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Guowei Sun, "On Standing Wave Solutions for Discrete Nonlinear Schrödinger Equations", Abstract and Applied Analysis, vol. 2013, Article ID 436919, 6 pages, 2013. https://doi.org/10.1155/2013/436919
On Standing Wave Solutions for Discrete Nonlinear Schrödinger Equations
The purpose of this paper is to study a class of discrete nonlinear Schrödinger equations. Under a weak superlinearity condition at infinity instead of the classical Ambrosetti-Rabinowitz condition, the existence of standing waves of the equations is obtained by using the Nehari manifold approach.
The discrete nonlinear Schrödinger (DNLS) equation was first derived in the context of nonlinear optics by Christodoulides and Joseph ; see also [2–5]. DNLS equation is one of the most important inherently discrete models, having a crucial role in the modeling of a great variety of phenomena, ranging from solid state and condensed matter physics to biology [6–10]. For example, Davydov  studied the equation in molecular biology and Su et al.  considered the equation in condensed matter physics. Eilbeck et al.  firstly pointed out the universal nature of the discrete nonlinear Schrödinger equation and reported a number of applications.
For the analytical study, many authors studied the existence results of standing wave solutions for DNLS equations. Much of the works concerns the periodic DNLS equations [12–14]. Recently, some authors considered the DNLS equations with infinitely growing potential. Zhang and Pankov [15, 16] devoted their efforts to the case of infinitely growing potential and power-like nonlinearity. In all these results, the nonlinearity is supposed to be either positive (self-focusing), or negative (defocusing). Pankov  studied the DNLS equatifvons with infinitely growing potential and sign-changing nonlinearity (a mixture of self-focusing and defocusing ones). Pankov and Zhang were concerned with the DNLS equations with infinitely growing potential and saturable nonlinearity in .
In this paper, we consider higher-dimensional generalizations of DNLS equation where and . The parameter characterizes the focusing properties of the following equation: if , the equation is self-focusing, while corresponds to the defocusing equation.
We assume that the nonlinearity is gauge invariant, that is, Then we can consider the special solutions of the form , for any . These solutions are called breather solutions or standing waves, due to their periodic time behavior. Inserting the ansatz of a breather solution into (1), it follows that satisfies the nonlinear system of algebraic equations We need the following assumptions.
The discrete potential satisfies
where is the length of multi-index .
, and there exists , such that
uniformly for .
uniformly for , where is the primitive function of , that is,
is strictly increasing on and .
We are concerned with the existence of ground state solutions, that is, solutions corresponding to the least positive critical value of the variational functional. To obtain the existence of ground states, usually besides the growth condition on the nonlinearity and a Nehari type condition, the following classical Ambrosetti-Rabinowitz superlinear condition (see, e.g., ) is assumed: It is easy to see that (8) implies that , for some constant and .
In this paper, instead of (8) we assume the super-quadratic condition . It is easy to see that (8) implies . It is well known that many nonlinearities such as do not satisfy (8). A crucial role that (8) plays is to ensure the boundedness of Palais-Smale sequences.
This paper is organized as follows. In Section 2, we establish the variational framework associated with (4). We then present the main results of this paper and compare them with the existing ones. Section 3 is devoted to prove some useful lemmas, and the proof of the main results is completed in Section 4.
In order to apply the critical point theory, we will establish the corresponding variational framework associated with (4).
For some positive integer , we consider the real sequence spaces Then the following embedding between spaces holds:
Let which is a self-adjoint operator defined on (see ).
Define the space Then is a Hilbert space equipped with the norm
Now we consider the variational functional defined on by where is the inner product in . Then . And for the derivative of , we have the following formula: Equation (16) implies that (4) is the corresponding Euler-Lagrange equation for . Thus, we have reduced the problem of finding a nontrivial solution of (4) to that of seeking a nonzero critical point of the functional on .
The following lemma plays an important role in this paper; it was established in .
Lemma 1. If satisfies the condition , then(1)for any , the embedding map from E into is compact,(2)the spectrum is discrete and consists of simple eigenvalues accumulating to .
By Lemma 1, we may assume that is the smallest eigenvalue of , that is
Now we are ready to state the main results.
Theorem 2. Suppose that conditions () and ()–() are satisfied. Then one has the following conclusions. (1)If , , (4) has no nontrivial solution.(2)If , , (4) has a nontrivial ground state solution.(3)If , , and is odd in for each , (4) has infinitely many pairs of solutions in .
Remark 3. In , the author considered the following DNLS equation:
where there exists a positive constant , such that for any , . Clearly, (18) corresponds (4) if we let . Therefore, (18) is a special case of (4).
In , the nonlinearity satisfies the following condition: which implies (8). So it is a stronger condition than . Therefore, our results generalize the corresponding ones.
Remark 4. In , the authors also considered (18) and assumed that the nonlinearity satisfies the classical Ambrosetti-Rabinowitz superlinear condition (8). Clearly, it is a stronger condition than .
Since , we may introduce an equivalent norm in by setting and then the functional can be rewritten as
To prove the multiplicity results, we need the following lemma.
Lemma 5 (see ). Let . If is a infinite-dimensional Hilbert space, is even and bounded below and satisfies the Palais-Smale condition. Then has infinitely many pairs of critical points.
3. Some Lemmas
In this section, we always assume that .
We define the Nehari manifold
To prove the main results, we need some lemmas.
Lemma 6. Suppose that conditions () and ()–() are satisfied. Then one has(1) and for all ,(2), for all .
Proof. (1) From and , it is easy to get that
By , we have
So for all .
(2) For all , by (1), we have
Lemma 7. Suppose that conditions () and ()–() are satisfied, and let Then one has the following.(1) as .(2) is strictly increasing for all and .(3) uniformly for on the weakly compact subsets of , as .
Proof. (1) and (2) are easy to be shown from and , respectively. Next, we verify (3). Let be weakly compact and let . It suffices to show that if as , then so does a subsequence of . Passing to a subsequence if necessary, and for every , as .
Since and , by and (23), we have
Lemma 8. Under the assumptions () and ()–(), for each , there exists a unique such that .
Proof. Let ,. Note that and from (2) of Lemma 7, then there exists a unique , such that whenever , whenever , and . So .
Remark 9. By (1) and (3) of Lemma 7, for small and for large. Together with Lemma 8, we have that is a unique maximum of and is the unique point on the ray () which intersects with . That is, is the unique maximum of on the ray. Therefore, we may define the mapping and by setting where .
Lemma 10. For each compact subset , there exists a constant such that for all .
Proof. Suppose that, by contradiction, as . By Lemma 6 and , we have This is a contradiction.
Lemma 11. (1) The mapping is continuous.
(2) The mapping is a homeomorphism between and , and the inverse of is given by .
Proof. (1) Suppose that . Since for each , we may assume that for all . Write . By Lemmas 8 and 10, is bounded, and hence after passing to a subsequence if needed. Since is closed and . Hence by the uniqueness of of Lemma 8. (2) This is an immediate consequence of .
Lemma 12. satisfies the Palais-Smale condition on .
Proof. Let be a sequence such that for some and as .
Firstly, we prove that is bounded. In fact, if not, we may assume by contradiction that as . Let . Then there exists a subsequence, still denoted by the same notation, such that in as .
Suppose that . For every , from Remark 9, we have This is a contradiction if . Therefore, .
According to Lemma 7(3), we have a contradiction again. Thus, is bounded.
Finally, we show that there exists a convergent subsequence of . Actually, there exists a subsequence, still denoted by the same notation, such that . By Lemma 1, for any , then Note that
The first term as because of the weak convergence.
By and , it is easy to show that for any , there exists , such that Then, Combining (32) and the boundedness of , the above inequality implies It follows from (33) that in ; that is, satisfies Palais-Smale condition.
The proof is complete.
Now we define the functional and by
Lemma 13. (1) , and
(2) , and
(3) is a Palais-Smale sequence for if and only if is a Palais-Smale sequence for .
(4) is a critical point of if and only if is a nontrivial critical point of . Moreover, the corresponding values of and coincide and .
Proof. (1) Let and . By Remark 9 and the mean value theorem, we obtain
where is small enough and . Similarly,
where . From the proof of Lemma 11, the function is continuous, combining these two inequalities that
Hence the Gâteaux derivative of is bounded linear in and continuous in . It follows that is a class of (see [19, Proposition 1.3]).
(2) follows from (1). Note only that since , .
(3) Let be a Palais-Smale sequence for , and let . Since for every we have an orthogonal splitting , using (2) we have Using (2) again, then Therefore, According to Lemma 6, for , , so there exists a constant such that . And since , . Together with Lemma 12, . Hence is a Palais-Smale sequence for if and only if is a Palais-Smale sequence for .
(4) By (45), if and only if . The other part is clear.
4. Proof of Main Results
Proof of Theorem 2. (1) If , , we suppose that (4) has a nontrivial solution . Then is a nonzero critical point of in and . But
This is a contradiction.
(2) If , . We firstly show that satisfies the Palais-Smale condition.
Let be a Palais-Smale sequence for ; then is a Palais-Smale sequence for by Lemma 13(3), where :. From Lemma 12, after passing to a subsequence and , so satisfies the Palais-Smale condition.
Let be a minimizing sequence for . By Ekeland's variational principle, we may assume that as , so is a Palais-Smale sequence for . By Palais-Smale condition, after passing to a subsequence if needed. Hence is a minimizer for and therefore a critical point of , and then is a critical point of and is also a minimizer for by Lemma 13. Therefore, is a ground state solution of (4).
(3) If , , and is odd in for each , then is even and so is . Since and satisfies the Palais-Smale condition, has infinitely many pairs of critical points by Lemma 5. It follows that (4) has infinitely many pairs of solutions in from Lemma 13.
This completes Theorem 2.
This work is supported by Program for the National Natural Science Foundation of China (no. 11071283) and Yuncheng University Science Foundation (nos. JY-2011026, JY-2011038, JY-2011039, and JC-2009024).
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Copyright © 2013 Guowei Sun. 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.