Abstract and Applied Analysis

Volume 2012 (2012), Article ID 703596, 15 pages

doi:10.1155/2012/703596

Research Article

Existence and Multiplicity Results of Homoclinic Solutions for the DNLS Equations with Unbounded Potentials

Defang Ma1,2 and Zhan Zhou1,2

¹School of Mathematics and Information Science, Guangzhou University, Guangdong, Guangzhou 510006, China

²Key Laboratory of Mathematics and Interdisciplinary Sciences of Guangdong Higher Education Institutes, Guangzhou University, Guangdong, Guangzhou 510006, China

Received 25 July 2012; Accepted 9 September 2012

Academic Editor: Wenming Zou

Copyright © 2012 Defang Ma and Zhan Zhou. 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.

Abstract.

A class of difference equations which include discrete nonlinear Schrödinger equations as special cases are considered. New sufficient conditions of the existence and multiplicity results of homoclinic solutions for the difference equations are obtained by making use of the mountain pass theorem and the fountain theorem, respectively. Recent results in the literature are generalized and greatly improved.

1. Introduction

Assume that is a positive integer. Consider the following difference equation in infinite dimensional lattices,

where , , is a real valued sequence, , is a Jacobi operator [1] given by

where and are real valued and bounded sequences.

We assume that for , then is a solution of (1.1), which is called the trivial solution. As usual, we say that a solution of (1.1) is homoclinic (to 0) if

where is the length of multiindex . In addition, if , then is called a nontrivial homoclinic solution. We are interested in the existence and multiplicity of the nontrivial homoclinic solutions for (1.1). This problem appears when we look for the discrete solitons of the following Discrete Nonlinear Schrödinger (DNLS) equation:

where

is the discrete Laplacian in spatial dimension. Moreover, assume that the nonlinearity is gauge invariant, that is,

Since solitons are spatially localized time-periodic solutions and decay to zero at infinity. Thus has the form

where is a real-valued sequence, and is the temporal frequency. Then (1.4) becomes

and (1.3) holds. Naturally, if we look for solitary solutions of (1.4), we just need to get the homoclinic solutions of (1.8). Obviously, (1.8) is a special case of (1.1) with , .

DNLS equation is one of the most important inherently discrete models, which models many phenomena in various areas of applications (see [2–4] and reference therein). For example, in nonlinear optics, DNLS equation appears as a model of infinite wave guide arrays. In the past decade, the existence and properties of mobile discrete solitons/breathers in DNLS equations have been considered in a number of studies [5–9].

When , , and , , and are -periodic in , the existence of homoclinic solutions for the (1.1) have been studied in [5, 6, 10] for the case where is with superlinear nonlinearity (kerr or cubic), in [9, 11–14] for the case where is with saturable nonlinearity, respectively. When , and are not periodic in , the existence of homoclinic solutions for some special case of (1.1) can be found in [7, 8, 15, 16]. Especially, in [17, 18], the authors obtained sufficient conditions for the existence of at least a pair of nontrivial homoclinic solutions for the special case of (1.1) when is unbounded by Nehari manifold method. It is worth pointing out that the so-called global Ambrosetti-Rabinowitz condition of plays a crucial role in [17, 18]. One aim of this paper is to replace the global Ambrosetti-Rabinowitz condition by a general one. The other aim of this paper is to obtain sufficient conditions for the existence of infinitely many nontrivial homoclinic solutions of (1.1). We will see that in Section 2, our results greatly improves those in [17, 18]. Our proofs of the main results are based on Mountain Pass Lemma and Fountain theorem. Our main ideas come from the papers [19–22].

This paper is organized as follows: in Section , we will first define some basic spaces. Then, we give the main results of this paper, and a comparison with the existing results is stated. Third, we establish the variational framework associated with (1.1) and transfer the problem of the existence and multiplicity of solutions in (defined in Section 2) of (1.1) into that of the existence and multiplicity of critical points of the corresponding functional. We also recall some basic results from critical point theory. Last, in Section 3, we present the proofs of our main results.

2. Preliminaries and Main Results

Let

Then the following embedding between spaces holds:

Assume the following condition on holds.

the discrete potential satisfies

Let

Since the operator is bounded and self-adjoint in the space with the norm (see [1]), and by the condition , we know that the potential is bounded below, without loss of generality, we suppose for all . Then the operator is an unbounded positive self-adjoint operator in .

Define the space

Then is a Hilbert space equipped with the norm

Since holds, we see that the spectrum is discrete and let be the smallest eigenvalue of , that is

Now, we present the following basic hypotheses in order to establish the main results in this paper:

, and there exists , such that

uniformly for .

uniformly for , where is the primitive function of , that is,

is increasing in and decreasing in , for all .

Under the above hypotheses, our results can be stated as follows.

Theorem 2.1. Assume that conditions , – hold. Then, we have the following conclusions.

(1)If , , (1.1) has no nontrivial solution in .(2)If , , (1.1) has at least one nontrivial solution in .(3)The solutions obtained in case (2) exponentially decay at infinity, that is, there exist two positive constants and such that

Theorem 2.2. Assume is odd in for each , and that conditions – hold. Then we have the following conclusions.

(1)If , , (1.1) has no nontrivial solution in .(2)If , , (1.1) has infinitely many solutions in satisfying

(3)The solutions obtained in case (2) exponentially decay at infinity, that is, (2.10) holds.

We notice that, in [17, 18], the authors consider the following DNLS equation

which is a special case of (1.1), where . They obtain the following results.

Theorem A. Assume that the DNLS (2.12) satisfies () and

() there exist two positive constants and , such that for any ,

The nonlinearity is odd and satisfies

() there are two positive constants , , and such that

() .

() there is a such that

Then we have the following conclusions.

(1)If , , (2.12) has no nontrivial solution.(2)If , , (2.12) has at least a pair of nontrivial solutions in .(3)The solutions obtained in case (2) exponentially decay at infinity, that is, (2.10) holds.

Remark 2.3. Clearly, (2.12) corresponds (1.1) if we let

Equations (2.13) and (2.14) imply (), conditions () and () imply (), and conditions (), () imply () and (). Equation (2.15) is unnecessary in Theorem 2.2. Thus, our Theorem 2.2 greatly improves Theorem A.

Remark 2.4. In (2.12), we define by

then does not satisfy (). However, if we let in (1.1), where satisfies (), then satisfies all conditions in Theorem 2.2.

Now, we will make some preparations for the proofs of our main results. Since the operator is bounded in , the following two norms are equivalent in the Hilbert space

The following theorem plays an important role in this paper, which gives a discrete version of compact embedding theorem [16–18].

Lemma 2.5. If V satisfies the condition , then for any , the embedding map from E into is compact, denote the best embedding constant .

Consider the function defined on by

Standard arguments show that the functional is well-defined functional on and (1.1) is easily recognized as the corresponding Euler-Lagrange equation for . Thus, to find nontrivial solutions of (1.1), we need only to look for nonzero critical points of .

For the derivative of we have the following formula:

Definition 2.6 (see [22, 23]). Let be a real Banach Space and . For some , we say satisfies the so-called condition if any sequence such that and , has a convergent subsequence.

Let be the open ball in with radius and center 0, and let denote its boundary. In order to obtain the existence of critical points of on , we cite some basic lemmas from [24], which will be used in the proof of Theorem 2.1. The first is the following Mountain Pass Lemma.

Lemma 2.7. Let be a real Banach Space, satisfies the condition for any , and

There exist such that .

There exist such that .

Then has a critical value .

In order to prove Theorem 2.2, we shall use the following fountain theorem [23, 25, 26]. Let be a real Banach Space with the norm and with dim for any . Set and .

Lemma 2.8. Let be even. If, for each sufficiently large , there exists such that

  .

.

J satisfies the condition for every .

Then J has an unbounded sequence of critical values.

3. Proofs of Main Results

Lemma 3.1. Suppose that , and – hold, then we have

there exists such that ;

there exists such that .

Proof. Let . According to and , it is easy to show that, there exists , such that,

for all and . This, together with the mean value theorem, leads to

By (3.2) and the Hölder inquality, it follows that

Noting that , we obtain the following estimate:

with .

It follows from that for any , there exists such that for all , we have

Notice that, from and , it is easy to get that

Let be the eigenvector of corresponding to the smallest eigenvalue , that is to say . Then, there exists , such that

Let

Taking large enough, such that for all , then, in view of (3.5)–(3.7), we have

Taking sufficiently large, for example, , we see that as . The proof is completed.

Lemma 3.2. Suppose that , , – hold. Then the functional satisfies the condition for any .

Proof. Let be a sequence of , that is,

To prove the functional satisfies the condition, first, we prove that is bounded in . In fact, if not, we may assume by contradiction that as . Set . Up to a sequence, we have

Case  1 (). By , where as , we have

Noticing that , we divide both sides of (3.13) by and get

Let , then it follows from (3.12) that

In view of , we have

Therefore,

This contradicts (3.14).

Case  2 (). We define

For any , let be large enough such that and .

By (3.2), (3.12), and , it is clear that

Thus, for large enough

where .

This implies that . Since and as  , attains its maximum at for large . Thus, .

On the other hand, from , we have that

where . In fact, for   or, we have

If , (3.21) is obvious.

By (3.10) and (3.21), we have

But (3.20) implies that

Thus, we get a contradiction. Combining the above arguments, we know that is bounded in .

Second, we show that there is a convergent subsequence of . Actually, there exists a subsequence, still denoted by the same notation, such that weakly converges to some . Applying Lemma 2.5, we see that that, for any ,

By a straightforward calculation, we have

Due to the weak convergence, it is clear that the first term   as  . It remains to show the second term in the right hand of equality (3.26) also tends to zero as .

Indeed, according to (2.2) and Hölder inequality, we have

Therefore, combining (3.25) and the boundedness of , the above inequality implies

So, from (3.26) we can conclude that in , and this means satisfies condition. The proof is completed.

Proof of Theorem 2.1. (1) By way of contradiction, assume that (1.1) has a nontrivial solution . Then is a nonzero critical point of   in  . Thus, . But

This is a contradiction, so the conclusion holds.

(2) By Lemma 3.1, the functional satisfies () and () of the mountain pass theorem. Lemma 3.2 implies that satisfies condition for any . It follows from Lemma 2.7 that has a critical value . Hence, (1.1) has at least one nontrivial solution .

(3) Assume is a nontrivial solution obtained in (2). Similar to [17], let and the operator defined by for , then (1.1) is equivalent to

Since , we have . The multiplication operator is compact in , and this implies that

Equation (2.10) follows from the standard theorem on exponential decay for the eigenfunction of Jacobi operator (see [1] for details).

This completes the proof of Theorem 2.1.

Assume that . Define by

Let for , , and , then we have

Lemma 3.3. Suppose that , – hold. Then there exists such that

(i), (ii).

Proof. (i) It follows from (3.2) that, for any

where . Since for ,

we see that as . Thus,

where . Notice that and as , so we have , as .

(ii) For any , let the dimension of be and . Assume that , then from (3.5), there exists a such that for . Thus, for ,

Taking sufficiently large, we have,

The proof is completed.

Proof of Theorem 2.2. The proofs for (1) and (3) are similar to that of (1) and (3) in Theorem 2.1, and we omit them. Now we give the proof of (2). By Lemma 3.3, the functional satisfies () and () of Lemma 2.8. Lemma 3.2 implies that satisfies condition for any . is odd implies that is even. It follows from Lemma 2.8 that has a sequence of critical points , such that . Hence, (1.1) has infinitely many high-energy solutions in . This completes the proof.

Acknowledgments

The authors are grateful to the anonymous referee for his/her valuable suggestions. This work is supported by the National Natural Science Foundation of China (no. 11171078) and the Specialized Fund for the Doctoral Program of Higher Education of China (no. 20114410110002).

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