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Homoclinic Solutions of a Class of Nonperiodic Discrete Nonlinear Systems in Infinite Higher Dimensional Lattices
By using critical point theory, we obtain a new sufficient condition on the existence of homoclinic solutions of a class of nonperiodic discrete nonlinear systems in infinite lattices. The classical Ambrosetti-Rabinowitz superlinear condition is improved by a general superlinear one. Some results in the literature are improved.
Assume that is a positive integer. Consider the following difference equation in infinite higher dimensional lattices: where is continuous in , , , is a positive real valued sequence, , and is a Jacobi operator  given by here, and are real valued bounded sequences.
Assume that for ; then is a solution of (1), which is called the trivial solution. As usual, we say that , a solution of (1), is homoclinic (to 0) if where is the length of multi-index . In addition, if , then is called a nontrivial homoclinic solution. We are interested in the existence of the nontrivial homoclinic solutions for (1). This problem appears when we seek the discrete solitons of nonperiodic discrete nonlinear Schrödinger (DNLS) equation where and is the discrete Laplacian in spatial dimension. Typical representatives of power nonlinearities are
Primarily, we are interested in spatially localized, or solitary, standing waves. Such waves are often called breathers or gap solitons. The origin of the last name is that typically such solutions do exist for frequencies in gaps of linear spectrum. Considering (4), we suppose that the nonlinearity is gauge invariant; that is, and, in addition, for for .
When has the form of (6), the homoclinic solutions of (9) were obtained by Karachalios in  by assuming that , , for some . We note that implies that . Moreover, (6) satisfies the classical Ambrosetti-Rabinowitz superlinear condition , and is nondecreasing with respect to , both of which played important roles in the existence of homoclinic solutions of in .
The aim of this paper is to improve both the monotone condition of and the classical Ambrosetti-Rabinowitz superlinear condition by general ones; see Remarks 8 and 9 for details. Moreover, in this paper, we only need . Particularly, our results improved the results in ; see Remarks 3 and 7 for details.
In the past years, there has been large growth in the study of DNLS equation, which is a nonlinear lattice system that appears in many areas of physics. Discrete solitons which exist in DNLS systems, that is, solitary waves and localized structures in spatially discrete media, are also of particular interest in their own right. Among these, one can mention photorefractive media , biomolecular chains , and Bose-Einstein condensates . The experimental observations of discrete solitons in nonlinear lattice systems have been reported [7–11]. To mention that, many authors have studied the existence of discrete solitons of the DNLS equations [12–17]. The fruitful methods include centre manifold reduction , variational methods [12, 14], the principle of anticontinuity [13, 17], and the Nehari manifold approach . However, most of the existing literature is devoted to the DNLS equations with constant coefficients or periodic coefficients. Results on such DNLS equations have been summarized in [19–23]. And we also want to mention that, in recent years, the existence of homoclinic solutions for difference equations has been studied by many authors, and we refer to [24–36].
Since the operator is bounded and self-adjoint in the space (defined in Section 2), we consider (1) as a nonlinear equation in with (3) being satisfied automatically. The spectrum of is closed. Thus, the complement consists of a finite number of open intervals called spectral gaps and two of them are semi-infinite which are denoted by and , respectively. In this paper, we consider the homoclinic solutions of (1) in for the case where and . The case where and is omitted, since, in this case, we can replace by .
The main idea in this paper is as follows. First, we assume that converges to zero at infinity; that is, . After that, we prove a compact inclusion between ordinary sequence spaces and weighted sequence spaces (defined in Section 2), in order to come over lack of compactness for the so-called condition (defined in Section 2). Finally, by making use of the Mountain Pass Lemma , we prove the existence of homoclinic solutions of (1) in .
In this section, we first establish the variational setting associated with (1). Let Then the following embedding between spaces holds: For , we get the usual Hilbert space of square-summable sequences, with the real scalar product
For a positive real valued bounded sequence , we define the weighted sequence spaces : It is not hard to see that is a Hilbert space, with the scalar product
For a certain class of weight , we have the following lemmas, which will play a crucial role in our analysis.
Lemma 1. Let be a multiplication operator from to defined by . If , then the operator is compact.
Proof of Lemma 1. Let We only need to prove that is precompact in . By assumption, for any , there exists such that for any . Define a cutting sequence by Denote by the anticutting sequence. Then for any For arbitrary and finite-dimensional and bounded, we know that is precompact. The proof is complete.
Lemma 2. One assumes positive sequence of real numbers with . Then with compact inclusion.
On the Hilbert space , we consider the functional where is the primitive function of . Then and
Equation (20) implies that (1) is the corresponding Euler-Lagrange equation for . Therefore, we have reduced the problem of finding a nontrivial homoclinic solution of (1) to that of seeking a nonzero critical point of the functional on .
Let be the distance from to the spectrum ; that is, Then, we have We also consider a norm in defined by Since norm (23) is an equivalent norm with the usual one of .
In order to obtain the existence of critical points of on , we cite some basic notations and some known results from critical point theory.
Let be a Hilbert space and denote the set of functionals that are Fréchet differentiable and their Fréchet derivatives are continuous on .
Let . A sequence is called a sequence for if for some and as . We say satisfies the condition if any sequence for possesses a convergent subsequence.
Let be the open ball in with radius and center 0, and let denote its boundary. The following lemma is taken from .
Lemma 4 (Mountain Pass Lemma). If and satisfies the following conditions: there exist and such that , then there exists a sequence for the mountain pass level which is defined by where
3. Main Results
In this section, we will establish some sufficient conditions on the existence of nontrivial solutions of (1) in .
Theorem 5. Assume that , , and the following conditions hold. is continuous in , as uniformly for .There exist , such that uniformly for and .There exists some such that , for , , and , where . for , and , uniformly for .Positive real valued sequence with .Then (1) has at least a nontrivial solution in and the solution decays exponentially at infinity. That is, there exist two positive constants and such that
Proposition 6. Assume that , , and when for all . Then (1) has no nontrivial solutions in .
Remark 7. It is easy to see that the function defined by where and , , for some , satisfies all conditions in Theorem 5. This case was studied by , and we find that Theorem 5 improves Theorem 2.3 in .
Remark 8. We will introduce another condition : is nondecreasing with respect to . We want to point out that condition is equivalent to when and gives “better monotony” when , since implies (see ). Moreover, we can find that satisfies but not for some .
Remark 9. As we know, the condition is often called Ambrosetti-Rabinowitz superlinear condition . Clearly, implies . Actually, is more general than . Let be a positive sequence, and . Then satisfies . However, does not satisfy .
The proof of Theorem 5 is based on a direct application of the following lemmas. The key points read as follows.
Lemma 10. Assume that the conditions of Theorem 5 hold; then one has the following. There exist two constants and such that .There exists an such that as .
Proof of Lemma 10. Let and . By and , there exists , such that
for all and , and (31) implies that
By (32) and the Hölder inequality, we have
Since , we have
Let be the eigenvector of corresponding to the eigenvalue ; that is to say, . There exists , such that Let By , for any , there exists such that Taking large enough, such that for all , then, combining (35), (36), and (37), we have where . Letting be large enough, such that , we obtain that as . The proof is complete.
Lemma 11. Assume that the conditions of Theorem 5 hold; then the functional satisfies the condition for any given .
Proof of Lemma 11. Let be a sequence of ; that is,
First, we prove that is bounded in . By way of contradiction, assume that as . Set . Up to a sequence, we have
Case 1 (). By , where as , we have Let . Obviously, is nonempty. Then, for some , it follows from (41) that Combining () and , we have However, as . This contradicts (42).
Case 2 (). Let For any given , let be large enough such that and . Combining (32), (41), and , it is easy to see that Thus, for large enough, we have
By (47), (48), and , we have Noting that and , as , then when is big enough. Thus, . In view of (), it follows that By (50), we have This contradicts (49), so is bounded in .
Second, we show that there exists a convergent subsequence of . In fact, there exists a subsequence, still denoted by the same notation, such that By Lemma 2, we have By direct calculation, we obtain Therefore, combining (11), (24), (52), (53), (54), and the boundedness of , it is clear that and this means satisfies condition. The proof is complete.
Now, we are ready to prove Theorem 5.
Proof of Theorem 5. Let , , and be obtained in Lemma 10.
Since as , there exists a real number such that Immediately, we obtain Now that we have verified all assumptions of Lemma 4, we know possesses a sequence for the mountain pass level with where
By Lemma 11, has a convergent subsequence such that as for some bounded . Since , we have as . By the uniqueness of limit and the fact that is bounded, we obtain that is a nontrivial critical point of as the corresponding critical value . Hence, (1) has at least one nontrivial solution in .
Finally, we show that satisfies (28). In fact, similar to , for , let then where Clearly, . Thus, the multiplication by is a compact operator in , which implies that where stands for the essential spectrum. Equation (62) means that is an eigenfunction that corresponds to the eigenvalue of finite multiplicity of the operator . Equation (28) follows from the standard theorem on exponential decay for such eigenfunctions . Now the proof of Theorem 5 is complete.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work is supported by the Program for Changjiang Scholars and Innovative Research Team in University (no. IRT1226), the National Natural Science Foundation of China (no. 11171078), the Specialized Fund for the Doctoral Program of Higher Education of China (no. 20114410110002), and the Project for High Level Talents of Guangdong Higher Education Institutes.
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Copyright © 2014 Genghong Lin 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.