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Nontrivial solutions for a quasilinear elliptic system

Xiyou Cheng* and Lu Yang

Author Affiliations

School of Mathematics and Statistics, Lanzhou University, Lanzhou, 730000, P.R. China

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Boundary Value Problems 2014, 2014:34  doi:10.1186/1687-2770-2014-34

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/34


Received:5 October 2013
Accepted:16 January 2014
Published:7 February 2014

© 2014 Cheng and Yang; 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 deal with the existence of three nontrivial solutions for the potential system of p-Laplacian equations with homogeneous Dirichlet boundary conditions. Applying the Nehari procedure and abstract linking theorem on a product space, we give a linking structure for our variational problem, and then, combining with the classical minimax principle, we obtain three nontrivial critical values for the relevant energy functional.

MSC: 35J20, 35J25.

Keywords:
elliptic equations; Nehari manifold; linking theorem; minimax principle

1 Introduction

In this paper, we are concerned with the multiplicity of nontrivial solutions for the following system of quasilinear equations:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M1">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M2">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M3">View MathML</a>), Ω is a bounded smooth domain in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M4">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M5">View MathML</a>), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M6">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M7">View MathML</a> are Carathéodory functions (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M8">View MathML</a>), and there exists a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M9">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M10">View MathML</a>

In recent years, many authors have studied the existence of nontrivial solutions for Laplacian systems and p-Laplacian systems, see [1-5] and the references therein. Usually the authors change the problem into the critical point problem of the corresponding energy functional and then apply the critical point theory or the variational method, or they change it into the fixed point problem of the corresponding compactly continuous mapping and then apply topological degree theory or the method of lower and upper solutions. For instance, in [2] Costa and Magalhaes unified the cooperative and noncooperative Laplacian systems, and they got the existence of nontrivial solutions via the variational approach; in [1] Conti et al. dealt with the competitive Laplacian system, and they established the existence of positive solutions by the Nehari procedure, critical point theory, and topological degree theory; in [3,4] the authors studied the sublinear p-Laplacian systems, and they obtained the existence of positive solutions by the method of lower and upper solutions and Leray-Schauder degree theory, respectively; in [5] Zhang and Zhang considered the existence of nontrivial solutions for nonlinear Laplacian systems and p-Laplacian systems applying the direct variational method. More recently, some authors discussed the multiplicity of nontrivial solutions for Laplacian systems and p-Laplacian systems, see [6-12] and references therein. In [6,11,12], the authors provided the existence results of three nontrivial solutions for system (1.1) with one parameter for the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M11">View MathML</a>, where the main methods used are the Nehari procedure and the linking theorem on product space. In [8] Motreanu and Zhang consider a general noncoercive quasilinear elliptic system, they establish the existence of two opposite constant sign solutions; in the case where the system has a variational structure, under the proper hypotheses, they obtain a third nontrivial solution, which is sign changing in the sense that one cannot have both components of the new solution of the same constant sign; their approach relies on a suitable method of sub-supersolutions combined with truncation and variational arguments that do not require a subcritical growth condition. In [9] Shen and Zhang established the existence of two positive solutions for multi-parameter p-Laplacian systems with critical exponents by use of the Nehari procedure and the variational approach. In [7,10], the present author and coauthors dealt with a class of Laplacian systems with superlinear and sublinear terms applying the fixed point index formula on a product cone, and they obtained the existence and multiplicity of positive solutions.

Motivated by some ideas in [1,7,9,12], we shall deal with the existence of nonnegative solutions (especially, positive solutions) for system (1.1) with superlinear and subcritical nonlinear terms. It is well known that the Ambrosetti-Rabinowitz type result (see [13,14]) can be extended to system (1.1) with superlinear and subcritical nonlinear terms by imposing the Ambrosetti-Rabinowitz conditions and other proper conditions on nonlinear terms. Now, we have a natural question of when system (1.1) has multiple nonnegative solutions. In order to obtain the multiplicity of nonnegative solutions for system (1.1), we need only to construct the multiple critical values of the corresponding energy functional. For this matter, first we get two ground states in view of the Nehari procedure, establish a linking structure by the abstract linking theorem on the product space, and then construct the third critical value by the classical minimax principle.

The present paper is organized as following. In Section 2, we provide a linking structure for our variational problem (see Theorem 2.2); in Section 3, we verify that the energy functional satisfies the (P.S.) condition (see Theorem 3.1); in Section 4, we prove the existence of three nonnegative solutions for system (1.1) as λ is small enough, and, in addition, one of them is positive if the equations excluding coupled terms have both a unique positive solution (see Theorem 4.1).

2 Linking structure

In this section, we list some preliminaries, including the abstract linking theorem on the product space and the concept of Nehari manifold, and then we give a linking structure, which is useful for constructing the critical value of the functional associated with system (1.1).

Definition 2.1[12]

Let f be a real functional on Banach space X and c be a real constant, we say that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M12">View MathML</a> has the sphere property, if the following hypotheses are satisfied:

(S1) f is continuous on X;

(S2) there is a homeomorphic mapping between <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M13">View MathML</a> and unit sphere of X;

(S3) for any fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M14">View MathML</a>, the equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M15">View MathML</a> has a unique solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M16">View MathML</a>;

(S4) X is separated into two open connected subsets by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M13">View MathML</a> and the origin is contained in one of the subsets.

Theorem 2.1[12]

LetX, Ybe Banach spaces with the following direct sum decomposition:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M18">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M19">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M20">View MathML</a>are finite dimensional subspaces ofX, Y, respectively. Letf, gbe the real functionals onX, Y, respectively, c, dbe two real constants and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M13">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M22">View MathML</a>have the sphere property. Take<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M23">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M24">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M25">View MathML</a>. Denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M26">View MathML</a>

then∂QlinksS.

Definition 2.2[15]

Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M27">View MathML</a> is such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M28">View MathML</a>, then the constraint set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M29">View MathML</a> is called a Nehari manifold of X.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M30">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M31">View MathML</a>

Denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M32">View MathML</a> the Nehari manifold of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M33">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M34">View MathML</a>; we have the following.

Lemma 2.1Assume that the following conditions are satisfied:

(F1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M35">View MathML</a>, and there exist constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M36">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M37">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M38">View MathML</a>

(F2) there exist constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M39">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M40">View MathML</a>

(F3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M41">View MathML</a>uniformly w.r.t. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M42">View MathML</a>, here<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M43">View MathML</a>is the first eigenvalue of the operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M44">View MathML</a>on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M45">View MathML</a>;

(F4) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M46">View MathML</a>is an increasing function ofuon<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M47">View MathML</a>.

Then the Nehari manifolds<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M48">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M49">View MathML</a>have the sphere property.

Proof It is very similar to the proof of Lemma 4.1 on page 72 in [15], so we omit it. □

Theorem 2.2Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M50">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M51">View MathML</a>satisfy conditions (F1)-(F4). Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M52">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M53">View MathML</a>, here<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M54">View MathML</a>. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M55">View MathML</a>, then∂QlinksS.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M56">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M57">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M58">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M59">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M60">View MathML</a>, and by conditions (F1), (F3) and (F4) it is easy to verify that there exists a unique <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M61">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M62">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M63">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M64">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M65">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M66">View MathML</a>. Thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M67">View MathML</a>

From Lemma 2.1 and Theorem 2.1, ∂Q links S. □

3 Palais-Smale condition

It is well known that the nontrivial critical points of the functional

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M68">View MathML</a>

correspond to nontrivial solutions of system (1.1). In the following, we show that the functional ψ satisfies the (P.S.) condition.

Theorem 3.1Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M50">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M51">View MathML</a>satisfy conditions (F1)-(F2), and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M71">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M72">View MathML</a>satisfy the conditions:

(H1) there exist constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M73">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M74">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M75">View MathML</a>

(H2) there exist constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M76">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M77">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M78">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M79">View MathML</a>

then the functionalψsatisfies (P.S.) condition.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M80">View MathML</a> be a sequence such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M81">View MathML</a>

(3.1)

we need to prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M82">View MathML</a> has a strongly convergent subsequence.

Claim I. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M82">View MathML</a> is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M84">View MathML</a>.

In fact, from (3.1) there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M85">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M86">View MathML</a>

(3.2)

and for n large enough

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M87">View MathML</a>

(3.3)

Taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M88">View MathML</a>, combining with (3.2) and (3.3), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M89">View MathML</a>

as n large enough, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M90">View MathML</a>’s are positive constants, which implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M82">View MathML</a> is bounded.

Claim II. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M82">View MathML</a> has a strongly convergent subsequence.

From Claim I, there exists a subsequence, relabel it as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M93">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M94">View MathML</a>

In what follows, we show that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M95">View MathML</a>

In fact, by (3.1) we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M96">View MathML</a>

(3.4)

that is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M97">View MathML</a>

(3.5)

From conditions (F1), (H1), and the compact embedding theorems (see [16]), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M98">View MathML</a>

(3.6)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M99">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M100">View MathML</a> are the conjugate numbers of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M101">View MathML</a>, q, respectively. In combination with (3.5) and (3.6), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M102">View MathML</a>

(3.7)

By the property (S+) of p-Laplacian operator (see [14,17]), we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M103">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M104">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M105">View MathML</a>. Similarly, we can prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M106">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M107">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M105">View MathML</a>. □

4 Nontrivial solutions

First, we consider the least energy critical point of the functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M33">View MathML</a>, which is very useful for the multiplicity of nontrivial solutions to system (1.1).

For that purpose, we define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M110">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M111">View MathML</a>

Similar to [15], we have the following.

Lemma 4.1Under conditions (F1)-(F4), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M112">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M113">View MathML</a>is a critical value of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M33">View MathML</a>.

Proof It is very similar to the proof of Theorem 4.2 on page 73 in [15], so we omit it. □

Lemma 4.2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M115">View MathML</a>satisfy conditions (F1)-(F4), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M116">View MathML</a>be of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M117">View MathML</a>class and satisfy<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M118">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M119">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M120">View MathML</a>a constant (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M121">View MathML</a>). Denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M122">View MathML</a>

then, on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M32">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M124">View MathML</a>

for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M125">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M126">View MathML</a>a constant.

Proof By the assumptions on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M116">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M128">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M129">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M130">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M131">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M132">View MathML</a>

due to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M133">View MathML</a>, which means that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M134">View MathML</a>

The proof is complete. □

Theorem 4.1Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M50">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M51">View MathML</a>satisfy conditions (F1)-(F4) and that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M71">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M72">View MathML</a>satisfy conditions (H1)-(H2) and the following condition:

(H3) for a.a. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M139">View MathML</a>and all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M140">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M141">View MathML</a>

Then, there exists a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M142">View MathML</a>such that system (1.1) has at least three nonnegative solutions for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M143">View MathML</a>. Furthermore, if the problems

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M144">View MathML</a>

have both a unique positive solution, then for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M145">View MathML</a>, system (1.1) has at least three nonnegative solutions in which there is a nontrivial positive solution.

Proof From Lemma 4.1, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M146">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M147">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M148">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M149">View MathML</a>), which implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M150">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M151">View MathML</a> are nonnegative solutions of the following system:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M152">View MathML</a>

(4.1)

Combining with condition (H3), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M150">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M154">View MathML</a> are nonnegative solutions of system (1.1) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M155">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M156">View MathML</a>.

Now, we prove the existence of the third nonnegative solution.

First, we consider <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M157">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M158">View MathML</a>, as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M159">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M90">View MathML</a>’s are positive constants, which implies that there exists an <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M161">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M162">View MathML</a>

(4.2)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M163">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M164">View MathML</a>), and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M165">View MathML</a>

then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M166">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M167">View MathML</a>, thus ∂Q links S by Theorem 2.2.

Second, we claim that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M168">View MathML</a> as λ is small enough. In fact, by Lemma 4.1,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M169">View MathML</a>

which, in connection with (4.2), implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M170">View MathML</a>

(4.3)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M125">View MathML</a>.

On the other hand, combining with Lemma 4.2, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M172">View MathML</a> we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M173">View MathML</a>

here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M174">View MathML</a>’s are proper positive constants. Now let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M175">View MathML</a>, then for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M143">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M177">View MathML</a>

(4.4)

Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M178">View MathML</a> is inferred from (4.3) and (4.4).

Finally, we define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M179">View MathML</a>

In combination with Theorem 3.1 and the classical minimax principle (see [15,18]), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M180">View MathML</a> is the third critical value of ψ.

Therefore, system (1.1) has at least three nonnegative solutions for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M143">View MathML</a>.

In addition, if the equations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M182">View MathML</a>

have both a unique positive solution (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M121">View MathML</a>), the third nonnegative solution of system (1.1) is actually positive. In fact, if one component of the third solution equals zero, its functional value equals <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M184">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/34/mathml/M185">View MathML</a>, which is a contradiction. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors read and approved the final manuscript.

Acknowledgements

This work is partly supported by NNSF (11101404, 11201204, 11361053) of China and the State Scholarship Fund (201308620021) of China Scholarship Council.

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