Homoclinic Solutions for a Class of Second Order Nonautonomous Singular Hamiltonian Systems

Abstract

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We are concerned with the existence of homoclinic solutions for the following second order nonautonomous singular Hamiltonian systems u ¨ + a t [subscript] W u [/subscript] u = 0 , (HS) where - ∞ < t < + ∞ , u = [subscript] u 1 [/subscript] , [subscript] u 2 [/subscript] ,   ... , [subscript] u N [/subscript] ∈ [superscript] [...] N [/superscript] N ≥ 3 , a : [...] [arrow right] [...] is a continuous bounded function, and the potential W : [superscript] [...] N [/superscript] \ { ξ } [arrow right] [...] has a singularity at 0 ≠ ξ ∈ [superscript] [...] N [/superscript] , and [subscript] W u [/subscript] u is the gradient of W at u . The novelty of this paper is that, for the case that N ≥ 3 and (HS) is nonautonomous (neither periodic nor almost periodic), we show that (HS) possesses at least one nontrivial homoclinic solution. Our main hypotheses are the strong force condition of Gordon and the uniqueness of a global maximum of W . Different from the cases that (HS) is autonomous a t ≡ 1 or (HS) is periodic or almost periodic, as far as we know, this is the first result concerning the case that (HS) is nonautonomous and N ≥ 3 . Besides the usual conditions on W , we need the assumption that [superscript] a [variant prime] [/superscript] t < 0 for all t ∈ [...] to guarantee the existence of homoclinic solution. Recent results in the literature are generalized and significantly improved.

[ProQuest: [...] denotes non US-ASCII text; see PDF]

Ziheng Zhang 1 and Fang-Fang Liao 2 and Patricia J. Y. Wong 3

Academic Editor:Jifeng Chu

1, Department of Mathematics, Tianjin Polytechnic University, Tianjin 300387, China
2, Department of Mathematics, College of Science, Hohai University, Nanjing 210098, China
3, School of ELectrical and Electronic Engineering, Nanyang Technological University, 50 Nanyang Avenue, 639798, Singapore

Received 3 December 2013; Accepted 10 January 2014; 19 February 2014

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

As is known to all, the search for periodic as well as homoclinic and heteroclinic solutions of Hamiltonian systems has a long and rich history. In present paper, we particularly focus our attention on the existence of homoclinic solutions of second order nonautonomous singular Hamiltonian systems. For the results on the literature of periodic solutions for such singular systems, we refer the reader to the book [1] of Ambreosetti and Zelati.

Second order Hamiltonian systems are systems of the following form: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . Roughly speaking, they are the Euler-Lagrange equations of the functional [figure omitted; refer to PDF] where the integration is taken over a finite interval [figure omitted; refer to PDF] or all real [figure omitted; refer to PDF] and the Lagrangian has the form [figure omitted; refer to PDF] Clearly, when the potential [figure omitted; refer to PDF] is [figure omitted; refer to PDF] -periodic in [figure omitted; refer to PDF] , it is natural to look for [figure omitted; refer to PDF] -periodic solutions of [figure omitted; refer to PDF] as critical points of the functional [figure omitted; refer to PDF] over a suitable space of [figure omitted; refer to PDF] -periodic functions. Also, in such a case, one can look for homoclinic solutions at the origin as limits of [figure omitted; refer to PDF] -periodic solutions (subharmonic solutions) as [figure omitted; refer to PDF] , or alternatively, as critical points of the functional [figure omitted; refer to PDF] over a suitable space of functions on the whole space [figure omitted; refer to PDF] (typically, [figure omitted; refer to PDF] ).

For singular systems, one assumes that [figure omitted; refer to PDF] and [figure omitted; refer to PDF] for some [figure omitted; refer to PDF] . Although the study of singular systems is perhaps as old as Kepler's classical problem in mechanics, [figure omitted; refer to PDF] (and, also, the [figure omitted; refer to PDF] -body problem), the interest in such problems was renewed by the pioneering papers [2] of Gordon in 1975 and [3] of Rabinowitz in 1978. In [2], the notion of strong force is introduced to deal with singular problems, while in [3], the use of variational methods is brought into the study of periodic solutions of Hamiltonian systems.

The present paper is concerned with the existence of homoclinic solutions for the following second order nonautonomous singular Hamiltonian systems: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ), [figure omitted; refer to PDF] is a continuous bounded function, and the potential [figure omitted; refer to PDF] has a singularity at [figure omitted; refer to PDF] ; [figure omitted; refer to PDF] is the gradient of [figure omitted; refer to PDF] at [figure omitted; refer to PDF] . We recall that a homoclinic solution of [figure omitted; refer to PDF] is a solution such that [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] and [figure omitted; refer to PDF] Throughout this paper, we assume that the following hypotheses are supposed:

(A) [figure omitted; refer to PDF] is a continuous function such that [figure omitted; refer to PDF]

where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are two positive constants and [figure omitted; refer to PDF]

( [figure omitted; refer to PDF] ): [figure omitted; refer to PDF] for some [figure omitted; refer to PDF] ;

( [figure omitted; refer to PDF] ): [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] and [figure omitted; refer to PDF] if and only if [figure omitted; refer to PDF] and [figure omitted; refer to PDF] ;

( [figure omitted; refer to PDF] ): there is a constant [figure omitted; refer to PDF] such that [figure omitted; refer to PDF]

: for all [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] ;

( [figure omitted; refer to PDF] ): [figure omitted; refer to PDF] ;

( [figure omitted; refer to PDF] ): there is a neighbourhood [figure omitted; refer to PDF] of [figure omitted; refer to PDF] in [figure omitted; refer to PDF] and a function [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] as [figure omitted; refer to PDF] and [figure omitted; refer to PDF]

Now we are in the position to state our main result.

Theorem 1.

Under the conditions of (A) and ( [figure omitted; refer to PDF] )-( [figure omitted; refer to PDF] ), [figure omitted; refer to PDF] has at least one nontrivial homoclinic solution.

Remark 2.

The assumption ( [figure omitted; refer to PDF] ) is the so-called strong force condition (see Gordon [2]), which is used to verify the Palais-Smale condition for the functional corresponding to the approximating problem [figure omitted; refer to PDF] defined below. For example, ( [figure omitted; refer to PDF] ) is satisfied when [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) in a neighbourhood of [figure omitted; refer to PDF] . The assumption ( [figure omitted; refer to PDF] ) is a kind of concavity condition for [figure omitted; refer to PDF] near [figure omitted; refer to PDF] . In particular, ( [figure omitted; refer to PDF] ) holds for small [figure omitted; refer to PDF] when [figure omitted; refer to PDF] is negative definite.

In the case of autonomous singular Hamiltonian systems, the first result on existence of a homoclinic orbit using variational methods was obtained by Tanaka [4] under essentially the same assumptions as above. In [4], Tanaka used a minimax argument from Bahri and Rabinowitz [5] in order to get approximating solutions of the following boundary value problem: [figure omitted; refer to PDF] as critical points of the corresponding functional and obtained uniform estimates to show that those solutions converge weakly to a nontrivial homoclinic solution of [figure omitted; refer to PDF] . Regarding multiplicity of homoclinics, still in the autonomous singular case, early results were obtained by Caldiroli [6], who showed existence of two homoclinic orbits, and by Bessi [7], who used Lyusternick-Schnirelman category to prove the existence of [figure omitted; refer to PDF] distinct homoclinics for potentials satisfying a pinching condition. For [figure omitted; refer to PDF] , Janczewska and Maksymiuk in [8], via use of variational methods and geometrical arguments, investigated the existence of homoclinic orbits. In addition, different kinds of multiplicity results were obtained in [9, 10] (still for conservative systems) by exploiting the topology of [figure omitted; refer to PDF] , the domain of the potential, when the set [figure omitted; refer to PDF] is such that the fundamental group of [figure omitted; refer to PDF] is nontrivial.

In the case of planar autonomous systems, more extensive existence and multiplicity results were obtained. Indeed, under essentially the same conditions as above with [figure omitted; refer to PDF] , Rabinowitz showed in [11] that [figure omitted; refer to PDF] has at least a pair of homoclinic solutions by exploiting the topology of the plane and minimizing the energy functional on classes of sets with a fixed winding number around the singularity [figure omitted; refer to PDF] . The result in [11] was substantially improved in [12] where, using the same idea, the authors showed that a nondegeneracy variational condition introduced in [12] is in fact necessary and sufficient for the minimum problem to have a solution in the class of sets with winding number greater than [figure omitted; refer to PDF] and, therefore, proved a result on existence of infinitely many homoclinic solutions. For the recent results, we refer the reader to [13, 14].

On the other hand, in the case of [figure omitted; refer to PDF] -periodic time dependent Hamiltonians in [figure omitted; refer to PDF] , existence of infinitely many homoclinic orbits was obtained for smooth Hamiltonians by using a variational procedure due to Séré in [15, 16] for the first systems and in [17, 18] for second order systems. In case [figure omitted; refer to PDF] , using theses ideas, Rabinowitz [19] constructed infinitely many multibump homoclinic solutions for [figure omitted; refer to PDF] of the form [figure omitted; refer to PDF] , with [figure omitted; refer to PDF] being almost periodic and [figure omitted; refer to PDF] satisfying (A) and ( [figure omitted; refer to PDF] )-( [figure omitted; refer to PDF] ). Recently, Izydorek and Janczewska in [20] investigated the existence of at least two connecting orbits in some more general sense. For the case that [figure omitted; refer to PDF] , recently, the authors in [21], using the category theory, for the first time considered the existence of infinitely many geometrically distinct homoclinic solutions, under the assumptions that [figure omitted; refer to PDF] is periodic and [figure omitted; refer to PDF] satisfies ( [figure omitted; refer to PDF] )-( [figure omitted; refer to PDF] ) and the following condition on [figure omitted; refer to PDF] at infinity:

( [figure omitted; refer to PDF] ): there exists [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] and [figure omitted; refer to PDF] for [figure omitted; refer to PDF] large.

Here we must point out that all the results mentioned above are obtained for the case that [figure omitted; refer to PDF] is autonomous or periodic or almost periodic. Motivated mainly by the works of [4, 21], in present paper, we focus our attention on the case that [figure omitted; refer to PDF] and [figure omitted; refer to PDF] is non-autonomous (neither periodic nor almost periodic). To the best of our knowledge, this is the first result on the existence of homoclinic solutions for second order nonautonomous singular Hamiltonian systems in [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ). The proof of Theorem 1 will be demonstrated in the following sections. To this end, we employ the technique used in [4]. Explicitly, considering the approximating problem: [figure omitted; refer to PDF] solutions of [figure omitted; refer to PDF] are obtained as critical points of the functional [figure omitted; refer to PDF] defined in Section 2. We show the existence of critical points of [figure omitted; refer to PDF] via a minimax argument, which is essentially due to Bahri and Rabinowitz [5]. Furthermore, we also get some estimates, which are uniform with respect to [figure omitted; refer to PDF] , for minimax values and corresponding critical points [figure omitted; refer to PDF] . These uniform estimates permit us to let [figure omitted; refer to PDF] ; for a suitable sequence [figure omitted; refer to PDF] and a subsequence [figure omitted; refer to PDF] , we see that [figure omitted; refer to PDF] converges weakly to a homoclinic solution of [figure omitted; refer to PDF] as [figure omitted; refer to PDF] .

The remaining part of this paper is organized as follows. In Section 2, via a minimax argument, for any [figure omitted; refer to PDF] , we show that [figure omitted; refer to PDF] has at least one nontrivial solution [figure omitted; refer to PDF] . In Section 3, some uniform estimates for solutions [figure omitted; refer to PDF] are obtained to investigate homoclinic solution of [figure omitted; refer to PDF] . In Section 4, we are devoted to accomplishing the proof of Theorem 1.

2. Approximating Problem [figure omitted; refer to PDF]

In this section, we investigate the approximating problem [figure omitted; refer to PDF] via a minimax argument. Denote by [figure omitted; refer to PDF] the usual Sobolev space on [figure omitted; refer to PDF] with values in [figure omitted; refer to PDF] under the norm [figure omitted; refer to PDF] Let [figure omitted; refer to PDF] Clearly, [figure omitted; refer to PDF] is an open subset of [figure omitted; refer to PDF] . Define the functional [figure omitted; refer to PDF] as follows: [figure omitted; refer to PDF] Under the assumptions of Theorem 1, as usual, one can show that [figure omitted; refer to PDF] and [figure omitted; refer to PDF] Then there is an one-to-one correspondence between critical points of [figure omitted; refer to PDF] and classical solutions of [figure omitted; refer to PDF] .

In order to obtain a critical point of [figure omitted; refer to PDF] , we use a minimax argument. To do so, [figure omitted; refer to PDF] must satisfy the Palais-Smale condition ((PS) condition) on [figure omitted; refer to PDF] ; that is, for any sequence [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] is bounded and [figure omitted; refer to PDF] as [figure omitted; refer to PDF] , [figure omitted; refer to PDF] possesses a subsequence converging to some [figure omitted; refer to PDF] .

Proposition 3.

If [figure omitted; refer to PDF] and [figure omitted; refer to PDF] satisfy (A), ( [figure omitted; refer to PDF] ), ( [figure omitted; refer to PDF] ), ( [figure omitted; refer to PDF] ), and ( [figure omitted; refer to PDF] ), then [figure omitted; refer to PDF] satisfies the (PS) condition.

Proof.

Let [figure omitted; refer to PDF] be a sequence such that [figure omitted; refer to PDF] is bounded and [figure omitted; refer to PDF] as [figure omitted; refer to PDF] . Then, by ( [figure omitted; refer to PDF] ) and the definition of [figure omitted; refer to PDF] , [figure omitted; refer to PDF] is bounded in [figure omitted; refer to PDF] . Hence, we can extract a subsequence of [figure omitted; refer to PDF] , still denoted by [figure omitted; refer to PDF] , such that [figure omitted; refer to PDF] converges to [figure omitted; refer to PDF] (the closure of [figure omitted; refer to PDF] in [figure omitted; refer to PDF] ) weakly in [figure omitted; refer to PDF] . On the other hand, by Lemma 2.1 in [22], if [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] That is, [figure omitted; refer to PDF] . Hence, [figure omitted; refer to PDF] . Subsequently, in view of ( [figure omitted; refer to PDF] ) and the Sobolev compact embedding theorem, it follows that [figure omitted; refer to PDF] strongly in [figure omitted; refer to PDF] .

Since [figure omitted; refer to PDF] satisfies the (PS) condition, we have the following deformation theorem (see [23]).

Lemma 4.

Suppose that [figure omitted; refer to PDF] is not a critical value of [figure omitted; refer to PDF] . Then, for all [figure omitted; refer to PDF] , there are an [figure omitted; refer to PDF] and [figure omitted; refer to PDF] such that

(1) [figure omitted; refer to PDF] if [figure omitted; refer to PDF] ;

(2) [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] ;

(3) [figure omitted; refer to PDF] ,

where [figure omitted; refer to PDF] denotes the level set defined by [figure omitted; refer to PDF]

Now, one introduces a minimax procedure for [figure omitted; refer to PDF] . Let [figure omitted; refer to PDF] For [figure omitted; refer to PDF] , one observes that [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] Since [figure omitted; refer to PDF] , one can consider for each [figure omitted; refer to PDF] a map [figure omitted; refer to PDF] defined by [figure omitted; refer to PDF] One denotes by [figure omitted; refer to PDF] the Brouwer degree of a map [figure omitted; refer to PDF] . Let [figure omitted; refer to PDF] It is obvious that [figure omitted; refer to PDF] . Define a minimax value of [figure omitted; refer to PDF] by [figure omitted; refer to PDF] Then one has the following.

Proposition 5.

[figure omitted; refer to PDF] is a critical value of [figure omitted; refer to PDF] .

Proof.

[figure omitted; refer to PDF] will be seen later in Proposition 6. Here we assume it and prove that [figure omitted; refer to PDF] is a critical value of [figure omitted; refer to PDF] . On the contrary, we suppose that [figure omitted; refer to PDF] is not a critical value. Taking [figure omitted; refer to PDF] and [figure omitted; refer to PDF] in Lemma 4, we have a deformation flow [figure omitted; refer to PDF] with the properties (1)-(3). Moreover, we can verify [figure omitted; refer to PDF] In fact, since [figure omitted; refer to PDF] (see (1)), we have [figure omitted; refer to PDF] for [figure omitted; refer to PDF] . On the other hand, due to (2), we have [figure omitted; refer to PDF] Hence, [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . Thus we have [figure omitted; refer to PDF] Therefore, [figure omitted; refer to PDF] for [figure omitted; refer to PDF] ; that is, (22) holds.

Choose [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] and consider [figure omitted; refer to PDF] . Then, in virtue of (3), we obtain [figure omitted; refer to PDF] which contradicts the definition of [figure omitted; refer to PDF] . Therefore, [figure omitted; refer to PDF] is a critical value of [figure omitted; refer to PDF] .

Proposition 6.

There is a constant [figure omitted; refer to PDF] independent of [figure omitted; refer to PDF] such that [figure omitted; refer to PDF]

Proof.

For any given [figure omitted; refer to PDF] , we define [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) by [figure omitted; refer to PDF] Then, we can easily see the following:

(1) [figure omitted; refer to PDF] , that is, [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] ;

(2) [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] and [figure omitted; refer to PDF] .

Therefore, we get [figure omitted; refer to PDF] In what follows, we prove the existence of a constant [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] . For any given [figure omitted; refer to PDF] , we have [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is defined in ( [figure omitted; refer to PDF] ). Otherwise, we can easily observe that [figure omitted; refer to PDF] . Hence, there is [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] Since [figure omitted; refer to PDF] , there is an [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] By the Schwarz inequality, we have for [figure omitted; refer to PDF] [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Thus, we have [figure omitted; refer to PDF] which yields that [figure omitted; refer to PDF] Combining with (28) and (35), we obtain the desired conclusion.

In view of Propositions 5 and 6, we deduce the following proposition concerned with the uniform boundness of critical value of [figure omitted; refer to PDF] .

Proposition 7.

For [figure omitted; refer to PDF] , [figure omitted; refer to PDF] possesses a solution [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are independent of [figure omitted; refer to PDF] .

3. Estimates for Solutions of [figure omitted; refer to PDF]

In this section, we give some estimates on the solutions [figure omitted; refer to PDF] of [figure omitted; refer to PDF] to allow [figure omitted; refer to PDF] go to [figure omitted; refer to PDF] in Section 4. Firstly, from the definition of [figure omitted; refer to PDF] and Proposition 7, we have the following.

Lemma 8.

There is a constant [figure omitted; refer to PDF] which is independent of [figure omitted; refer to PDF] such that [figure omitted; refer to PDF]

In what follows, we denote by [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , various constants which are independent of [figure omitted; refer to PDF] .

Proposition 9.

Consider [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] .

Proof.

Suppose that [figure omitted; refer to PDF] for some [figure omitted; refer to PDF] . On account of [figure omitted; refer to PDF] , we can find an interval [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] and [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] , which, combining with Lemma 8, implies that [figure omitted; refer to PDF] On the other hand, we have [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is defined in (33). Combining with (38) and (39), we deduce that [figure omitted; refer to PDF] which yields that [figure omitted; refer to PDF] As a result, we conclude that [figure omitted; refer to PDF]

Lemma 10.

Define the function [figure omitted; refer to PDF] as follows: [figure omitted; refer to PDF] Then, [figure omitted; refer to PDF] is nondecreasing on [figure omitted; refer to PDF] . Furthermore, one has [figure omitted; refer to PDF]

Proof.

In view of (A) and ( [figure omitted; refer to PDF] ), it deduces that [figure omitted; refer to PDF] which implies that [figure omitted; refer to PDF] is nondecreasing on [figure omitted; refer to PDF] . Thus, we have [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] . Integrating (43) over [figure omitted; refer to PDF] and by Lemma 8, we have [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] . We observe the fact that [figure omitted; refer to PDF] . Otherwise, [figure omitted; refer to PDF] and then we have [figure omitted; refer to PDF] by the uniquemess of the solution of the initial value problem: [figure omitted; refer to PDF] which contradicts the fact that [figure omitted; refer to PDF] . Consequently, combining with [figure omitted; refer to PDF] and in view of (46), we have [figure omitted; refer to PDF]

It only remains to show that [figure omitted; refer to PDF] as [figure omitted; refer to PDF] . On the contrary, due to the facts that [figure omitted; refer to PDF] is increasing on [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] it occurs that [figure omitted; refer to PDF] However, this contradicts the fact that [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] .

The following proposition gives us an [figure omitted; refer to PDF] -bound from below on [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] . Here, it must be pointed that the condition ( [figure omitted; refer to PDF] ) is used only in this proposition.

Proposition 11.

Consider [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] .

Proof.

Using (43), we get [figure omitted; refer to PDF] Due to the fact that [figure omitted; refer to PDF] (the reason has been explained in Lemma 10) and the condition ( [figure omitted; refer to PDF] ), we obtain that [figure omitted; refer to PDF] Suppose that [figure omitted; refer to PDF] takes its maximum at [figure omitted; refer to PDF] . From the above inequality, we deduce that [figure omitted; refer to PDF] . Thus, we have [figure omitted; refer to PDF]

By Proposition 11, we can find two numbers [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] . For [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , we need the following property.

Lemma 12.

Consider [figure omitted; refer to PDF] , [figure omitted; refer to PDF] as [figure omitted; refer to PDF] .

Proof.

Let [figure omitted; refer to PDF] be a solution of the following initial value problem: [figure omitted; refer to PDF] By the continuous dependence of [figure omitted; refer to PDF] on the initial data, for any [figure omitted; refer to PDF] , there is an [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] By (44), for any [figure omitted; refer to PDF] , we can find [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] That is, [figure omitted; refer to PDF] for [figure omitted; refer to PDF] . Therefore, we have [figure omitted; refer to PDF] as [figure omitted; refer to PDF] . Similarly, we can obtain [figure omitted; refer to PDF] as [figure omitted; refer to PDF] .

4. Proof of Theorem 1

In this section, we construct a homoclinic solution of [figure omitted; refer to PDF] as a limit of the solutions [figure omitted; refer to PDF] of [figure omitted; refer to PDF] when [figure omitted; refer to PDF] goes to [figure omitted; refer to PDF] and complete the proof of Theorem 1. In order to accomplish such a process, for each [figure omitted; refer to PDF] , we define [figure omitted; refer to PDF] by [figure omitted; refer to PDF] Then it directly follows from Lemma 8 and Proposition 9 that

(i) [figure omitted; refer to PDF] is a solution of [figure omitted; refer to PDF] in [figure omitted; refer to PDF] ;

(ii) [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] ;

(iii): [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] are uniformly bounded for [figure omitted; refer to PDF] .

By (iii), we can extract a subsequence [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] converges to some [figure omitted; refer to PDF] with [figure omitted; refer to PDF] in the following sense: [figure omitted; refer to PDF] Moreover, we have [figure omitted; refer to PDF] Due to the strong force condition ( [figure omitted; refer to PDF] ), similar to Lemma 2.1 in [22], we can also obtain that [figure omitted; refer to PDF]

In what follows, we focus our attentions to show that [figure omitted; refer to PDF] is exactly right the homiclinic solution of [figure omitted; refer to PDF] that we need.

Proposition 13.

[figure omitted; refer to PDF] is a nontrivial solution of [figure omitted; refer to PDF] on [figure omitted; refer to PDF] .

Proof.

In view of (61), it is sufficient to verify that for any [figure omitted; refer to PDF] [figure omitted; refer to PDF] By Lemma 12, we can choose [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] . By property (i) of [figure omitted; refer to PDF] , we have for all [figure omitted; refer to PDF] [figure omitted; refer to PDF] On account of (58) and (59), we get (62) as [figure omitted; refer to PDF] . On the other hand, as a direct consequence of (58) and the property (ii) of [figure omitted; refer to PDF] , we have [figure omitted; refer to PDF] ; that is, [figure omitted; refer to PDF] is nontrivial.

As the last step of the proof of Theorem 1, we show that [figure omitted; refer to PDF] and [figure omitted; refer to PDF] satisfy the following property.

Proposition 14.

Consider [figure omitted; refer to PDF] , [figure omitted; refer to PDF] as [figure omitted; refer to PDF] .

Proof.

Here, we just check the case that [figure omitted; refer to PDF] , since it is the same as [figure omitted; refer to PDF] . First, we prove [figure omitted; refer to PDF] as [figure omitted; refer to PDF] . On the contrary, we assume that [figure omitted; refer to PDF] as [figure omitted; refer to PDF] . Then, for some sequence [figure omitted; refer to PDF] and for some [figure omitted; refer to PDF] , we have [figure omitted; refer to PDF] On the other hand, in view of ( [figure omitted; refer to PDF] ) and (60), it follows that [figure omitted; refer to PDF] Hence, there is a sequence [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] . Thus, [figure omitted; refer to PDF] must intersect [figure omitted; refer to PDF] and [figure omitted; refer to PDF] infinitely as [figure omitted; refer to PDF] . However, this contradicts [figure omitted; refer to PDF] and (60). In fact, suppose that [figure omitted; refer to PDF] is an interval such that [figure omitted; refer to PDF] Then, we obtain [figure omitted; refer to PDF] where [figure omitted; refer to PDF] . On account of Schwarz inequality, it follows that [figure omitted; refer to PDF] If [figure omitted; refer to PDF] intersects [figure omitted; refer to PDF] and [figure omitted; refer to PDF] infinitely as [figure omitted; refer to PDF] , we can find infinitely many disjoint intervals [figure omitted; refer to PDF] with the property (66). Thus, we have [figure omitted; refer to PDF] which contradicts [figure omitted; refer to PDF] and (60). Therefore, we obtain [figure omitted; refer to PDF] as [figure omitted; refer to PDF] .

Since [figure omitted; refer to PDF] satisfies [figure omitted; refer to PDF] , [figure omitted; refer to PDF] is bounded on each compact interval by (iii) and (61). Thus, [figure omitted; refer to PDF] converges to [figure omitted; refer to PDF] in [figure omitted; refer to PDF] . Hence, [figure omitted; refer to PDF] Since [figure omitted; refer to PDF] is increasing, [figure omitted; refer to PDF] as [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , it is obvious that [figure omitted; refer to PDF] as [figure omitted; refer to PDF] .

Up to now, we are in the position to give the proof of our main result.

Proof.

In view of Propositions 13 and 14, it is obvious that [figure omitted; refer to PDF] is one nontrivial homoclinic solution of [figure omitted; refer to PDF] .

Acknowledgment

The project is supported by the National Natural Science Foundation of China (Grant no. 11101304).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

References

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[3] P. H. Rabinowitz, "Periodic solutions of Hamiltonian systems," Communications on Pure and Applied Mathematics , vol. 31, no. 2, pp. 157-184, 1978.

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