Research Article | Open Access
On the Existence of Ground State Solutions of the Periodic Discrete Coupled Nonlinear Schrödinger Lattice
We study the existence of ground state solutions of the periodic discrete coupled nonlinear Schrödinger lattice by using the Nehari manifold approach combined with periodic approximations. We show that both of the components of the ground state solutions are not zero.
In this paper, we consider the coupled discrete Schrödinger system where is a positive constant, is a real valued -periodic sequence, , and . is the discrete Laplacian operator defined as .
It is well known that coupled nonlinear Schrödinger equations arise quite naturally in nonlinear optics  and Bose-Einstein condensates. Bose-Einstein condensation for a mixture of different interaction atomic species with the same mass was realized in 1997 (see ), which stimulated various analytical and numerical results on the ground state solutions of 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 ground states. We refer to the continuation methods in [4, 5], which have been proved to be powerful for both theoretical considerations and numerical computations (see ), to , which exploits spatial dynamics and centre manifold reduction, to the variational methods in [8–15], which rely on critical point techniques (linking theorems, the Nehari manifold), and to the Krasnoselskii fixed point theorem together with a suitable compactness criterion .
The aim of this paper is to study the discrete solitons of (1), that is, solutions of the form where the amplitudes and are supposed to be real. Inserting the ansatz of the discrete solitons (3) into (1), we obtain the following equivalent algebraic equations: and (4) becomes
In fact, we consider the following more general equations: where and are the second-order difference operators defined by where , , , and are real valued -periodic sequences, and , , and are positive numbers. Obviously, (5) is a special case of (7) with , , , and .
Since, for , the operator is a bounded and self-adjoint operator in , its spectrum has a band structure; that is, is a union of a finite number of closed intervals . 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 two types of solutions to (7) as follows: (i) -periodic, that is, , , and (ii) discrete solitons. Actually, in case (ii), we look for solutions in the space ; then (6) holds naturally. System (7) has a trivial solution , . We are looking for nontrivial solutions.
The main idea in this paper is as follows. First, we consider (7) in a finite -periodic sequence space, and is not a spectrum of the corresponding operator , . By using the Nehari manifold approach, we obtain the existence of -periodic solutions. Then we show that these solutions have upper and lower bounds. Finally, by an approximation technique, we prove that the limit of these solutions exists and is the solution of (7) in . Compared with the existence of ground state solutions of the DNLS, the difficulty is that we need to show that both of the components of the ground state solutions are not zero.
The remaining of this paper is organized as follows. First, in Section 2, we establish the variational framework associated with (7) and introduce the Nehari manifolds. Then, in Section 3, we present a sufficient condition on the existence of -periodic solutions and nontrivial solutions in of (7).
2. Preliminaries and the Nehari Manifold
In this section, we first establish the variational framework associated with (7).
Let be the set of the following form:
For any fixed positive integer , we define the subspace of as Obviously, is isomorphic to and hence can be equipped with the inner product and norm as respectively. Sometimes, we will consider norm on as We also define a norm on by The symbol stands for the space with the norm and the inner product We also consider norm on as We mention that where .
Consider the functionals on and on defined by respectively. Then and its derivative is given by and and its derivative is given by
Let be the distance from to the spectrum ; that is, Let . Then, we have Furthermore, we let Then, obviously, and .
Next, we study the main properties of the Nehari manifolds with the functionals and .
Let Then and are functionals and their derivatives are given by respectively. The Nehari manifolds are defined as follows: Note that contains all critical points of in .
To prove the main results, we need some lemmas on the Nehari manifolds.
Lemma 1. Assume that and hold. Then the sets and are nonempty closed submanifolds in and , respectively. The derivatives and are nonzero on the corresponding Nehari manifolds. Moreover, there exists such that , and , .
Proof. The proofs for both cases are similar. We only provide the proof for the case of as an illustration.
First, we show that .
Let . Then by (21) Noticing that and , by (33), we see that for small enough and for large enough. As a consequence, there exists such that ; that is, .
Let . By (21), (30), and the definition of , we have Hence, , and the implicit function theorem implies that is a submonifold in .
Now let us prove the last statement of the lemma. Let . By the assumption of Lemma 1 and the definition of , we have where . Let . Then, by (24) and (35), it is easy to see that which implies that
Closedness of is obvious. The proof is completed.
Lemma 2. Assume that and hold. Then there exists such that for all .
Lemma 3. For , the function , , has a unique critical point at , which is, actually, a global maximum. The same statement holds for and .
Proof. Let , , . Computing the derivative of , we have This shows that is a unique maximum point. The proof is completed.
Lemma 4. Let be a minimizer of the functional constrained on the Nehari manifold ; that is, and then is a nontrivial -periodic solution to (7), which is called a nontrivial periodic ground state solution to (7).
Proof. According to Lagrange multiplier method, is the critical point of the functional . Thus , and for arbitrary , After taking , we obtain But Thus, and for any . Take and in (44) for , where We see that . Thus, is a nontrivial -periodic ground state solution to (7). The proof is completed.
Through a similar argument to the proof of Lemma 4, we get the following lemma.
Lemma 5. Let be a minimizer of the functional constrained on the Nehari manifold ; that is, and then is a nontrivial discrete soliton to (7), which is called a nontrivial ground state solution to (7).
3. Main Results
In this section, we will establish some sufficient conditions on the existence of -periodic solutions and nontrivial solutions in of (7).
We start with the following.
Lemma 6. Assume that and hold. Then the minimum value in (40) is attained.
Proof. Let be a minimizing sequence for ; that is, as . The fact that shows that is bounded. Since the space is finite dimensional, so the norm is equivalent to the Euclidean norm on , and the sequence is bounded. Passing to a subsequence, we can assume that converges to . Since the set is closed and the functional is continuous, we obtain that and . The proof is completed.
To obtain a nontrivial solutions in of (7), we need the following lemma.
Lemma 7. Assume that and hold. Let be a -periodic ground state solution, that is, a solution of (40). Then the sequences and are bounded above and below away from zero.
Proof. By Lemma 2, is obviously bounded below away from zero. Let
so , as in the beginning of the proof of Lemma 1; there exists such that ; that is, . Therefore, . And is bounded above.
Next, we prove that are bounded above and below away from zero. By the previous proof, we see that is bounded above and below away from zero; that is, there exist and such that . Then This implies that The proof is completed.
Lemma 8. Assume that and hold. Let be a -periodic ground state solution, that is, a solution of (40). Then there exist positive constants and such that
Proof. By Lemmas 4 and 6, we know that is a nontrivial critical point of . Therefore, we have
Let with , such that and . By the fact that
If one of the components of , say , is equal to 0, then . Thus, by (54), we obtain
By (56), we get
Similarly, if , then we have
If and , then, by (54) and (55), we obtain
By (59), we get
Let Then, by (57), (58), and (60), we get (51). The proof is completed.
Now we are ready to state our main results.
Theorem 9. Assume that and hold. Then for every positive integer , (7) possesses a nontrivial -periodic ground state solution in . Moreover, there are other three nontrivial -periodic ground state solutions , , and to (7).
Proof. Consider the sequence of -periodic solutions found in Lemma 6. By Lemma 8, without loss of generality, we can assume that the subsequence of satisfies . Thus, there exists such that
By the periodicity of the coefficients in (7), we see that is also a solution to (7). Making some shifts if necessary, without loss of generality, we can assume that in (63). Moreover, passing to a subsequence of , we can also assume that and . It follows from (37), (50), (51), and (63) that we can choose a subsequence, still denoted by and , such that and for all . Notice that . Then with . Furthermore, (7) possesses pointwise limits and hence is a nontrivial solution to (7). By the way, , , and are nontrivial solutions to (7). Since , if is a ground state solution, then , , and are the ground state solutions.
Now we prove that the solution is a ground state solution. By Lemma 5, we have to show that . Actually, we have proven that, for every sequence , passing to a subsequence still denoted by and making appropriate shifts, we can suppose that and pointwise, where is a nontrivial solution. For any positive integer , we have Let ; we obtain that and, hence, Now we prove that By Lemma 7, we know that the sequence is bounded above and below away from zero. We extract a subsequence, still denoted by , and hence we prove that Given , let be such that Choose sufficiently close to 1 such that We also have that . By density argument, we can find a finitely supported sequence sufficiently close to in such that and Then there exists such that and Let be such that and if . If is large enough, then and This implies (68). Hence, by (66) and (68), we have .
Finally, we will show that and . From the above arguments, system (7) has a nontrivial ground state solution in . If one of the components of , say , is equal to 0, then (the proof for the other case is similar). For small enough, we consider . By a similar argument to the proof of Lemma 1, we know that there exists such that ; that is . By , we have , where Moreover, . Notice that , , and For the sake of simplicity, we let If , then, by (28) and (75), This, combined with and (77) yields . If , then, by (28) and (75), This, together with and (78) gives . Thus, if , then we have . For small enough, we have Hence, by (79), we have This is a contradiction. So, . The proof is completed.
The authors would like to thank the anonymous referee for his/her valuable suggestions. 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), the Specialized Fund for the Doctoral Program of Higher Education of China (no. 20114410110002), and SRF of Guangzhou Education Bureau (no. 10A012).
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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.