Open Access Research

Existence of homoclinic solutions for a class of second-order Hamiltonian systems with subquadratic growth

Dan Zhang

Author Affiliations

Department of Mathematics, Hunan University of Science and Engineering, Yongzhou, Hunan, 425100, People’s Republic of China

Boundary Value Problems 2012, 2012:132  doi:10.1186/1687-2770-2012-132


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/132


Received:6 July 2012
Accepted:25 October 2012
Published:13 November 2012

© 2012 Zhang; 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 properly constructing a functional and by using the critical point theory, we establish the existence of homoclinic solutions for a class of subquadratic second-order Hamiltonian systems. Our result generalizes and improves some existing ones. An example is given to show that our theorem applies, while the existing results are not applicable.

Keywords:
homoclinic solutions; critical point theory; Hamiltonian systems; nontrivial solution

1 Introduction

Consider the following second-order Hamiltonian system:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M1">View MathML</a>

(HS)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M3">View MathML</a> is a symmetric matrix-valued function, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M5">View MathML</a> is the gradient of W about q. As usual we say that a solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M6">View MathML</a> of (HS) is homoclinic (to 0) if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M7">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M8">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M9">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M10">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M11">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M6">View MathML</a> is called a nontrivial homoclinic solution.

By now, the existence and multiplicity of homoclinic solutions for second-order Hamiltonian systems have been extensively investigated in many papers (see, e.g., [1-17] and the references therein) via variational methods. More precisely, many authors studied the existence and multiplicity of homoclinic solutions for (HS); see [5-17]. Some of them treated the case where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M13">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M14">View MathML</a> are either independent of t or periodic in t (see, for instance, [5-7]), and a more general case is considered in the recent paper [7]. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M13">View MathML</a> is neither constant nor periodic in t, the problem of the existence of homoclinic solutions for (HS) is quite different from the one just described due to the lack of compactness of the Sobolev embedding. After the work of Rabinowitz and Tanaka [8], many results [9-17] were obtained for the case where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M13">View MathML</a> is neither constant nor periodic in t.

Recently, Zhang and Yuan [15] obtained the existence of a nontrivial homoclinic solution for (HS) by using a standard minimizing argument. In this paper, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M17">View MathML</a> denotes the standard inner product in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M18">View MathML</a>, and subsequently, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M19">View MathML</a> is the induced norm. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M20">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M21">View MathML</a>.

Theorem 1.1 (See [[15], Theorem 1.1])

Assume thatLandWsatisfy the following conditions:

(H1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M22">View MathML</a>is a symmetric matrix for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M23">View MathML</a>, and there is a continuous function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M24">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M25">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M23">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M27">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M28">View MathML</a>as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M29">View MathML</a>.

(H2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M30">View MathML</a>where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M31">View MathML</a>is a positive continuous function such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M32">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M33">View MathML</a>is a constant.

Then (HS) possesses at least one nontrivial homoclinic solution.

In [15-17], the authors considered the case where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M34">View MathML</a> is subquadratic as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M35">View MathML</a>. However, there are many functions with subquadratic growth but they do not satisfy the condition (H2) in [15] and the corresponding conditions in [16,17]. For example,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M36">View MathML</a>

(1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M37">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M38">View MathML</a> are positive continuous functions such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M39">View MathML</a>.

In this paper, our aim is to revisit (HS) and study the subquadratic case which is not included in [15-17]. Now, we state our main result.

Theorem 1.2Let the above condition (H1) hold. Moreover, assume that the following conditions hold:

(H3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M40">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M41">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M31">View MathML</a>is a positive continuous function such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M32">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M44">View MathML</a>is a constant.

(H4) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M45">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M46">View MathML</a>where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M47">View MathML</a>are positive continuous functions such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M48">View MathML</a>.

Then (HS) possesses at least one nontrivial homoclinic solution.

Remark 1.1 Obviously, the condition (H2) is a special case of (H3)-(H4). If (H2) holds, so do (H3)-(H4); however, the reverse is not true. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M34">View MathML</a> defined in (1) can satisfy the conditions (H3) and (H4), but <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M34">View MathML</a> cannot satisfy the condition (H2). So, we generalize and significantly improve Theorem 1.1 in [15].

Remark 1.2 We still consider the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M34">View MathML</a> defined in (1),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M52">View MathML</a>

Due to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M53">View MathML</a>, there are no constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M54">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M55">View MathML</a>

so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M34">View MathML</a> does not satisfy the conditions (W2) and (W3) in [16]. Moreover, for any given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M57">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M34">View MathML</a> does not satisfy the condition (W2) in [17]. Therefore, we also extend Theorem 1.2 in [16] and Theorem 1.1 in [17].

Example 1.1 Consider the following second-order Hamiltonian system with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M59">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M60">View MathML</a>

(2)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M61">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M62">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M63">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M64">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M65">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M66">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M67">View MathML</a>. Clearly, (H1), (H3), and (H4) hold. Therefore, by applying Theorem 1.2, the Hamiltonian system (2) possesses at least one nontrivial homoclinic solution.

Remark 1.3 It is easy to see that (H2) in Theorem 1.1 is not satisfied, so we cannot obtain the existence of homoclinic solutions for the Hamiltonian system (2) by Theorem 1.1. On the other hand, W does not satisfy the conditions (W2) and (W5) of [17], then we cannot obtain the existence of homoclinic solutions for the Hamiltonian system (2) by Theorem 1.1 in [17].

The remainder of this paper is organized as follows. In Section 2, some preliminary results are presented. In Section 3, we give the proof of Theorem 1.2.

2 Preliminary results

In order to establish our result via the critical point theory, we firstly describe some properties of the space on which the variational associated with (HS) is defined. Like in [15], let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M68">View MathML</a>

Then the space E is a Hilbert space with the inner product

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M69">View MathML</a>

and the corresponding norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M70">View MathML</a>. Note that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M71">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M72">View MathML</a> with the embedding being continuous. Here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M73">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M74">View MathML</a>) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M75">View MathML</a> denote the Banach spaces of functions on ℝ with values in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M18">View MathML</a> under the norms

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M77">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M78">View MathML</a>

respectively. In particular, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M79">View MathML</a>, there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M80">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M81">View MathML</a>

(3)

here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M82">View MathML</a>.

Lemma 2.1There exists a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M83">View MathML</a>such that if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M84">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M85">View MathML</a>

(4)

Proof From (H1), we can imply that there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M83">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M87">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M88">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M89">View MathML</a>. By the above inequality, one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M90">View MathML</a>

So, the lemma is proved. □

Lemma 2.2 ([[9], Lemma 1])

Suppose thatLsatisfies (H1). Then the embedding ofEin<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M91">View MathML</a>is compact.

Lemma 2.3Suppose that (H1) and (H4) are satisfied. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M92">View MathML</a> (weakly) inE, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M93">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M91">View MathML</a>.

Proof Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M92">View MathML</a> in E. Then there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M96">View MathML</a> such that, by the Banach-Steinhaus theorem and (3),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M97">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M37">View MathML</a>, by (H4) there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M99">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M100">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M101">View MathML</a>and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M88">View MathML</a>. Hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M103">View MathML</a>

On the other hand, by Lemma 2.2, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M104">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M105">View MathML</a>, passing to a subsequence if necessary, which implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M106">View MathML</a> for almost every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M88">View MathML</a> . Then using Lebesgue’s convergence theorem, the lemma is proved. □

Now, we introduce more notation and some necessary definitions. Let E be a real Banach space, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M108">View MathML</a>, which means that I is a continuously Fréchet-differentiable functional defined on E. Recall that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M108">View MathML</a> is said to satisfy the (PS) condition if any sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M110">View MathML</a>, for which <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M111">View MathML</a> is bounded and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M112">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M113">View MathML</a>, possesses a convergent subsequence in E.

Lemma 2.4 ([[18], Theorem 2.7])

LetEbe a real Banach space, and let us have<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M108">View MathML</a>satisfying the (PS) condition. IfIis bounded from below, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M115">View MathML</a>

is a critical value ofI.

3 Proof of Theorem 1.2

Now, we are going to establish the corresponding variational framework to obtain homoclinic solutions of (HS). Define the functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M116">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M117">View MathML</a>

(5)

Lemma 3.1Under the assumptions of Theorem 1.2, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M118">View MathML</a>

(6)

which yields that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M119">View MathML</a>

(7)

Moreover, Iis a continuously Fréchet-differentiable functional defined onE, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M108">View MathML</a>and any critical point ofIonEis a classical solution of (HS) with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M121">View MathML</a>.

Proof We firstly show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M108">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M84">View MathML</a>, by (3), (H4), and the Hölder inequality, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M124">View MathML</a>

(8)

Combining (5) and (8), we show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M116">View MathML</a>. Next, we prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M108">View MathML</a>. Rewrite I as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M127">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M128">View MathML</a>

It is easy to check that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M129">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M130">View MathML</a>

(9)

Thus, it is sufficient to show that this is the case for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M131">View MathML</a>. In the process we will see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M132">View MathML</a>

(10)

which is defined for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M133">View MathML</a>. For any given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M84">View MathML</a>, let us define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M135">View MathML</a> as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M136">View MathML</a>

It is obvious that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M137">View MathML</a> is linear. Now, we show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M137">View MathML</a> is bounded. Indeed, for any given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M84">View MathML</a>, by (3) and (H4), there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M140">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M141">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M88">View MathML</a>, which yields that by (4) and the Hölder inequality,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M143">View MathML</a>

(11)

Moreover, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M144">View MathML</a>, by the mean value theorem, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M145">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M146">View MathML</a>. Therefore, by Lemma 2.3 and the Hölder inequality, one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M147">View MathML</a>

(12)

as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M148">View MathML</a> in E. Combining (11) and (12), we see that (10) holds. It remains to prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M149">View MathML</a> is continuous. Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M150">View MathML</a> in E and note that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M151">View MathML</a>

By Lemma 2.3 and the Hölder inequality, we obtain that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M152">View MathML</a>

as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M150">View MathML</a>, which implies the continuity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M149">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M155">View MathML</a>.

Lastly, we check that critical points of I are classical solutions of (HS) satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M8">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M157">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M29">View MathML</a>. We know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M159">View MathML</a>, the space of continuous functions q on ℝ such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M8">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M29">View MathML</a>. Moreover, if q is one critical point of I, by (6) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M162">View MathML</a>

which yields that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M7">View MathML</a>, i.e., q is a classical solution of (HS). Since q is one critical point of I, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M164">View MathML</a>

It follows from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M8">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M29">View MathML</a> and the above equality that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M167">View MathML</a>

Hence, q satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M157">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M169">View MathML</a>. This proof is complete. □

Lemma 3.2Under the assumptions of Theorem 1.2, Isatisfies the (PS) condition.

Proof In fact, assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M170">View MathML</a> is a sequence such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M171">View MathML</a> is bounded and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M172">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M173">View MathML</a>. Then there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M174">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M175">View MathML</a>

(13)

for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M176">View MathML</a>.

We firstly prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M177">View MathML</a> is bounded in E. By (5) and (8), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M178">View MathML</a>

(14)

Combining (13) and (14), we obtain that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M179">View MathML</a>

(15)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M37">View MathML</a>, the above inequality shows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M177">View MathML</a> is bounded in E. By Lemma 2.2, the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M177">View MathML</a> has a subsequence, again denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M177">View MathML</a>, and there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M84">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M185">View MathML</a>, weakly in E,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M186">View MathML</a>, strongly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M91">View MathML</a>.

Hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M188">View MathML</a>

as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M189">View MathML</a>. Moreover, an easy computation shows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M190">View MathML</a>

So, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M191">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M113">View MathML</a>, i.e., I satisfies the Palais-Smale condition. □

Now, we can give the proof of Theorem 1.2.

Proof of Theorem 1.2 By (5) and (8), for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M193">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M194">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M195">View MathML</a>

(16)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M196">View MathML</a>, (16) implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M197">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M198">View MathML</a>. Consequently, I is a functional bounded from below. By Lemmas 3.2 and 2.4, I possesses a critical value <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M199">View MathML</a>, i.e., there is a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M84">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M201">View MathML</a>

On the other hand, take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M202">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M203">View MathML</a>, and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M204">View MathML</a> be given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M205">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M206">View MathML</a>. Then we obtain that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M207">View MathML</a>

which yields that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M208">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M209">View MathML</a> small enough since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/132/mathml/M210">View MathML</a>, i.e., the critical point obtained above is nontrivial. □

Competing interests

The authors declare that they have no competing interests.

Author’s contributions

The authors declare that the work was realized in collaboration with same responsibility. All authors read and approved the final manuscript.

Acknowledgements

This work is supported by the Research Foundation of Education Bureau of Hunan Province, China (No.11C0594). The authors would like to thank the anonymous referees very much for helpful comments and suggestions which led to the improvement of presentation and quality of the work.

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