Research Article | Open Access
Existence of Standing Waves for a Generalized Davey-Stewartson System
The purpose of this paper is to investigate the existence of standing waves for a generalized Davey-Stewartson system. By reducing the system to a single Schrödinger equation problem, we are able to establish some existence results for the system by variational methods.
1. Introduction and Main Results
In this paper, we are going to consider the existence of standing waves for a generalized Davey-Stewartson system in Here is the Laplacian operator in and is the imaginary unit, , , and satisfy some additional assumptions.
The Davey-Stewartson system is a model for the evolution of weakly nonlinear packets of water waves that travel predominantly in one direction, but in which the amplitude of waves is modulated in two spatial directions. They are given as where , is the complex amplitude of the shortwave and is the real longwave amplitude . The physical parameters and play a determining role in the classification of this system. Depending on their signs, the system is elliptic-elliptic, elliptic-hyperbolic, hyperbolic-elliptic, and hyperbolic-hyperbolic , although the last case does not seem to occur in the context of water waves.
As we know, the system can be reduced to a single Schrödinger equation by using Fourier transforms. Indeed, let be the singular integral operator defined by where , , and denotes the Fourier transform: Then the generalized Davey-Stewartson system can be reduced to the following single nonlocal Schrödinger equation In this paper, we are interested in the existence of standing waves for the above equation, that is, solutions in the form of where , . Then if is a solution of (1), then we can see that must satisfy the following Schrödinger problem:
We will consider the generalized Davey-Stewartson system with perturbation. Under suitable assumptions on the coefficients , the problem can be viewed as the perturbation of the generalized Davey-Stewartson system considered in [2, 3]. Here we will not use the critical point theory or the minimizing methods to establish the existence results. Moreover, we will not use Lion's Concentration-compactness principle to overcome the difficulty of losing compactness. Instead, we will apply the perturbation method developed by Ambrosetti and Badiale in [4, 5] to show the existence of solutions of (8) and (9). In [4, 5], Ambrosetti and Badiale established an abstract theory to reduce a class of perturbation problems to a finite dimensional one by some careful observation on the unperturbed problems and the Lyapunov-Schmit reduction procedure. This method has also been successfully applied to many different problems, see  for examples. In this paper we are going to consider the following two types of perturbed problems for generalized Davey-Stewartson system. Consider
The main results of the paper are the following theorems.
Theorem 2. Let , suppose , and there exists a positive constant such that satisfy is continuous, bounded and with ; is continuous, bounded and .
Then for small enough, there exists a solution in for problem (9). Moreover, if , then as .
Theorem 3. Let , suppose and satisfy and is continuous and there exist and such that as .
Then for small enough, there exists a solution in for problem (9). Moreover, if , then as .
Throughout this paper, we denote the norm of by and by we denote the usual -norm; stand for different positive constants.
The paper is organized as follows. In Section 2, we outline the abstract critical point theory for perturbed functionals and give some properties for the singular operator . In Section 3, we prove the main results by some lemmas.
2. The Abstract Theorem
To prove the main results, we need the following known propositions.
Proposition 4. For any positive constant , consider the following problem, : There is a unique positive radial solution , which satisfies the following decay property: where is a constant. The function is a critical point of functional defined by Moreover, possesses a 3-dimensional manifold of critical points Set and denote . We have (1), (2), (3), for all .
In the following, we outline the abstract theorem of a variational method to study critical points of perturbed functionals. Let be a real Hilbert space, we will consider the perturbed functional defined on it of the form where and . We need the following hypotheses and assume that (1) and are with respect to ; (2) is continuous in and for all ; (3) and are continuous maps from and , respectively, and is the space of linear continuous operators from to . (4) There is a -dimensional manifold , , consisting of critical points of , and such a will be called a critical manifold of . (5) let denote the tangent space to at , the manifold is nondegenerate in the following sense: and is an index-0 Fredholm operator for any . (6) There exists and a continuous function such that
Consider the existence of critical points of the perturbed problem We want to look for solutions of the form with and . Then we can reduce the problem to a finite-dimensional one by Lyapunov-Schmit procedure, that is, it is equivalent to solve the following system: Here is the orthogonal projection onto . Under the conditions above, the first equation in this system can be solved by implicit function theorem, and then by using the Taylor expansion, we obtain for In [4, 5] the following abstract theorem is proved.
Lemma 5. Suppose assumptions (1)–(6) are satisfied, and there exists and such that Then for any small, there exists which is a critical point of .
We give some facts about the singular integral in Cipolatti .
Lemma 6. Let be the singular integral operator defined in Fourier variable by where , , and denotes the Fourier transform: For , satisfies the following properties: (1). (2) if , then . (3) preserves the following operations: translation: , . dilation: , . conjugation: , is the complex conjugate of .
3. Proof of the Main Results
In this section, we would apply the abstract tools of the previous section to prove the main results. First let us consider (8), the corresponding energy functional can be defined as It is easy to see that is of , and thus is a solution of (8) if and only if is a critical point of the action functional .
Proof of Theorem 1. Set
then can be rewritten as
Thus and are both with respect to . To apply Lemma 5, by Proposition 4, we need only to check that
From the fact that , for any , we have Since exponentially decays at infinity, we know the right side of the equality goes to 0, if .
Let be the quadratic functional on defined by it follows from the Parseval identity that and in particular we have Then for any , we have Since exponentially decays at infinity, the right side of the inequality (32) goes to . Thus from (29) and (32) above we soon get Then by assumption (10) that we know . Thus, the conclusion follows from Lemma 5 that any strict maximum or minimum of gives rise to a critical point of the perturbed functional and hence to a solution of (8).
We are going to consider problem (9). Set We have It can be proved that is a solution of system (9) if and only if is a critical point of the functional defined by Set Then can be rewritten as Define and for
Lemma 7. Under assumptions and , is continuous in .
Proof. From the proof of Lemma 4.1 in , we know is continuous in , and hence we only need to prove that is continuous in .
If , with . Then we can estimate that It is obvious that , as . At the same time, we know Estimating the first term , by Hölder inequality, we know Since is bounded and continuous, the operator , the dominated convergence theorem implies that Similarly, we can deduce that vanishes, as . Hence as .
If , by definition, . Since is also bounded, we know , applying Parseval identity and Hölder inequality, we get therefore , as . Hence is continuous and the lemma is proved.
Lemma 8. Under assumptions and , and are continuous in .
Proof. and are continuous in , see [7, Lemma 4.2] for the details. Here we only prove that and are continuous in .
If with , then Estimating the second term, since , by Hölder inequality, we know Thus as . Estimating the first term , we know As in Lemma 7, by Hölder inequality again, we can prove that , as , . Therefore as .
If , from the definition of , we know . Hence And we know , as . From the above arguments, we know is continuous in .
In the following we prove that is continuous in . As we know
If with , then where We estimate only, and can be estimated in a similar way, indeed Similar to the proof in Lemma 7, we know as and . Thus we know as .
If , then from the definition of , we know Using Hölder inequality, we know Therefore From the above arguments, we know is continuous in and the proof is complete.
Lemma 9. Assume and are satisfied. Define Then
Proof. By changing of variable, we know
Since is continuous and bounded, the dominated convergence theorem implies that
On the other hand, since is bounded and , then, changing of variable, we know
since . Thus we obtain
Now we are ready to prove From the proof of , we first know Also, since is bounded, it is easy to check that Moreover, recall that is bounded and use Hölder inequality, we get since , we have From the above arguments, we know and the proof is completed.
Proof of Theorem 2. By the exponential decay property of proposition , it is easy to check that is a compact perturbation of the identity map, and so it is an index-0 Fredholm operator. By Proposition 4, we know that is a nondegenerate 3-dimensional critical manifold. From Lemmas 7 to 9, we know all the assumptions of Lemma 5 are satisfied. Since has a strict (global) maximum at , has a strict (global) maximum or minimum at depending on the sign of . By the abstract theorem, we know the existence of family solutions . If , it is easy to check that as .
In the following we prove Theorem 3.
Lemma 11. Assume and are satisfied. Then , , and are continuous in .
Proof . Keeping the exponentially decay property of in mind, the continuity of , , and in can be proved similarly as in . We can also repeat the proof in Lemma 7 to know the continuity of . Thus the lemma is concluded.
Lemma 12. Assume and are satisfied. Define Then for all , we have
Proof. As we know
By assumption () and the decay property of ,
Moreover, by the boundedness of , we know
Since we obtain
To study the property of , since and exponentially decays at infinity, from the proof in , we know On the other hand, from the boundedness of and , we have Since , we get From the above arguments, we know
Proof of Theorem 3. From Lemmas 11 and 12, we know that all the assumptions of Lemma 5 are satisfied. Since and , we know that there is such that either By the abstract Theorem 2, we know the existence of family solutions . If , it is easy to check that as .
This work is supported by ZJNSF (Y7080008, R6090109, LQ12A01015, Y201016244, 2012C31025), SRPWZ (G20110004), and NSFC (10971194, 11005081, 21207103).
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