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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 842594, 6 pages
Standing Wave Solutions for the Discrete Coupled Nonlinear Schrödinger Equations with Unbounded Potentials
1School of Mathematics and Information Science, Guangzhou University, Guangzhou, Guangdong 510006, China
2Key Laboratory of Mathematics and Interdisciplinary Sciences of Guangdong Higher Education Institutes, Guangzhou University, Guangzhou, Guangdong 510006, China
Received 18 January 2013; Accepted 25 February 2013
Academic Editor: Chuangxia Huang
Copyright © 2013 Meihua Huang 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.
We demonstrate the existence of standing wave solutions of the discrete coupled nonlinear Schrödinger equations with unbounded potentials by using the Nehari manifold approach and the compact embedding theorem. Sufficient conditions are established to show that the standing wave solutions have both of the components not identically zero.
Consider the coupled discrete Schrödinger system where , are real valued sequences, . is the discrete Laplacian operator defined as .
In this paper, we will study the standing wave solutions of (1), that is, solutions of the form where the amplitude are supposed to be real. Inserting the ansatz of the standing wave solutions (3) into (1), we obtain the following equivalent algebraic equations:
Since Bose-Einstein condensation for a mixture of different interaction atomic species with the same mass was realized in 1997 (see ), this stimulated various analytical and numerical results on the standing wave solutions of the system (2). The discrete nonlinear Schrödinger equations (DNLS) have a crucial role in the modeling of a great variety of phenomena, ranging from solid-state and condensed-matter physics to biology. During the last years, there has been a growing interest in approaches to the existence problem for standing waves. We refer to the continuation methods in [3, 4], which have been proved powerful for both theoretical considerations and numerical computations (see ), to , which exploits spatial dynamics and centre manifold reduction, and to the variational methods in [7–11], which rely on critical point techniques (linking theorems and Nehari manifold).
We noticed that most works on the existence of standing waves solutions are for single discrete nonlinear Schrödinger equation, and less is known for discrete nonlinear Schrödinger system. In the recent paper , the authors considered the standing wave solutions of the following system: which is more general than the system (1). However, they make a mistake to obtain the equivalent algebraic equations because may be different from . Hence, there are two ways to correct this mistake. The first method is to study the special standing wave solutions (3) of the system (5) with . The second method is to study the standing wave solutions (3) of the system (5) with . In this paper, we consider the second method. By the way, the proof of the main results in  is also not fully corrected.
The paper is organized as follows. In Section 2, we introduce some preliminaries and a discrete version of compact embedding theorem. Some key lemmas on the Nehari manifold are proved in Section 3. In Section 4, the main results are stated and proved.
In this section we describe the functional setting needed for the treatment of the infinite nonlinear system (4). We first introduce a compact embedding theorem.
Consider the real sequence spaces Between spaces the following elementary embedding relation holds: For the case , we need the usual Hilbert space of , endowed with the real scalar product Let us point out that the spectrum of in coincides with the interval . Obviously, we have Assume that the potential , satisfies Without loss of generality we assume that ; that is for . Let which are self-adjoint operators defined on , and
The following lemma can be found in .
Lemma 1. If , satisfy the condition (10), then for any , and are compactly embedded into and denote the best embedding constant and , respectively. Furthermore, the spectra and are discrete, respectively.
3. Some Lemmas on the Nehari Manifold
To prove the main results, we need some lemmas on the Nehari manifold.
Lemma 2. Assume that , and (10) holds. Then the Nehari manifold is nonempty in . Furthermore, for , attains a unique maximum point at .
Proof. First we show that .
From (15) and (16), we rewrite Let ; then by (19) Notice that and ; by (20), we see that for small enough and for large enough. As a consequence, there exists such that ; that is, .
Let , . Computing the derivative of , we have This shows that is a unique maximum point. The proof is completed.
Lemma 3. Assume that , and (10) holds. Then there exists such that , for all .
Proof. Since is the smallest eigenvalue of and is the smallest eigenvalue of , from the definition of the constant and , we get and . For any , we have
where and .
By (22), it is easy to see that and this implies that Moreover, we have Let ; then we get , for all . The proof is completed.
4. Main Results
Now we state our main results in this paper as follows.
In order to prove Theorem 4, we consider the following constrained minimization problem: From the standard variational method, the proof of Theorem 4 is changed into finding a solution to the minimization problem (27). Now we are ready to prove Theorem 4.
Proof. Let be given by (27). By Lemma 2, is nonempty and there exists a sequence such that
By Lemma 3, and . By virtue of (26), we have Thus, sequences and are bounded in Hilbert spaces and , respectively. Therefore, there exist subsequences of and (denoted by itself) that weakly converge to some and , respectively. By Lemma 1, we get, for any , By virtue of (15) and (16), we have First, we claim that
According to (30), it suffices to show that
In fact, Thus Hölder inequality and (30) imply the (33) holds.
Next, we show that and .
Since and are Hilbert spaces, by (32) we have which implies . Through a similar argument to the proof of Lemma 2, we know that is positive as is small enough. Therefore there exists such that which implies . Thus we have and by (32), , where Clearly, is strictly increasing on . Therefore by (27), This implies that and .
Finally, we will prove is a nontrivial solution to system (13).
Since is an energy minimizer on Nehari manifold , there exists a Lagrange multiplier such that for any . Let in (38). implies that but Thus, and for any . Take and in (41) for , where We see that . Thus, is a nontrivial solution to system (13). The proof is completed.
By Theorem 4, the system (1) has a nontrivial solution. However, it is uncertain if two components of this solution are nonzero. Therefore, we want to find solutions of the system (1) which have both of the components not identically zero. In order to achieve this goal, we consider the system (1) with ; that is, In system (43), we know that , where , is given by (11). By the definition of , in Section 2 of this paper, we obtain that . Hence, . For the sake of simplicity, we let , , and . The notations in Section 2, such as are the same.
Now, we give the second result of this paper as follows.
Proof. By Theorem 4, we know that system (43) has a nontrivial standing wave solution in .
Now we will prove that and .
Since , we know that . If one of the components , say , then . For small enough, we consider ; by a similar argument to the proof of Lemma 2, we know that there exists such that ; that is, .
By (20) and , we have , where and .
We noticed that and
For the sake of simplicity, we let If , then Thus, and (47) yields .
If , then by (45), Thus, and (48) yields .
From the above arguments, if , then .
For small enough, we have Hence, by (49), we have This is a contradiction. So, .
Similarly, if and , then . The proof is completed.
This work is supported by Program for Changjiang Scholars and Innovative Research Team in University (no. IRT1226), 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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