Abstract and Applied Analysis

Volume 2013 (2013), Article ID 848690, 9 pages

http://dx.doi.org/10.1155/2013/848690

Research Article

Existence of Standing Waves for a Generalized Davey-Stewartson System

Xiaoxiao Hu,¹ Xiao-xun Zhou,² Wu Tunhua,¹ and Min-Bo Yang³

¹School of Information and Engineering, Wenzhou Medical College, Wenzhou, Zhejiang 325035, China

²School of Marxism, Tongji University, Shanghai 200092, China

³Department of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang 321004, China

Received 13 September 2012; Revised 26 December 2012; Accepted 14 January 2013

Academic Editor: Norimichi Hirano

Copyright © 2013 Xiaoxiao Hu et al. 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.

Abstract.

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 [1]. 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 [2], 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 [6] 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 1. Assume that , and . Take the function from Proposition 4 in Section 2, if there holds

then for any small, there exists at least one solution of problem (8).

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 [2].

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 [7], 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 [7], 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 .

Remark 10. The hypothesis is used to apply Lemma 5 and has been already used in [4, 7]. If is identically zero, we can not conclude that there exist critical points of .

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 [7]. 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 [7], 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 .

Acknowledgments

This work is supported by ZJNSF (Y7080008, R6090109, LQ12A01015, Y201016244, 2012C31025), SRPWZ (G20110004), and NSFC (10971194, 11005081, 21207103).

References

A. Davey and K. Stewartson, “On three-dimensional packets of surface waves,” Proceedings of the Royal Society of London A, vol. 338, pp. 101–110, 1974.

R. Cipolatti, “On the existence of standing waves for a Davey-Stewartson system,” Communications in Partial Differential Equations, vol. 17, no. 5-6, pp. 967–988, 1992.

R. Cipolatti, “On the instability of ground states for a Davey-Stewartson system,” Annales de l'Institut Henri Poincaré, Physique Théorique, vol. 58, no. 1, pp. 85–104, 1993.

A. Ambrosetti and M. Badiale, “Homoclinics: Poincaré-Melnikov type results via a variational approach,” Annales de l'Institut Henri Poincaré: Analyse Non Linéaire, vol. 15, no. 2, pp. 233–252, 1998.

A. Ambrosetti and M. Badiale, “Variational perturbative methods and bifurcation of bound states from the essential spectrum,” Proceedings of the Royal Society of Edinburgh A, vol. 128, no. 6, pp. 1131–1161, 1998.

A. Ambrosetti and A. Malchiodi, Perturbation Methods and Semilinear Elliptic Problems on Rn, vol. 240 of Progress in Mathematics, Birkhäuser, Basel, Switzerland, 2006.

M. Badiale and A. Pomponio, “Bifurcation results for semilinear elliptic problems in ,” Proceedings of the Royal Society of Edinburgh A, vol. 134, no. 1, pp. 11–32, 2004.