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Existence of homoclinic orbits for a class of p-Laplacian systems in a weighted Sobolev space

Xiubo Shi1, Qiongfen Zhang1* and Qi-Ming Zhang2

Author Affiliations

1 College of Science, Guilin University of Technology, Guilin, Guangxi, 541004, P.R. China

2 College of Science, Hunan University of Technology, Zhuzhou, Hunan, 412000, P.R. China

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Boundary Value Problems 2013, 2013:137  doi:10.1186/1687-2770-2013-137


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/137


Received:5 December 2012
Accepted:7 May 2013
Published:24 May 2013

© 2013 Shi 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

By applying the mountain pass theorem and symmetric mountain pass theorem in critical point theory, the existence of at least one or infinitely many homoclinic solutions is obtained for the following p-Laplacian system:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M1">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M6">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M7">View MathML</a> are not periodic in t.

MSC: 34C37, 35A15, 37J45, 47J30.

Keywords:
homoclinic solutions; variational methods; weighted <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M8">View MathML</a> space; p-Laplacian systems

1 Introduction

Consider homoclinic solutions of the following p-Laplacian system:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M9">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M12">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M13">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M14">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M15">View MathML</a>. As usual, we say that a solution u of (1.1) is a nontrivial homoclinic (to 0) if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M16">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M17">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M18">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M19">View MathML</a>.

When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M20">View MathML</a>, (1.1) reduces to the following second-order Hamiltonian system:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M21">View MathML</a>

(1.2)

If we take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M20">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M23">View MathML</a>, then (1.2) reduces to the following second-order Hamiltonian system:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M24">View MathML</a>

(1.3)

The existence of homoclinic orbits for Hamiltonian systems is a classical problem and its importance in the study of the behavior of dynamical systems has been recognized by Poincaré [1]. Up to the year of 1990, a few of isolated results can be found, and the only method for dealing with such a problem was the small perturbation technique of Melnikov.

Recently, the existence and multiplicity of homoclinic solutions and periodic solutions for Hamiltonian systems have been extensively studied by critical point theory. For example, see [2-20] and references therein. However, few results [21,22] have been obtained in the literature for system (1.2). In [22], by introducing a suitable Sobolev space, Salvatore established the following existence results for system (1.2) when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M3">View MathML</a>.

Theorem A[22]

Assume thataandWsatisfy the following conditions:

(A) Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M27">View MathML</a>is a continuous, positive function onsuch that for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M4">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M29">View MathML</a>

(W1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M30">View MathML</a>and there exists a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M31">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M32">View MathML</a>

(W2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M33">View MathML</a>as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M34">View MathML</a>uniformly with respect to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M12">View MathML</a>.

(W3) There exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M36">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M37">View MathML</a>

Then problem (1.2) has one nontrivial homoclinic solution.

When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M38">View MathML</a> is an even function in x, Salvatore [22] obtained the following existence theorem of an unbounded sequence of homoclinic orbits for problem (1.2) by the symmetric mountain pass theorem.

Theorem B[22]

Assume thataandWsatisfy (A), (W1)-(W3) and the following condition:

(W4) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M39">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M40">View MathML</a>.

Then problem (1.2) has an unbounded sequence of homoclinic solutions.

In [21], Chen and Tang improved Theorem A and Theorem B by relaxing conditions (W1) and (W2) and removing condition (W3). Motivated mainly by the ideas of [18,21-23], we will consider homoclinic solutions of (1.1) by the mountain pass theorem and symmetric mountain pass theorem. Precisely, we obtain the following main results.

Theorem 1.1Suppose thataandWsatisfy the following conditions:

(A)′ Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M2">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M27">View MathML</a>is a continuous, positive function onsuch that for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M4">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M45">View MathML</a>

(W5) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M46">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M47">View MathML</a>, and there exists a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M48">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M49">View MathML</a>

uniformly in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M50">View MathML</a>.

(W6) There is a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M51">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M52">View MathML</a>

(W7) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M53">View MathML</a>and there exists a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M54">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M55">View MathML</a>

Then problem (1.1) has one nontrivial homoclinic solution.

Theorem 1.2Suppose thataandWsatisfy (A)′, (W6) and the following conditions:

(W5)′ <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M46">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M47">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M58">View MathML</a>

uniformly in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M4">View MathML</a>.

(W7)′ <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M53">View MathML</a>and there exists a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M54">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M62">View MathML</a>

Then problem (1.1) has one nontrivial homoclinic solution.

Theorem 1.3Suppose thataandWsatisfy (A)′ and (W4)-(W7). Then problem (1.1) has an unbounded sequence of homoclinic solutions.

Theorem 1.4Suppose thataandWsatisfy (A)′, (W4), (W5)′, (W6), (W7)′. Then problem (1.1) has an unbounded sequence of homoclinic solutions.

Remark 1.1 When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M20">View MathML</a>, condition (A)′ reduces to condition (A). Obviously, Theorem 1.1-Theorem 1.4 generalize and improve Theorem A, Theorem B and the corresponding results in [21]. It is easy to see that our results hold true even if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M20">View MathML</a>. To the best of our knowledge, similar results for problem (1.1) cannot be seen in the literature; from this point, our results are new.

Remark 1.2 If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M20">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M23">View MathML</a>, then problem (1.1) reduces to problem (1.3). As pointed out in [23], Theorem A can be proved by replacing (A) with the more general assumption: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M67">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M68">View MathML</a>.

The rest of this paper is organized as follows. In Section 2, some preliminaries are presented and we establish an embedding result. In Section 3, we give the proofs of our results. In Section 4, some examples are given to illustrate our results.

2 Preliminaries

We set, for any real number <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M69">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M70">View MathML</a>

with the usual norms

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M71">View MathML</a>

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M72">View MathML</a>

be the Sobolev space with the norm given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M73">View MathML</a>

If σ is a positive, continuous function on ℝ and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M74">View MathML</a>, let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M75">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M76">View MathML</a> equipped with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M77">View MathML</a>

is a reflexive Banach space. When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M78">View MathML</a>, we set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M79">View MathML</a>

with the norm given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M80">View MathML</a>

Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M81">View MathML</a>, where a is the function given in condition (A)′. Then E with its standard norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M82">View MathML</a> is a reflexive Banach space. The functional φ corresponding to (1.1) on E is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M83">View MathML</a>

(2.1)

Clearly, it follows from (W5) or (W5)′ that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M84">View MathML</a>. By Theorem 2.1 of [24], we can deduce that the map

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M85">View MathML</a>

is continuous from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M86">View MathML</a> in the dual space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M87">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M88">View MathML</a>. As the embeddings <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M89">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M90">View MathML</a> are continuous, if (A)′ and (W5) or (W5)′ hold, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M91">View MathML</a> and one can easily check that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M92">View MathML</a>

(2.2)

Furthermore, the critical points of φ in E are classical solutions of (1.1) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M93">View MathML</a>.

To prove our results, we need the following generalization of the Lebesgue dominated convergence theorem.

Lemma 2.1[25]

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M94">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M95">View MathML</a>be two sequences of measurable functions on a measurable setA, and let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M96">View MathML</a>

If

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M97">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M98">View MathML</a>

then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M99">View MathML</a>

The following lemma is an improvement result of [23] in which the author considered the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M20">View MathML</a>.

Lemma 2.2Ifasatisfies assumption (A)′, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M101">View MathML</a>

(2.3)

Moreover, there exists a Sobolev spaceZsuch that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M102">View MathML</a>

(2.4)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M103">View MathML</a>

(2.5)

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M104">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M105">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M106">View MathML</a>

where from (A)′, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M107">View MathML</a>. Then (2.3) holds.

By (A)′, there exists a continuous positive function ρ such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M108">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M68">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M110">View MathML</a>

Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M111">View MathML</a>

(2.4) holds by taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M112">View MathML</a>.

Finally, as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M113">View MathML</a> is the weighted Sobolev space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M114">View MathML</a>, it follows from [24] that (2.5) holds. □

The following two lemmas are the mountain pass theorem and symmetric mountain pass theorem, which are useful in the proofs of our theorems.

Lemma 2.3[26]

LetEbe a real Banach space and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M115">View MathML</a>satisfying (PS)-condition. Suppose<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M116">View MathML</a>and:

(i) There exist constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M117">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M118">View MathML</a>.

(ii) There exists an<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M119">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M120">View MathML</a>.

ThenIpossesses a critical value<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M121">View MathML</a>which can be characterized as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M122">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M123">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M124">View MathML</a>is an open ball inEof radiusρcentered at 0.

Lemma 2.4[26]

LetEbe a real Banach space and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M115">View MathML</a>withIeven. Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M116">View MathML</a>andIsatisfies (PS)-condition, (i) of Lemma 2.3 and the following condition:

(iii) For each finite dimensional subspace<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M127">View MathML</a>, there is<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M128">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M129">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M130">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M131">View MathML</a>is an open ball inEof radiusrcentered at 0.

ThenIpossesses an unbounded sequence of critical values.

Lemma 2.5Assume that (W6) and (W7) or (W7)′ hold. Then, for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M132">View MathML</a>,

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M133">View MathML</a>is nondecreasing on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M134">View MathML</a>;

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M135">View MathML</a>is nonincreasing on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M134">View MathML</a>.

The proof of Lemma 2.5 is routine and we omit it.

3 Proofs of theorems

Proof of Theorem 1.1 Step 1. The functional φ satisfies (PS)-condition. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M137">View MathML</a> satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M138">View MathML</a> be bounded and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M139">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M140">View MathML</a>. Hence, there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M141">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M142">View MathML</a>

(3.1)

It is well known [27] that there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M143">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M144">View MathML</a>

(3.2)

From (2.1), (2.2), (3.1), (W6) and (W7), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M145">View MathML</a>

It follows from Lemma 2.2, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M146">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M51">View MathML</a> and the above inequalities that there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M148">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M149">View MathML</a>

(3.3)

Now we prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M150">View MathML</a> in E. Passing to a subsequence if necessary, it can be assumed that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M151">View MathML</a> in E. From Lemma 2.2, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M150">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M153">View MathML</a>. From (3.2) and (3.3), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M154">View MathML</a>

(3.4)

Inequality (3.4) implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M155">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M12">View MathML</a>. By (W5), we know that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M157">View MathML</a>

which implies that for any given constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M158">View MathML</a>, there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M159">View MathML</a> related to C such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M160">View MathML</a>

Hence, there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M161">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M162">View MathML</a>

(3.5)

Hence, from (3.5), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M163">View MathML</a>

(3.6)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M164">View MathML</a>. Moreover, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M27">View MathML</a> is a positive continuous function on ℝ, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M166">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M167">View MathML</a> for almost every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M12">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M169">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M170">View MathML</a>

It follows from Lemma 2.1, (3.6) and the above inequalities that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M171">View MathML</a>

This shows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M172">View MathML</a>

(3.7)

From (2.2), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M173">View MathML</a>

(3.8)

It is easy to see that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M174">View MathML</a> there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M175">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M176">View MathML</a>

(3.9)

Inequality (3.9) implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M177">View MathML</a>

(3.10)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M178">View MathML</a>

(3.11)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M179">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M180">View MathML</a> are positive constants. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M139">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M182">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M183">View MathML</a> in E and the embeddings <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M89">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M90">View MathML</a> are continuous, it follows from Lemma 2.2, (3.7), (3.8), (3.10) and (3.11) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M186">View MathML</a>

(3.12)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M187">View MathML</a>

(3.13)

Hence, by (3.12) and (3.13), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M150">View MathML</a> in E. This shows that φ satisfies (PS)-condition.

Step 2. From (W5), there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M189">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M190">View MathML</a>

(3.14)

By (3.14), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M191">View MathML</a>

(3.15)

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M192">View MathML</a>

(3.16)

Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M193">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M194">View MathML</a>, it follows from (3.2) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M195">View MathML</a>

which shows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M196">View MathML</a>. From Lemma 2.5(i) and (3.16), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M197">View MathML</a>

(3.17)

It follows from (W7), (3.15), (3.17) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M198">View MathML</a>

Therefore, we can choose a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M199">View MathML</a> depending on ρ such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M200">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M201">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M202">View MathML</a>.

Step 3. From Lemma 2.5(ii) and (3.2), we have for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M201">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M204">View MathML</a>

(3.18)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M205">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M206">View MathML</a>. Take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M207">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M208">View MathML</a>

(3.19)

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M209">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M210">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M211">View MathML</a>, from Lemma 2.5(i) and (3.19), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M212">View MathML</a>

(3.20)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M213">View MathML</a>. From (W7), (2.1), (3.18), (3.19), (3.20), we get for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M211">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M215">View MathML</a>

(3.21)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M216">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M217">View MathML</a>, it follows from (3.21) that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M218">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M219">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M220">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M221">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M222">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M223">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M224">View MathML</a>. By Lemma 2.3, φ has a critical value <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M225">View MathML</a> given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M226">View MathML</a>

(3.22)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M227">View MathML</a>

Hence, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M228">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M229">View MathML</a>

The function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M230">View MathML</a> is a desired solution of problem (1.1). Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M231">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M230">View MathML</a> is a nontrivial homoclinic solution. The proof is complete. □

Proof of Theorem 1.2 In the proof of Theorem 1.1, the condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M233">View MathML</a> in (W7) is only used in the proofs of (3.3) and Step 2. Therefore, we only need to prove that (3.3) and Step 2 still hold if we use (W5)′ and (W7)′ instead of (W5) and (W7). We first prove that (3.3) holds. From (W6), (W7)′, (2.1), (2.2) and (3.1), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M234">View MathML</a>

which implies that there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M148">View MathML</a> such that (3.3) holds. Next, we prove Step 2 still holds. From (W5)′, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M189">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M237">View MathML</a>

(3.23)

By (3.23), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M238">View MathML</a>

(3.24)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M239">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M194">View MathML</a>, it follows from (3.2) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M241">View MathML</a>

which shows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M196">View MathML</a>. It follows from (2.1) and (3.24) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M243">View MathML</a>

Therefore, we can choose a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M199">View MathML</a> depending on ρ such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M200">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M201">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M202">View MathML</a>. The proof of Theorem 1.2 is complete. □

Proof of Theorem 1.3 Condition (W4) shows that φ is even. In view of the proof of Theorem 1.1, we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M248">View MathML</a> and satisfies (PS)-condition and assumptions (i) of Lemma 2.3. Now, we prove that (iii) of Lemma 2.4. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M249">View MathML</a> be a finite dimensional subspace of E. Since all norms of a finite dimensional space are equivalent, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M250">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M251">View MathML</a>

(3.25)

Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M252">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M253">View MathML</a> is a base of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M249">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M255">View MathML</a>

(3.26)

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M256">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M257">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M258">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M259">View MathML</a>

(3.27)

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M260">View MathML</a>

(3.28)

It is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M261">View MathML</a> is a norm of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M249">View MathML</a>. Hence, there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M263">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M264">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M265">View MathML</a>, by Lemma 2.2, we can choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M266">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M267">View MathML</a>

(3.29)

where δ is given in (3.24). Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M268">View MathML</a>

(3.30)

Hence, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M269">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M270">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M271">View MathML</a>

(3.31)

Then by (3.25)-(3.28), (3.30) and (3.31), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M272">View MathML</a>

(3.32)

This shows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M273">View MathML</a> and there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M274">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M275">View MathML</a>, which together with (3.29), implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M276">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M277">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M278">View MathML</a>

(3.33)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M279">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M12">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M281">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M282">View MathML</a>, it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M283">View MathML</a>. For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M201">View MathML</a>, from Lemma 2.5(i) and (3.2), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M285">View MathML</a>

(3.34)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M286">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M287">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M288">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M258">View MathML</a>, it follows that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M290">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M291">View MathML</a>

(3.35)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M292">View MathML</a>. Then, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M269">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M294">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M295">View MathML</a>, it follows from (3.27), (3.30), (3.31), (3.32) and (3.35) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M296">View MathML</a>

(3.36)

On the other hand, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M297">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M269">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M299">View MathML</a>

(3.37)

Therefore, from (3.33), (3.36) and (3.37), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M300">View MathML</a>

(3.38)

From (3.29) and (3.30), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M301">View MathML</a>

(3.39)

By (2.1), (3.15), (3.34), (3.38), (3.39) and Lemma 2.5, we have for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M269">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M303">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M304">View MathML</a>

(3.40)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M305">View MathML</a>, we deduce that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M306">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M307">View MathML</a>

It follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M308">View MathML</a>

which shows that (iii) of Lemma 2.4 holds. By Lemma 2.4, φ possesses an unbounded sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M309">View MathML</a> of critical values with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M310">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M311">View MathML</a> is such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M312">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M313">View MathML</a> . If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M314">View MathML</a> is bounded, then there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M315">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M316">View MathML</a>

(3.41)

By (3.2) and (3.41), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M317">View MathML</a>

(3.42)

From (W5), we can choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M318">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M319">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M320">View MathML</a>

which implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M321">View MathML</a>

(3.43)

Hence, by (2.1) and (3.43), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M322">View MathML</a>

which, together with (3.41), implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M323">View MathML</a>

This contradicts the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M309">View MathML</a> is unbounded, and so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M314">View MathML</a> is unbounded. The proof is complete. □

Proof of Theorem 1.4 In view of the proofs of Theorem 1.2 and Theorem 1.3, the conclusion of Theorem 1.4 holds. The proof is complete. □

4 Examples

Example 4.1 Consider the following system:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M326">View MathML</a>

(4.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M327">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M328">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M12">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M13">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M331">View MathML</a> and a satisfies (A)′. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M332">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M333">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M334">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M335">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M336">View MathML</a>. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M337">View MathML</a>

Then it is easy to check that all the conditions of Theorem 1.3 are satisfied with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M338">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M339">View MathML</a>. Hence, problem (4.1) has an unbounded sequence of homoclinic solutions.

Example 4.2 Consider the following system:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M340">View MathML</a>

(4.2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M341">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M342">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M12">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M13">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M331">View MathML</a> and a satisfies (A)′. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M346">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M347">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M348">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M349">View MathML</a>. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M350">View MathML</a>

Then it is easy to check that all the conditions of Theorem 1.4 are satisfied with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M351">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/137/mathml/M339">View MathML</a>. Hence, by Theorem 1.4, problem (4.2) has an unbounded sequence of homoclinic solutions.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors read and approved the final manuscript.

Acknowledgements

XS and QZ are supported by the Scientific Research Foundation of Guangxi Education Office (No. 201203YB093), Guangxi Natural Science Foundation (Nos. 2013GXNSFBA019004 and 2012GXNSFBA053013) and the Scientific Research Foundation of Guilin University of Technology. QMZ is supported by the NNSF of China (No. 11201138) and the Scientific Research Fund of Hunan Provincial Education Department (No. 12B034).

References

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