Existence of Standing Waves for a Generalized Davey-Stewartson System

and Applied Analysis 3 (3) G(ε, u) and G(ε, u) are continuous maps from R × E → 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 manifold Z, d ≥ 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 (z)) = T θ Z and I 0 (z θ ) is an index-0 Fredholm operator for any z θ ∈ Z. (6) There exists α > 0 and a continuous function Γ : Z → R such that Γ (z) = lim ε→0 G (ε, z) ε , G 󸀠 (ε, z) = o (ε α/2 ) . (18) Consider the existence of critical points of the perturbed problem I 󸀠 ε (u) = 0. (19) Wewant to look for solutions of the form u = z+wwith z ∈ Z and w ∈ W = (T θ Z) ⊥. 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: PI 󸀠 ε (z + w) = 0, (I − P) I 󸀠 ε (z + w) = 0. (20) 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(ε, z) I ε (u) = I 0 (z) + ε α Γ (z) + o (ε α ) . (21) In [4, 5] the following abstract theorem is proved. Lemma5. Suppose assumptions (1)–(6) are satisfied, and there exists δ > 0 and z ∈ Z such that either min ‖z−z ∗ ‖=δ Γ (z) > Γ (z ∗ ) or max ‖z−z ∗ ‖=δ Γ (z) < Γ (z ∗ ) . (22) Then for any ε small, there exists u ε which is a critical point of I ε . 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 F {E 1 (ψ)} (ξ) = σ 1 (ξ)F (ψ) (ξ) , (23) where σ 1 (ξ) = ξ 2 1 /|ξ| 2, ξ ∈ R, and F denotes the Fourier transform: F (ψ) (ξ) = ( 1 2π ) 3/2 ∫ e −iξx ψ (x) dx. (24) For 1 < p < ∞, E 1 satisfies the following properties: (1) E 1 ∈ L(L, L). (2) if ψ ∈ H(R), then E 1 (ψ) ∈ H 1 (R). (3) E 1 preserves the following operations: translation: E 1 (ψ(⋅ + y))(x) = E 1 (ψ)(x + y), y ∈ R. 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 ε : H 1 (R) → R can be defined as I ε (φ) = 1 2 󵄩󵄩󵄩󵄩φ 󵄩󵄩󵄩󵄩 2 − ε 4 ∫ b (x) E 1 (b (x) 󵄨󵄨󵄨󵄨φ 󵄨󵄨󵄨󵄨 2 ) 󵄨󵄨󵄨󵄨φ 󵄨󵄨󵄨󵄨 2 − 1 p ∫ (1 + εa (x)) 󵄨󵄨󵄨󵄨φ 󵄨󵄨󵄨󵄨 p . (25) It is easy to see that I ε : H 1 (R) → R is of C, and thus φ is a solution of (8) if and only if φ is a critical point of the action functional I ε (φ). Proof of Theorem 1. Set I 0 (φ) = 1 2 󵄩󵄩󵄩󵄩φ 󵄩󵄩󵄩󵄩 2 − 1 p ∫ 󵄨󵄨󵄨󵄨φ 󵄨󵄨󵄨󵄨 p , G (φ) = − 1 p ∫a (x) 󵄨󵄨󵄨󵄨φ 󵄨󵄨󵄨󵄨 p − 1 4 ∫ b (x) E 1 (b (x) 󵄨󵄨󵄨󵄨φ 󵄨󵄨󵄨󵄨 2 ) 󵄨󵄨󵄨󵄨φ 󵄨󵄨󵄨󵄨 2 , (26) then I ε (u) can be rewritten as I ε (φ) = I 0 (φ) + εG (φ) . (27) Thus I 0 (φ) and G(φ) are both C with respect to φ. To apply Lemma 5, by Proposition 4, we need only to check that lim |θ|→∞ Γ (θ) = 0, Γ (θ) := G|Z = − 1 p ∫a (x) 󵄨󵄨󵄨󵄨zθ 󵄨󵄨󵄨󵄨 p


Introduction and Main Results
In this paper, we are going to consider the existence of standing waves for a generalized Davey Here Δ is the Laplacian operator in R 3 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 ,  1 ,  2 ∈ R, (, , ) 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, hyperbolicelliptic, 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  1 be the singular integral operator defined by where  1 () =  2 1 /|| 2 ,  ∈ R 3 , and F denotes the Fourier transform: Then the generalized Davey-Stewartson system can be reduced to the following single nonlocal Schrödinger equation −  − Δ =  ()          −2  +  ()  1 ( ()          2 ) . (5) In this paper, we are interested in the existence of standing waves for the above equation, that is, solutions in the form of  (, ) =    () ,  (, ) =  () , where  > 0, ,  ∈  1 (R 3 ).Then if (, ) is a solution of (1), then we can see that  must satisfy the following Schrödinger problem: −Δ +  =  ()  1 ( ()          2 )  +  ()          −2 .(7) 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 Concentrationcompactness 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.
Throughout this paper, we denote the norm of  1 (R 3 ) 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  1 .In Section 3, we prove the main results by some lemmas.

The Abstract Theorem
To prove the main results, we need the following known propositions.
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  0 :  → R and  : R ×  → R. We need the following hypotheses and assume that (1)  0 and  are  2 with respect to ; (2)  is continuous in (, ) and (0, ) = 0 for all ; (3)   (, ) and   (, ) are continuous maps from R ×  →  and (, ), respectively, and (, ) is the space of linear continuous operators from  to .
(4) There is a -dimensional  2 manifold ,  ≥ 1, consisting of critical points of  0 , and such a  will be called a critical manifold of  0 .
(5) let    denote the tangent space to  at   , the manifold  is nondegenerate in the following sense: Ker(  0 ()) =    and   0 (  ) is an index-0 Fredholm operator for any   ∈ .
(6) There exists  > 0 and a continuous function Γ :  → R 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  > 0 and  * ∈  such that ℎ min Then for any  small, there exists   which is a critical point of   .

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   :  1 (R 3 ) → R can be defined as It is easy to see that   :  1 (R 3 ) → R is of  2 , and thus  is a solution of (8) if and only if  is a critical point of the action functional   ().
Proof of Theorem 1.
then   () can be rewritten as Thus  0 () and () are both  2 with respect to . . (29) Since  exponentially decays at infinity, we know the right side of the equality goes to 0, if  → ∞.
Let  1 be the quadratic functional on  2 defined by it follows from the Parseval identity that and in particular we have 0 < Since  exponentially decays at infinity, the right side of the inequality (32) goes to 0. Thus from (29) and (32) above we soon get lim Then by assumption (10) that we know Γ(0) ̸ = 0. Thus, the conclusion follows from Lemma 5 that any strict maximum or minimum of Γ gives rise to a critical point of the perturbed functional and hence to a solution of (8).
If (, ) → (0,  0 ), then from the definition of Now we are ready to prove From the proof of [7], we first know   1 (,   ) =  ( From the above arguments, we know lim and the proof is completed. Proof of Theorem 2. By the exponential decay property of proposition , it is easy to check that   0 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  = 0, Γ has a strict (global) maximum or minimum at  = 0 depending on the sign of ∫(() − ).By the abstract theorem, we know the existence of family solutions {(,   )} ⊂ R ×  1 (R 3 ).If 2 <  < 2+4/3, it is easy to check that   → 0 as  → 0.
In the following we prove Theorem 3.
Proof .Keeping the exponentially decay property of  in mind, the continuity of  1 ,   1 , and   1 in (, ) can be proved similarly as in [7].We can also repeat the proof in Lemma 7 to know the continuity of  2 .Thus the lemma is concluded.