Homoclinic Solutions for a Class of Nonlinear Difference Equations

Abstract

We prove the existence of homoclinic solutions of a class of nonlinear difference equations with superlinear nonlinearity by using the generalized Nehari manifold approach. For the case where the nonlinearity is odd, we obtain infinitely many homoclinic solutions of the equations. Recent results in the literature are generalized and improved.

1. Introduction

In this paper, we consider the following difference equation: where , , and is the discrete Laplacian in spatial dimension.

Assume that satisfies the following condition: where is the length of multi-index . Without loss of generality, we assume that for all .

Assume further that ; then is a solution of (1), which is called the trivial solution. As usual, we call that a solution of (1) is homoclinic (to 0) if In addition, we are interested in the existence of nontrivial homoclinic solution for (1), that is, solutions that are not equal to identically. This problem appears when we look for the discrete solutions of the discrete nonlinear Schrödinger (DNLS) equation in dimensional lattices: where the nonlinearity is gauge invariant; that is,

The parameter characterizes the focusing properties of the equation: if , the equation is self-focusing, while corresponds to the defocusing equation.

Due to the definition of solutions, we know that has the form where is a real-valued sequence and is the temporal frequency. Then (5) becomes (1) and (4) holds. Therefore, the problem on the existence of solitons of (5) has been reduced to that on the existence of solutions of the boundary value problem (1)–(4).

DNLS equation is one of the most important inherently discrete models. It plays a crucial role in the modeling of a great variety of phenomena, ranging from solid state and condensed matter physics to biology (see [1–3] and reference therein). As we know, Davydov [4] studied the discrete nonlinear Schrödinger equation in molecular biology and Su et al. [5] considered the equation in condensed matter physics.

The existence of discrete solutions for DNLS equations has been studied by many authors. When the potential is unbounded, some existence results were obtained by using various methods. For example, the authors obtained the existence of discrete solutions for DNLS equations by Nehari manifold method in [6–8] and by the mountain pass theorem and fountain theorem in [9], respectively. In [10], Zhang and Pankov obtained the existence of infinitely many nontrivial solutions for DNLS equations by the linking theorem. When the potential is periodic, the existence of solutions for the periodic DNLS equations with superlinear nonlinearity [11–15] and with saturable nonlinearity [16–20] has been studied, respectively.

As it is well known, the Ambrosetti-Rabinowitz condition plays a crucial role in proving the boundedness of the Palais-Smale sequence [6, 7]. In this paper, we assume that the nonlinearity satisfies more general superlinear conditions than the classical Ambrosetti-Rabinowitz superlinearity condition [6–8, 15, 21], and we investigate the existence and multiplicity results of homoclinic solutions for the case by the generalized Nehari manifold approach. One aim of this paper is to find ground state homoclinic solutions, that is, nontrivial homoclinic solutions corresponding to the least positive critical value of the variational functional. The other aim of this paper is to obtain sufficient conditions for the existence of infinitely many pairs of homoclinic solutions of (1).

This paper is organized as follows. The assumptions on the nonlinearity and the main results are summarized in Section 2. We mention that our results improve the corresponding results in [6, 7, 9, 10]. The proofs of the main theorems are completed in Section 3.

2. Preliminaries and Main Results

Assume that the following conditions hold.(f₁), and there exist , such that (f₂) as uniformly for .(f₃) uniformly for , where is the primitive function of , that is, (f₄) is strictly increasing on and for all .Let Then the following embedding between spaces holds: Let be positive self-adjoint operator defined on .

Define the space Then is a Hilbert space equipped with the norm

Consider the functional defined on by The hypotheses on imply that the functional . Then the derivative of has the following formula: Equation (16) implies that (1) is easily recognized as the corresponding Euler-Lagrange equation for . Thus, to find nontrivial solutions of (1), we need only to look for nonzero critical points of .

The following lemma plays an important role in this paper; it was established in [6].

Lemma 1. If satisfies the condition , then(1)for any , the embedding map from into is compact;(2)the spectrum is discrete and consists of simple eigenvalues accumulating to .

Let , where , , and correspond to the positive, zero, and negative part of the spectrum of in , respectively. More precisely, by Lemma 1, we can assume that are all eigenvalues of and a corresponding orthogonal (in ) set of eigenfunctions by . Suppose , where . Then We also admit the cases and which, respectively, correspond to and . For , we take .

For any , letting with , , we can define an equivalent norm on by respectively. So can be rewritten as where .

We define for the subspace and the convex subset of , where, as usual, and . Let the generalized Nehari manifold

Now we are ready to state the main results.

Theorem 2. Suppose that conditions ,   (f₁)–(f₄) are satisfied. Then one has the following conclusions.(1)If , , (1) has no nontrivial solution.(2)If , , (1) has a nontrivial ground state homoclinic solution.

Theorem 3. Suppose thatconditions ,  (f₁)–(f₄) are satisfied; let and if is odd in for each . Then (1) has infinitely many pairs of homoclinic solutions in satisfying

Remark 4. In [9], the authors considered (1); they obtained the existence of nontrivial solutions for the case . In our paper, we consider more general case . Thus, our results extend their corresponding ones.

Remark 5. In [6, 7], the authors considered the following DNLS equation: which is a special case of (1). They obtained the existence of solutions for the case .
They additionally assumed that and satisfies the following condition: there is a such that which implies This is the classical Ambrosetti-Rabinowitz superlinear condition. It is easy to see that (27) implies that , for some constant and , so it is a stronger condition than (f₃).
In [6], Zhang obtained a minimizer of the corresponding functional on the Nehari manifold . It is crucial to require that is of class . However, in our paper, we do not assume that is of class , so the generalized Nehari manifold may not be a smooth manifold and it is not clear that the minimizer on is a critical point of . Our assumptions do not require this smoothness condition. Therefore, our results extend those of [6].
In [10], the authors considered (25) for the case ; they also assumed that and satisfies (27). We define by then satisfies all conditions in Theorems 2 and 3, but does not satisfy (27). Therefore, our results improve and extend their corresponding ones.

We recall some basic results from critical point theory. The following lemma plays an important role in the proof of multiplicity results. Let .

Lemma 6 (see [22]). 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. Proofs of Main Results

Throughout this paper, we always assume that and (f₁)–(f₄) are satisfied. In this section, we consider . To continue the discussion, we need the following technical lemma.

Lemma 7 (see [23]). Let be numbers with and . Then

Lemma 8. If , then Hence is the unique global maximum of .

Proof. Let ; then we rewrite by Since , we have Together with Lemma 7, we know that The proof is complete.

Lemma 9. For each , the set consists of precisely one point which is the unique global maximum of .

Proof. By Lemma 8, it suffices to show that . Since , we may assume that . To end this proof, we should show two key conclusions.
Firstly, we claim that there exists such that where .
In fact, by (f₁) and (f₂), it is easy to show that, for any , there exists , such that Since is equivalent to the norm on and for with , for any and , we have which implies for some (small enough).
The first inequality is a consequence of Lemma 8, since for every there is such that .
Secondly, we claim that there exists such that where , for fixed with .
Indeed, suppose by contradiction that there exists such that for all and as , where and . Set with and ; then Note that, from (f₂) and (f₄), it is easy to get that So we have Passing to a subsequence if necessary, we assume that , , , , and for every . Hence . We distinguish two cases to finish the proof of (37).
Case 1. If , then there exists such that , as . Then by (f₃), we have which contradicts (38).
Case 2. If , then and therefore Hence because is finite dimensional space. Consequently, and which contradicts (38). Hence, (37) holds.
By (34), we have for small . Together with (37), we have
Finally, we show that is weakly upper semicontinuous on .
Let in . Then as , for all after passing to a subsequence if needed. Hence . Then that is, is weakly lower semicontinuous. From the weak lower semicontinuous of the norm, it is easy to see that is weakly upper semicontinuous on .
From above, we have for some . By (37), is a critical point of . Hence . Consequently, .
This completes the proof.

Lemma 10. Suppose that conditions ,  (f₁)–(f₄) are satisfied. Then satisfies the Palais-Smale condition on .

Proof. Suppose is 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 .
If , since for some , as , it follows again from (f₃) and Fatou's lemma that this is a contradiction. Hence .
Note that by (39), we have hence . If , then and Hence because is finite dimensional space. Consequently, , a contradiction again. Therefore ; thus there exists such that for all after passing to a subsequence.
Since , . Applying Lemma 1, we see that, for any , By (35), for any , which implies that  as .
Since for all , Lemma 8 implies that as . This is a contradiction if . Therefore, 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 . Applying Lemma 1, we see that, for any , Note that Due to the weak convergence, it is clear that the first term as . It remains to show that the second term in the right hand of equality (53) also tends to be zero as .
Indeed, according to (35) and Hölder inequality, we have Therefore, combining (52) and the boundedness of , the above inequality implies It follows from (53) that in, and this means that satisfies Palais-Smale condition. The proof is complete.

Proof of Theorem 2. (1) If and , we suppose that (1) has a nontrivial solution . Then is a nonzero critical point of in and . But This is a contradiction.
(2) Let and . Suppose that conditions ,  (f₁)–(f₄) are satisfied; then may not be of class of ; nevertheless, is still a topological manifold, naturally homeomorphic to the unit sphere in . So, we may define a homeomorphism between and , where . We distinguish five steps to end this proof of Theorem 2.
Step 1. We define a homeomorphism between and.
According to Lemma 9, for each , we may define mapping where is the unique point of . Then mapping is continuous.
In fact, let be a sequence with . Since , without loss of generality, we may assume that for all . Then , where . By (37) there exists such that It follows from Lemma 10 that is bounded. Passing to a subsequence if needed, we may assume that where by (47). Let . Moreover, by Lemma 9, Therefore, using the weak lower semicontinuity of the norm and , we get Hence all inequalities above must be equalities and it follows that and . By Lemma 9, and hence .
Next, we define mapping and the inverse of is given by . It is easy to show that is a homeomorphism between and from above.
Step 2. We consider the functional and defined by
Then and
Moreover, and
In fact, we put , so we have . Let . Choose such that for and put . We may write with . From above, we know that the function is continuous. Then . By Lemma 9 and the mean value theorem, we have with some . Similarly, with some . Combining these inequalities and the continuity of function , we have Hence the Gâteaux derivative of is bounded linear in and continuous in . It follows that is of class (see [21]).
Note only that since , we have , so (65) holds.
Step 3. We will show that is a Palais-Smale sequence for if and only if is a Palais-Smale sequence for .
Indeed, let be a Palais-Smale sequence for , and let . Since for every we have an orthogonal splitting ; by Step 2, we have because for all and is orthogonal to . Then Therefore According to (47) and Lemma 10, . Hence is a Palais-Smale sequence for if and only if is a Palais-Smale sequence for .
Step 4. By (71), if and only if . Obviously, we have is a critical point of if and only if is a nontrivial critical point of . Moreover, the corresponding values of and coincide and .
Step 5. We claim that satisfies the Palais-Smale condition.
Let be a Palais-Smale sequence for ; it follows from Step 3 that is a Palais-Smale sequence for , where . By Lemma 10, we have after passing to a subsequence and ; this implies that has a convergent subsequence. Therefore, satisfies the Palais-Smale condition.
Finally, let be a minimizing sequence for . By Ekeland's variational principle we may assume as ; then is a Palais-Smale sequence for . By Palais-Smale condition, has a convergent subsequence, still denoted by such that . Hence is a minimizer for and therefore a critical point of ; then is a critical point of and also is a minimizer for . That is, is a ground state homoclinic solution of (1).
This completes Theorem 2.

Proof of Theorem 3. If is odd in for each , then is even, so is . Since by (34), is bounded from below. Note that satisfies the Palais-Smale condition in proof of Theorem 2. This combining with Lemma 6, has infinitely many pairs of critical points. Consequently, (1) has infinitely many pairs of solutions in .
Let where denotes the usual Kranoselskii genus ([22]). Since , for all . Now standard arguments using the deformation lemma imply that all are critical values of and ; that is, as .
This completes Theorem 3.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

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 the Project for High Level Talents of Guangdong Higher Education Institutes.