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Existence of nontrivial weak homoclinic orbits for second-order impulsive differential equations

Hui Fang* and Hongbo Duan

Author Affiliations

Department of Mathematics, Kunming University of Science and Technology, Kunming, Yunnan, 650500, China

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Boundary Value Problems 2012, 2012:138  doi:10.1186/1687-2770-2012-138


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/138


Received:25 July 2012
Accepted:7 November 2012
Published:26 November 2012

© 2012 Fang and Duan; 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

A sufficient condition is obtained for the existence of nontrivial weak homoclinic orbits of second-order impulsive differential equations by employing the mountain pass theorem, a weak convergence argument and a weak version of Lieb’s lemma.

1 Introduction

Fečkan [1], Battelli and Fečkan [2] studied the existence of homoclinic solutions for impulsive differential equations by using perturbation methods. Tang et al.[3-6] studied the existence of homoclinic solutions for Hamiltonian systems via variational methods. In recent years, many researchers have paid much attention to multiplicity and existence of solutions of impulsive differential equations via variational methods (for example, see [7-12]). However, few papers have been published on the existence of homoclinic solutions for second-order impulsive differential equations via variational methods.

In this paper, we consider the following impulsive differential equations:

(1.1)

(1.2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M3">View MathML</a> is of class <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M5">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M6">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M7">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M8">View MathML</a>. ℤ denotes the set of all integers, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M9">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M10">View MathML</a>) are impulsive points. Moreover, there exist a positive integer p and a positive constant T such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M11">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M12">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M13">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M14">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M15">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M16">View MathML</a> represent the right and left limits of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M17">View MathML</a> at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M18">View MathML</a> respectively.

We say that a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M19">View MathML</a> is a weak homoclinic orbit of Eqs. (1.1) and (1.2) if q satisfies (1.1) and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M20">View MathML</a>

Motivated by the works of Nieto and Regan [7], Smets and Willem [13], in this paper we study the existence of nontrivial weak homoclinic orbits of (1.1)-(1.2) by using the mountain pass theorem, a weak version of Lieb’s lemma and a weak convergence argument. Our method is different from those of [8,9].

The main result is the following.

Theorem 1.1Assume that Eqs. (1.1) and (1.2) satisfy the following conditions:

(H1) There exists a positive numberTsuch that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M21">View MathML</a>

(H2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M22">View MathML</a>uniformly for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M23">View MathML</a>;

(H3) There exists a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M24">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M25">View MathML</a>

(H4) There exist constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M26">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M27">View MathML</a>such that

(H5) There exists a constantb, with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M29">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M30">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M31">View MathML</a>

Then there exists a nontrivial weak homoclinic orbit of Eqs. (1.1) and (1.2).

Remark 1.1 (H2) implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M32">View MathML</a> is an equilibrium of (1.1)-(1.2).

Remark 1.2 Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M33">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M34">View MathML</a>. It is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M35">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M36">View MathML</a> satisfy (H1)-(H5).

2 Proof of main results

Lemma 2.1 (Mountain pass lemma [14])

LetEbe a Banach space and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M37">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M38">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M39">View MathML</a>be such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M40">View MathML</a>and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M41">View MathML</a>

Let

Then, for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M43">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M44">View MathML</a>, there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M45">View MathML</a>such that

(V1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M46">View MathML</a>;

(V2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M47">View MathML</a>;

(V3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M48">View MathML</a>.

In what follows, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M49">View MathML</a> denotes the space of sequences whose second powers are summable on ℤ (the set of all integers), that is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M50">View MathML</a>

The space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M49">View MathML</a> is equipped with the following norm:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M52">View MathML</a>

We now prove some technical lemmas.

Lemma 2.2The space

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M53">View MathML</a>

(2.1)

is a Hilbert space with the inner product

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M54">View MathML</a>

(2.2)

and the corresponding norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M55">View MathML</a>

(2.3)

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M56">View MathML</a> be a Cauchy sequence in H, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M57">View MathML</a> is a Cauchy sequence in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M58">View MathML</a> and there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M59">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M57">View MathML</a> converges to y in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M58">View MathML</a>. Define the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M19">View MathML</a> as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M63">View MathML</a>

It is easy to see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M64">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M65">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M66">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M67">View MathML</a>

which implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M68">View MathML</a>, that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M69">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M66">View MathML</a>. Therefore, q is continuous. Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M71">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M72">View MathML</a>.

Noticing that, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M73">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M74">View MathML</a>

which implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M75">View MathML</a>. On the other hand, since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M76">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M77">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M14">View MathML</a>), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M79">View MathML</a>

Therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M80">View MathML</a>

Consequently, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M81">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M56">View MathML</a> converges to q in H. The proof is complete. □

Lemma 2.3For any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M81">View MathML</a>, the following inequalities hold:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M84">View MathML</a>

Furthermore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M85">View MathML</a>and

Proof For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M23">View MathML</a>, there exists an integer k such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M88">View MathML</a>. Then it follows from Cauchy-Schwarz inequality that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M89">View MathML</a>

which implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M90">View MathML</a>.

Furthermore, from the above argument, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M91">View MathML</a>

that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M92">View MathML</a>.

Since

Finally, we obtain that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M94">View MathML</a>

The proof is complete. □

Define the functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M95">View MathML</a> as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M96">View MathML</a>

(2.4)

Lemma 2.4If (H1)-(H5) hold, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M97">View MathML</a>and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M98">View MathML</a>

(2.5)

Proof From the continuity of V, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M99">View MathML</a> and (H2)-(H3), we see that, for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M100">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M101">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M102">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M103">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M104">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M105">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M106">View MathML</a>

Therefore, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M107">View MathML</a>

It follows from (H5) that, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M108">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M109">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M110">View MathML</a>

(2.6)

Thus, φ and the right hand of (2.5) is well defined on H. By the definition of Fréchet derivative, it is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M97">View MathML</a> and (2.5) holds. □

Lemma 2.5If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M81">View MathML</a>is a critical point of the functionalφ, thenqsatisfies (1.1).

Proof If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M81">View MathML</a> is a critical point of the functional φ, then for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M114">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M115">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M116">View MathML</a>, take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M114">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M118">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M119">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M120">View MathML</a>. Therefore, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M121">View MathML</a>

by the definition of the weak derivative, which implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M122">View MathML</a>

(2.7)

Hence, the critical point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M81">View MathML</a> of the functional φ satisfies (1.1). The proof is complete. □

Lemma 2.6Under the assumptions (H1)-(H5), there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M124">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M39">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M126">View MathML</a>and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M127">View MathML</a>

Proof If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M81">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M129">View MathML</a>, then, by Lemma 2.3, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M130">View MathML</a>. Hence, by (H5) and Lemma 2.3, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M131">View MathML</a>

(2.8)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M132">View MathML</a>

(2.9)

It follows from (2.8), (H4) and Lemma 2.3 that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M133">View MathML</a>

Therefore, as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M24">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M135">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M39">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M137">View MathML</a>.

Now, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M138">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M139">View MathML</a>. Then there exists a subset <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M140">View MathML</a> of ℝ and λ large enough such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M141">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M142">View MathML</a>, by (2.4), (H4) and Lemma 2.3, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M143">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M24">View MathML</a>, the right-hand member is negative of λ sufficiently large, and there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M145">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M126">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M147">View MathML</a>. The proof is complete. □

Lemma 2.7Under the assumptions (H1)-(H5), there exists a bounded sequence<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M56">View MathML</a>inHsuch that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M149">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M150">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M151">View MathML</a>. Furthermore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M152">View MathML</a>does not converge to 0 in measure.

Proof All we have to prove is that any sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M56">View MathML</a> obtained by taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M154">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M155">View MathML</a> in Lemma 2.1 is bounded and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M152">View MathML</a> does not converge to 0 in measure. For n sufficiently large, it follows from (H3), (H5), (2.4), (2.5), (2.8) and (2.9) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M157">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M158">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M56">View MathML</a> is bounded in H.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M160">View MathML</a>. By (H2) and (H3), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M161">View MathML</a>

which implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M162">View MathML</a>

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M43">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M44">View MathML</a> such that, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M165">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M166">View MathML</a>

Therefore, by Lemma 2.3, we have

(2.10)

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M152">View MathML</a> converges to 0 in measure on R, then it follows from (H5) and (2.10) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M169">View MathML</a>

a contradiction. The proof is complete. □

The following lemma is similar to a weak version of Lieb’s lemma [15], which will play an important role in the proof of Theorem 1.1.

Lemma 2.8If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M170">View MathML</a>is bounded inHand<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M171">View MathML</a>does not converge to 0 in measure, then there exist a sequence<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M172">View MathML</a>and a subsequence<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M173">View MathML</a>of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M174">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M175">View MathML</a>

Proof If

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M176">View MathML</a>

then, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M43">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M178">View MathML</a> such that, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M179">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M180">View MathML</a>

Therefore, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M23">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M179">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M183">View MathML</a>

which implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M184">View MathML</a>

a contradiction. Therefore, there exist a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M185">View MathML</a> and a subsequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M186">View MathML</a> of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M187">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M188">View MathML</a>

where ℕ denotes the set of all positive integers. So, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M189">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M190">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M191">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M192">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M23">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M170">View MathML</a> is bounded in H, by Lemma 2.3, it is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M195">View MathML</a> is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M196">View MathML</a>. Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M197">View MathML</a> has a subsequence which weakly converges to u in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M196">View MathML</a>. Without loss of generality, we assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M199">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M196">View MathML</a>. Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M201">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M202">View MathML</a>. Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M203">View MathML</a> uniformly converges to u in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M204">View MathML</a>. Noticing that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M205">View MathML</a>

we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M206">View MathML</a>

that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M207">View MathML</a>. □

Proof of Theorem 1.1 By Lemma 2.7, there exists a bounded <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M208">View MathML</a> in H such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M209">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M56">View MathML</a> does not converge to 0 in measure on ℝ, where d is the mountain pass value. By Lemma 2.8, there exists a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M211">View MathML</a> in ℤ such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M212">View MathML</a>

For any fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M189">View MathML</a>, set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M214">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M215">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M216">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M10">View MathML</a>) are impulsive points and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M218">View MathML</a>

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M114">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M220">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M221">View MathML</a>

Hence, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M222">View MathML</a>

which implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M223">View MathML</a>

(2.11)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M224">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M225">View MathML</a> in H, therefore

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M226">View MathML</a>

(2.12)

As <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M227">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M196">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M229">View MathML</a> is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M196">View MathML</a> and hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M231">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M232">View MathML</a> and all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M189">View MathML</a>. Also, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M234">View MathML</a> uniformly converges to ω on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M235">View MathML</a> and, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M99">View MathML</a> being uniformly continuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M237">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M238">View MathML</a> uniformly converges to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M239">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M237">View MathML</a>. By the Lebesgue dominated convergence theorem, this implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M241">View MathML</a>

(2.13)

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M242">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M43">View MathML</a>, take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M244">View MathML</a> sufficiently large such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M245">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M227">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M196">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M227">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M249">View MathML</a>, therefore <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M250">View MathML</a> uniformly converges to ω in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M251">View MathML</a>. By the continuity of I, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M252">View MathML</a> such that, when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M253">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M254">View MathML</a>

Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M255">View MathML</a>

it follows from Lemma 2.3 that

Similarly, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M257">View MathML</a>

By the Cauchy-Schwarz inequality, we have

Therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M259">View MathML</a>

(2.14)

From (2.11)-(2.14), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M260">View MathML</a>

Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/138/mathml/M261">View MathML</a> and ω is a nontrivial weak homoclinic orbit of (1.1)-(1.2). □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the manuscript and read and approved the final manuscript.

Acknowledgements

This research is supported by the National Natural Science Foundation of China (Grant No. 10971085). The authors would like to thank the anonymous reviewers for their valuable comments and suggestions to improve the manuscript.

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