Existence of Standing Waves for a Generalized Davey-Stewartson System

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.

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Xiaoxiao Hu 1 and Xiao-xun Zhou 2 and Wu Tunhua 1 and Min-Bo Yang 3

Recommended by Norimichi Hirano

1, School of Information and Engineering, Wenzhou Medical College, Wenzhou, Zhejiang 325035, China
2, School of Marxism, Tongji University, Shanghai 200092, China
3, Department of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang 321004, China

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

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.

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 ^{... 3} [figure omitted; refer to PDF] Here Δ is the Laplacian operator in ^{... 3} and i is the imaginary unit, a (x ) , b (x ) , and p 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 [figure omitted; refer to PDF] where a , _{b 1} , _{b 2} ∈ ... , ψ (t ,x ,y ) is the complex amplitude of the shortwave and [straight phi] (t ,x ,y ) is the real longwave amplitude [ 1]. The physical parameters δ and m 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 _{E 1} be the singular integral operator defined by [figure omitted; refer to PDF] where _{σ 1} ( ξ ) = ^{ξ 1 2} / | ξ ^{| 2} , ξ ∈ ^{... 3} , and ... denotes the Fourier transform: [figure omitted; refer to PDF] Then the generalized Davey-Stewartson system can be reduced to the following single nonlocal Schrödinger equation [figure omitted; refer to PDF] In this paper, we are interested in the existence of standing waves for the above equation, that is, solutions in the form of [figure omitted; refer to PDF] where ω >0 , [varphi] ,v ∈ ^{H 1} ( ^{... 3} ) . Then if ( ψ , [straight phi] ) is a solution of ( 1), then we can see that [varphi] must satisfy the following Schrödinger problem: [figure omitted; refer to PDF]

We will consider the generalized Davey-Stewartson system with perturbation. Under suitable assumptions on the coefficients a (x ) , b (x ) , 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 [figure omitted; refer to PDF]

The main results of the paper are the following theorems.

Theorem 1.

Assume that 2 <p <6 , a (x ) ∈ ^{L 6 / ( 6 -p )} ( ^{... 3} ) and b (x ) ∈ ^{L 6} ( ^{... 3} ) . Take the function U from Proposition 4in Section 2, if there holds [figure omitted; refer to PDF] then for any [straight epsilon] small, there exists at least one solution of problem ( 8).

Theorem 2.

Let ω = ^{[straight epsilon] 2} , suppose 2 <p <4 , and there exists a positive constant A such that a (x ) , b (x ) satisfy

( _{a 1} ) : a (x ) -A is continuous, bounded and a (x ) -A ∈ ^{L 1} ( ^{... 3} ) with ∫ ... (a (x ) -A ) ...0;0 ;

( _{b 1} ) : b (x ) is continuous, bounded and b ∈ ^{L 2} ( ^{... 3} ) .

Then for [straight epsilon] >0 small enough, there exists a solution _{[varphi] [straight epsilon]} in ^{H 1} ( ^{... 3} ) for problem ( 9). Moreover, if 2 <p <2 +4 /3 , then _{[varphi] [straight epsilon]} [arrow right]0 as [straight epsilon] [arrow right]0 .

Theorem 3.

Let ω = ^{[straight epsilon] 2} , suppose 2 <p <4 and a (x ) , b (x ) satisfy ( _{b 1} ) and

( _{a 2} ) : a -A is continuous and there exist L ...0;0 and 0 < γ <3 such that ^{| x | γ} (a (x ) -A ) [arrow right]L as |x | [arrow right] ∞ .

Then for [straight epsilon] >0 small enough, there exists a solution _{[varphi] [straight epsilon]} in ^{H 1} ( ^{... 3} ) for problem ( 9). Moreover, if 2 <p <2 +4 /3 , then _{[varphi] [straight epsilon]} [arrow right]0 as [straight epsilon] [arrow right]0 .

Throughout this paper, we denote the norm of ^{H 1} ( ^{... 3} ) by [figure omitted; refer to PDF] and by _{| · | s} we denote the usual ^{L s} -norm; C , _{C i} 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 _{E 1} . 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 A , consider the following problem, 2 <p < ^{2 *} : [figure omitted; refer to PDF] There is a unique positive radial solution U , which satisfies the following decay property: [figure omitted; refer to PDF] where C >0 is a constant. The function U is a critical point of ^{C 2} functional _{I 0} : ^{H 1} ( ^{... 3} ) [arrow right] ... defined by [figure omitted; refer to PDF] Moreover, _{I 0} possesses a 3-dimensional manifold of critical points [figure omitted; refer to PDF] Set [figure omitted; refer to PDF] and denote X =span { ∂U / ∂ _{x i} ,1 ...4;i ...4;3 } . We have

(1) Q (U ) = ( 2 -p ) A _{∫ ^{... 3}} ... ^{U p} dx <0 ,

(2) KerQ =X ,

(3) Q (w ) ...5;C ^{|| w || 2} , for all w ∈ ^{( ...U [ecedil]5;X ) [perpendicular]} .

In the following, we outline the abstract theorem of a variational method to study critical points of perturbed functionals. Let E be a real Hilbert space, we will consider the perturbed functional defined on it of the form [figure omitted; refer to PDF] where _{I 0} :E [arrow right] ... and G : ... ×E [arrow right] ... . We need the following hypotheses and assume that

(1) _{I 0} and G are ^{C 2} with respect to u ;

(2) G is continuous in ( [straight epsilon] ,u ) and G (0 ,u ) =0 for all u ;

(3) ^{G [variant prime]} ( [straight epsilon] ,u ) and ^{G [variant prime][variant prime]} ( [straight epsilon] ,u ) are continuous maps from ... ×E [arrow right]E and L (E ,E ) , respectively, and L (E ,E ) is the space of linear continuous operators from E to E .

(4) There is a d -dimensional ^{C 2} manifold Z , d ...5;1 , consisting of critical points of _{I 0} , and such a Z will be called a critical manifold of _{I 0} .

(5) let _{T θ} Z denote the tangent space to Z at _{z θ} , the manifold Z is nondegenerate in the following sense:

: Ker ( ^{I 0 [variant prime][variant prime]} (z ) ) = _{T θ} Z and ^{I 0 [variant prime][variant prime]} ( _{z θ} ) is an index-0 Fredholm operator for any _{z θ} ∈Z .

(6) There exists α >0 and a continuous function Γ :Z [arrow right] ... such that [figure omitted; refer to PDF]

Consider the existence of critical points of the perturbed problem [figure omitted; refer to PDF] We want to look for solutions of the form u =z +w with z ∈Z and w ∈W = ( _{T θ} Z ^{) [perpendicular]} . 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: [figure omitted; refer to PDF] Here P is the orthogonal projection onto W . 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 u =z +w ( [straight epsilon] ,z ) [figure omitted; refer to PDF] In [ 4, 5] the following abstract theorem is proved.

Lemma 5.

Suppose assumptions (1)-(6) are satisfied, and there exists δ >0 and ^{z *} ∈Z such that [figure omitted; refer to PDF] Then for any [straight epsilon] small, there exists _{u [straight epsilon]} which is a critical point of _{I [straight epsilon]} .

We give some facts about the singular integral _{E 1} in Cipolatti [ 2].

Lemma 6.

Let _{E 1} be the singular integral operator defined in Fourier variable by [figure omitted; refer to PDF] where _{σ 1} ( ξ ) = ^{ξ 1 2} / ^{| ξ | 2} , ξ ∈ ^{... 3} , and ... denotes the Fourier transform: [figure omitted; refer to PDF] For 1 <p < ∞ , _{E 1} satisfies the following properties:

(1) _{E 1} ∈ [Lagrangian (script capital L)] ( ^{L p} , ^{L p} ) .

(2) if ψ ∈ ^{H 1} ( ^{... 3} ) , then _{E 1} ( ψ ) ∈ ^{H 1} ( ^{... 3} ) .

(3) _{E 1} preserves the following operations:

: translation: _{E 1} ( ψ ( · +y ) ) (x ) = _{E 1} ( ψ ) (x +y ) , y ∈ ^{... 3} .

: dilation: _{E 1} ( ψ ( λ · ) ) (x ) = _{E 1} ( ψ ) ( λx ) , λ >0 .

: conjugation: _{E 1} ( ψ ) ¯ = _{E 1} ( ψ ¯ ) , ψ ¯ is the complex conjugate of ψ .

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 _{I [straight epsilon]} : ^{H 1} ( ^{... 3} ) [arrow right] ... can be defined as [figure omitted; refer to PDF] It is easy to see that _{I [straight epsilon]} : ^{H 1} ( ^{... 3} ) [arrow right] ... is of ^{C 2} , and thus [varphi] is a solution of ( 8) if and only if [varphi] is a critical point of the action functional _{I [straight epsilon]} ( [varphi] ) .

Proof of Theorem 1.

Set [figure omitted; refer to PDF] then _{I [straight epsilon]} (u ) can be rewritten as [figure omitted; refer to PDF] Thus _{I 0} ( [varphi] ) and G ( [varphi] ) are both ^{C 2} with respect to [varphi] . To apply Lemma 5, by Proposition 4, we need only to check that [figure omitted; refer to PDF]

From the fact that a ∈ ^{L 6 / ( 6 -p )} ( ^{... 3} ) , for any T >0 , we have [figure omitted; refer to PDF] Since U exponentially decays at infinity, we know the right side of the equality goes to 0, if θ [arrow right] ∞ .

Let _{B 1} be the quadratic functional on ^{L 2} defined by [figure omitted; refer to PDF] it follows from the Parseval identity that [figure omitted; refer to PDF] and in particular we have [figure omitted; refer to PDF] Then for any T >0 , we have [figure omitted; refer to PDF] Since U exponentially decays at infinity, the right side of the inequality ( 32) goes to 0 . Thus from ( 29) and ( 32) above we soon get [figure omitted; refer to PDF] Then by assumption ( 10) that [figure omitted; refer to PDF] we know Γ (0 ) ...0;0 . Thus, the conclusion follows from Lemma 5that any strict maximum or minimum of Γ 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 [figure omitted; refer to PDF] We have [figure omitted; refer to PDF] It can be proved that [varphi] (x ) = ^{[straight epsilon] 2 / ( p -2 )} u ( [straight epsilon]x ) ∈ ^{H 1} ( ^{... 3} ) is a solution of system ( 9) if and only if u ∈ ^{H 1} ( ^{... 3} ) is a critical point of the functional _{I [straight epsilon]} : ^{H 1} ( ^{... 3} ) [arrow right] ... defined by [figure omitted; refer to PDF] Set [figure omitted; refer to PDF] Then _{I [straight epsilon]} (u ) can be rewritten as [figure omitted; refer to PDF] Define [figure omitted; refer to PDF] and for i =1,2 [figure omitted; refer to PDF]

Lemma 7.

Under assumptions ( _{a 1} ) and ( _{b 1} ) , G = _{G 1} + _{G 2} is continuous in ( [straight epsilon] ,u ) .

Proof.

From the proof of Lemma 4.1 in [ 7], we know _{G 1} is continuous in ( [straight epsilon] ,u ) ∈ ... × ^{H 1} ( ^{... 3} ) , and hence we only need to prove that _{G 2} is continuous in ( [straight epsilon] ,u ) .

If ( [straight epsilon] ,u ) [arrow right] ( _{[straight epsilon] 0} , _{u 0} ) , with _{[straight epsilon] 0} ...0;0 . Then we can estimate that [figure omitted; refer to PDF] It is obvious that _{I 2} [arrow right]0 , as [straight epsilon] [arrow right] _{[straight epsilon] 0} . At the same time, we know [figure omitted; refer to PDF] Estimating the first term _{Π 1} , by Hölder inequality, we know [figure omitted; refer to PDF] Since b (x ) is bounded and continuous, the operator _{E 1} ∈ [Lagrangian (script capital L)] ( ^{L 2} , ^{L 2} ) , the dominated convergence theorem implies that [figure omitted; refer to PDF] Similarly, we can deduce that _{Π 2} , _{Π 3} , _{Π 4} vanishes, as ( [straight epsilon] ,u ) [arrow right] ( _{[straight epsilon] 0} , _{u 0} ) . Hence [figure omitted; refer to PDF] as ( [straight epsilon] ,u ) [arrow right] ( _{[straight epsilon] 0} , _{u 0} ) .

If ( [straight epsilon] ,u ) [arrow right] (0 , _{u 0} ) , by definition, _{G 2} (0 ,u ) =0 . Since b (x ) ∈ ^{L 2} ( ^{... 3} ) is also bounded, we know b (x ) ∈ ^{L 6} ( ^{... 3} ) , applying Parseval identity and Hölder inequality, we get [figure omitted; refer to PDF] therefore _{G 2} ( [straight epsilon] ,u ) [arrow right]0 , as ( [straight epsilon] ,u ) [arrow right] (0 ,u ) . Hence G = _{G 1} + _{G 2} is continuous and the lemma is proved.

Lemma 8.

Under assumptions ( _{a 1} ) and ( _{b 1} ) , ^{G [variant prime]} and ^{G [variant prime][variant prime]} are continuous in ( [straight epsilon] ,u ) .

Proof.

^{G 1 [variant prime]} and ^{G 1 [variant prime][variant prime]} are continuous in ( [straight epsilon] ,u ) , see [ 7, Lemma 4.2] for the details. Here we only prove that ^{G 2 [variant prime]} and ^{G 2 [variant prime][variant prime]} are continuous in ( [straight epsilon] ,u ) .

If ( [straight epsilon] ,u ) [arrow right] ( _{[straight epsilon] 0} , _{u 0} ) with _{[straight epsilon] 0} ...0;0 , then [figure omitted; refer to PDF] Estimating the second term, since _{E 1} ∈ [Lagrangian (script capital L)] ( ^{L 2} , ^{L 2} ) , by Hölder inequality, we know [figure omitted; refer to PDF] Thus _{sup || v || =1 I 2} [arrow right]0 as [straight epsilon] [arrow right] _{[straight epsilon] 0} . Estimating the first term _{I 1} , we know [figure omitted; refer to PDF] As in Lemma 7, by Hölder inequality again, we can prove that _{A i} [arrow right]0 , as ( [straight epsilon] ,u ) [arrow right] ( _{[straight epsilon] 0} , _{u 0} ) , i =1,2 ,3,4 . Therefore || ^{G 2 [variant prime]} ( [straight epsilon] ,u ) - ^{G 2 [variant prime]} ( _{[straight epsilon] 0} , _{u 0} ) || [arrow right]0 as ( [straight epsilon] ,u ) [arrow right] ( _{[straight epsilon] 0} , _{u 0} ) .

If _{[straight epsilon] 0} =0 , from the definition of _{G 2} , we know || ^{G 2 [variant prime]} ( 0 , _{u 0} ) || =0 . Hence [figure omitted; refer to PDF] And we know | | ^{G 2 [variant prime]} ( [straight epsilon] ,u ) | | [arrow right]0 , as [straight epsilon] [arrow right]0 . From the above arguments, we know ^{G [variant prime]} = ^{G 1 [variant prime]} + ^{G 2 [variant prime]} is continuous in ( [straight epsilon] ,u ) .

In the following we prove that ^{G [variant prime][variant prime]} is continuous in ( [straight epsilon] ,u ) . As we know [figure omitted; refer to PDF]

If ( [straight epsilon] ,u ) [arrow right] ( _{[straight epsilon] 0} , _{u 0} ) with _{[straight epsilon] 0} ...0;0 , then [figure omitted; refer to PDF] where [figure omitted; refer to PDF] We estimate _{I 1} only, and _{I 2} can be estimated in a similar way, indeed [figure omitted; refer to PDF] Similar to the proof in Lemma 7, we know _{I 1} [arrow right]0 as [straight epsilon] [arrow right] _{[straight epsilon] 0} and u [arrow right] _{u 0} . Thus we know || ^{G 2 [variant prime][variant prime]} ( [straight epsilon] ,u ) - ^{G 2 [variant prime][variant prime]} ( _{[straight epsilon] 0} , _{u 0} ) || [arrow right]0 as ( [straight epsilon] ,u ) [arrow right] ( _{[straight epsilon] 0} , _{u 0} ) .

If ( [straight epsilon] ,u ) [arrow right] (0 , _{u 0} ) , then from the definition of _{G 2} , we know [figure omitted; refer to PDF] Using Hölder inequality, we know [figure omitted; refer to PDF] Therefore [figure omitted; refer to PDF] From the above arguments, we know ^{G [variant prime][variant prime]} is continuous in ( [straight epsilon] ,u ) and the proof is complete.

Lemma 9.

Assume ( _{a 1} ) and ( _{b 1} ) are satisfied. Define [figure omitted; refer to PDF] Then [figure omitted; refer to PDF]

Proof.

By changing of variable, we know [figure omitted; refer to PDF] Since a (x ) is continuous and bounded, the dominated convergence theorem implies that [figure omitted; refer to PDF] On the other hand, since _{z θ} is bounded and b (x ) ∈ ^{L 2} ( ^{... 3} ) , then, changing of variable, we know [figure omitted; refer to PDF] since 4 >p >2 . Thus we obtain [figure omitted; refer to PDF]

Now we are ready to prove [figure omitted; refer to PDF] From the proof of [ 7], we first know [figure omitted; refer to PDF] Also, since z is bounded, it is easy to check that [figure omitted; refer to PDF] Moreover, recall that b (x ) ∈ ^{L 2} is bounded and use Hölder inequality, we get [figure omitted; refer to PDF] since 4 >p , we have [figure omitted; refer to PDF] From the above arguments, we know [figure omitted; refer to PDF] and the proof is completed.

Proof of Theorem 2.

By the exponential decay property of proposition U , it is easy to check that ^{I 0 [variant prime][variant prime]} is a compact perturbation of the identity map, and so it is an index-0 Fredholm operator. By Proposition 4, we know that Z is a nondegenerate 3-dimensional critical manifold. From Lemmas 7to 9, we know all the assumptions of Lemma 5are satisfied. Since U has a strict (global) maximum at x =0 , Γ has a strict (global) maximum or minimum at θ =0 depending on the sign of ∫ ... (a (x ) -A ) . By the abstract theorem, we know the existence of family solutions { ( [straight epsilon] , _{u [straight epsilon]} ) } ⊂ ... × ^{H 1} ( ^{... 3} ) . If 2 <p <2 +4 /3 , it is easy to check that _{ψ [straight epsilon]} [arrow right]0 as [straight epsilon] [arrow right]0 .

Remark 10.

The hypothesis ∫ ... (a (x ) -A ) ...0;0 is used to apply Lemma 5and has been already used in [ 4, 7]. If ∫ ... (a (x ) -A ) is identically zero, we can not conclude that there exist critical points of _{I [straight epsilon]} .

In the following we prove Theorem 3.

Lemma 11.

Assume ( _{a 2} ) and ( _{b 1} ) are satisfied. Then G , ^{G [variant prime]} , and ^{G [variant prime][variant prime]} are continuous in ( [straight epsilon] ,u ) .

Proof .

Keeping the exponentially decay property of U in mind, the continuity of _{G 1} , ^{G 1 [variant prime]} , and ^{G 1 [variant prime][variant prime]} in ( [straight epsilon] ,u ) can be proved similarly as in [ 7]. We can also repeat the proof in Lemma 7to know the continuity of _{G 2} . Thus the lemma is concluded.

Lemma 12.

Assume ( _{a 2} ) and ( _{b 1} ) are satisfied. Define [figure omitted; refer to PDF] Then for all θ ∈ ^{R 3} , we have [figure omitted; refer to PDF]

Proof.

As we know [figure omitted; refer to PDF] By assumption ( _{a 2} ) and the decay property of U , [figure omitted; refer to PDF] Moreover, by the boundedness of _{z θ} , we know [figure omitted; refer to PDF] Since 3 > γ we obtain [figure omitted; refer to PDF]

To study the property of ^{G [variant prime]} ( [straight epsilon] , _{z θ} ) , since γ <3 and U exponentially decays at infinity, from the proof in [ 7], we know [figure omitted; refer to PDF] On the other hand, from the boundedness of _{Z θ} and b (x ) , we have [figure omitted; refer to PDF] Since γ <3 , we get [figure omitted; refer to PDF] From the above arguments, we know [figure omitted; refer to PDF]

Proof of Theorem 3.

From Lemmas 11and 12, we know that all the assumptions of Lemma 5are satisfied. Since _{lim | θ | [arrow right] ∞} Γ ( θ ) =0 and Γ (0 ) ...0;0 , we know that there is R >0 such that either [figure omitted; refer to PDF] By the abstract Theorem 2, we know the existence of family solutions { ( [straight epsilon] , _{u [straight epsilon]} ) } ⊂ ... × ^{H 1} ( ^{... 3} ) . If 2 <p <2 +4 /3 , it is easy to check that ( _{[varphi] [straight epsilon]} , _{ψ [straight epsilon]} ) [arrow right]0 as [straight epsilon] [arrow right]0 .

Acknowledgments

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

References

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