Theorem 2.2Assume that (A1)-(A3) hold, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M6">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M7">View MathML</a>. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M86">View MathML</a>has a positive critical point<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M110">View MathML</a>and a negative critical point<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M111">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M110">View MathML</a> (resp.,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M113">View MathML</a>) converges to a positive (resp., negative) solution for system (1.1) as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M93">View MathML</a>.

Notation We denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M115">View MathML</a> the norm of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M116">View MathML</a> and by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M117">View MathML</a> the norm of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M118">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M119">View MathML</a>).

3 Compactness of the perturbed functional

In this section, we verify the Palais-Smale condition (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M80">View MathML</a> condition in short) for the perturbed functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M121">View MathML</a>. We have the following proposition.

Proposition 3.1For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M92">View MathML</a>fixed, the functional<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M123">View MathML</a>satisfies<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M80">View MathML</a>condition for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M125">View MathML</a>. That is, any sequence<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M126">View MathML</a>satisfying, for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M125">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M128">View MathML</a>

(3.1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M129">View MathML</a>

(3.2)

has a strongly convergent subsequence in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M85">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M131">View MathML</a>is the dual space of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M85">View MathML</a>.

To give the proof of Proposition 3.1, we need the following lemma firstly.

Lemma 3.2Suppose that a sequence<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M133">View MathML</a>satisfies (3.1) and (3.2). Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M134">View MathML</a>

Proof It follows from (3.1) and (3.2) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M135">View MathML</a>

Thus we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M136">View MathML</a>

This completes the proof of Lemma 3.2. □

Now we give the proof of Proposition 3.1.

Proof of Proposition 3.1 From Lemma 3.2 , we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M137">View MathML</a> is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M85">View MathML</a>. So there exists a subsequence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M137">View MathML</a>, still denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M137">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M141">View MathML</a>

Now we prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M142">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M85">View MathML</a>. In (2.5), choosing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M144">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M145">View MathML</a>

(3.3)

We may estimate the terms involved as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M146">View MathML</a>

Returning to (3.3), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M147">View MathML</a>

which implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M148">View MathML</a>, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M149">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M85">View MathML</a>. This completes the proof of Proposition 3.1. □

4 Some asymptotic behavior

Proposition 3.1 enables us to apply minimax argument to the functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M151">View MathML</a>. In this section, we also study the behavior of the sequences <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M60">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M61">View MathML</a> satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M154">View MathML</a>

(4.1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M155">View MathML</a>

(4.2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M156">View MathML</a>

(4.3)

The following proposition is the key of this section.

Proposition 4.1Assume that the sequences<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M60">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M158">View MathML</a>satisfy (4.1)-(4.3). Then, after extracting a sequence, still denoted byn, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M159">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M160">View MathML</a>

as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M103">View MathML</a>.

Proof Similar to the proof of Lemma 3.2, by (4.1)-(4.3), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M162">View MathML</a>

(4.4)

Thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M163">View MathML</a>

(4.5)

for some C independent of n. Then, up to a subsequence, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M164">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M165">View MathML</a>

as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M103">View MathML</a>. This completes the proof of Proposition 4.1. □

5 Proof of main results

In this section, we give the proof of our main results. Firstly, we prove Theorem 2.1.

Proof of Theorem 2.1 Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M167">View MathML</a> satisfies the following equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M168">View MathML</a>

(5.1)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M169">View MathML</a>. Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M170">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M171">View MathML</a>

By Moser’s iteration, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M172">View MathML</a>

(5.2)

Hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M173">View MathML</a>

(5.3)

for some C independent of n. To show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M67">View MathML</a> is a critical point of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M66">View MathML</a>, we use some arguments in [22,23] (see more references therein). In (5.1), we choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M176">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M177">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M178">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M179">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M180">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M181">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M182">View MathML</a> is a constant. Substituting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M88">View MathML</a> into (5.1), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M184">View MathML</a>

(5.4)

Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M185">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M186">View MathML</a> are positive for M large enough. By Fatou’s lemma, the weak convergence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M187">View MathML</a> and the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M188">View MathML</a> is bounded, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M189">View MathML</a>

(5.5)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M190">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M191">View MathML</a>. We may choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M192">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M193">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M194">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M195">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M196">View MathML</a>. Then we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M197">View MathML</a>

(5.6)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M190">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M191">View MathML</a>.

Similarly, we may obtain an opposite inequality. Thus we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M200">View MathML</a>

(5.7)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M201">View MathML</a>. That is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M67">View MathML</a> is a critical point of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M66">View MathML</a> and a solution for system (1.1). By doing approximations, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M67">View MathML</a> in the place of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M205">View MathML</a> of (5.7)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M206">View MathML</a>

(5.8)

Setting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M207">View MathML</a> in (5.1), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M208">View MathML</a>

(5.9)

Using <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M209">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M103">View MathML</a>, (5.8), (5.9) and lower semi-continuity, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M211">View MathML</a>

as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M103">View MathML</a>.

In particular, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M213">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M214">View MathML</a>

as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M103">View MathML</a>. This completes the proof of Theorem 2.1. □

Next, we apply the mountain pass theorem to obtain the existence of critical points of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M86">View MathML</a>. Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M217">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M218">View MathML</a>.

Let us consider the functional

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M219">View MathML</a>

(5.10)

Here and in what follows, we denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M220">View MathML</a>. The functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M86">View MathML</a> satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M80">View MathML</a> condition. Similarly, we may verify that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M223">View MathML</a> satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M80">View MathML</a> condition. By the ε-Young inequality, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M225">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M226">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M227">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M228">View MathML</a>

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M229">View MathML</a>

for ε, ρ small. Thus we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M230">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M231">View MathML</a> and for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M218">View MathML</a> small enough. Choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M233">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M191">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M235">View MathML</a>. Define a path <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M236">View MathML</a> by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M237">View MathML</a>. When T is large enough, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M238">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M239">View MathML</a>

for some m independent of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M82">View MathML</a>.

Define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M241">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M242">View MathML</a>

From the mountain pass theorem we obtain that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M243">View MathML</a>

is a critical value of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M223">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M110">View MathML</a> be a critical point corresponding to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M246">View MathML</a>. We have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M247">View MathML</a>. Thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M110">View MathML</a> is a positive critical point of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M86">View MathML</a> by the strong maximum principle. In summary, we have the following.

Proposition 5.1There exist positive constantsρandmindependent ofμsuch that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M86">View MathML</a>has a positive critical point<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M110">View MathML</a>satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M252">View MathML</a>

Finally, we give the proof of Theorem 2.2.

Proof of Theorem 2.2 For a positive solution of system (1.1), the proof follows from Proposition 5.1 and Theorem 2.1. A similar argument gives a negative solution of system (1.1). This completes the proof of Theorem 2.2. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All the authors were involved in carrying out this study. All authors read and approved the final manuscript.

Acknowledgements

This paper was finished while the first author was a visiting fellow at the School of Mathematical Sciences of Beijing Normal University, and the first author would like to express her gratitude for their hospitality during her visit. This work is supported by the National Science Foundation of China (11061031), Fundamental Research Funds for the Central Universities (31920130004) and Fundamental Research Funds for the Gansu University.

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