Existence of single peak solutions for a nonlinear Schrödinger system with coupled quadratic nonlinearity

which arises fromsecond-harmonic generation in quadraticmedia.Here ε > 0 is a small parameter, 2 ≤ N < 6, μ > 0 and μ > γ, P(x), Q(x) are positive function potentials. By applying reduction method, we prove that if x0 is a non-degenerate critical point of Δ(P + Q) on some closed N − 1 dimensional hypersurface, then the system above has a single peak solution (vε , wε) concentrating at x0 for ε small enough.


Introduction and main result
In this paper, we consider the following Schrödinger system with coupled quadratic nonlinearity where ε > 0 is a small parameter, 2 ≤ N < 6, μ > 0 and μ > γ, P(x), Q(x) are positive potentials. System (1.1) arises from the cubic nonlinear Schrödinger equation 2) which appears in the nonlinear optic theory and can be used to describe the formation and propagation of optical solutions in Kerr-type materials [6,19]. Here ϕ is a slowly varying envelope of electric field, the realvalued parameter r and χ represent the relative strength and sign of dispersion/diffraction and nonlinearity respectively, and z is the propagation distance coordinate. The Laplacian operator ∇ 2 can either be ∂ 2 ∂τ 2 for temporal solitons with τ is the normalized retarded time, or , where x = (x 1 , · · · , x N ) is in the direction orthogonal to z. Solitary wave solutions to (1.2) and its generations have been studied in [4,18]. Also, (1.1) appears in the study of standing waves for the following nonlinear system ⎧ ⎨ ⎩ i ∂ϕ1 ∂t = −ε 2 Δϕ 1 + (P(x) + μ)ϕ 1 − μ|ϕ 1 |ϕ 2 , ( x, t) ∈ R N × R + , i ∂ϕ2 ∂t = −ε 2 Δϕ 2 + (Q(x) + μ)ϕ 2 − μ 2 |ϕ 1 | 2 − γ|ϕ 2 | 2 , (x, t) ∈ R N × R + , (1.3) with the form ϕ 1 (x, t) = v(x)e iμt , ϕ 2 (x, t) = w(x)e iμt , where i is the imaginary unit and ε is the Planck constant. When ε = 1 and γ = 0, the existence of ground state solutions of (1.3) was proved in [27]. Besides, (1.1) is closely related to the general parabolic system with coupled nonlinearity and the nonlinear evolution equations. For this information, we can refer to [11,16,23,25,26] and references therein. By contrast with the coupled Schrödinger system (1.3) with χ (2) nonlinearities, the following χ 2 nonlinear Schrödinger system ⎧ ⎨ has been extensively investigated. There are many interesting results about (1.4) under various assumptions of V 1 (x) and V 2 (x), one can refer to [1-4, 7, 8, 12-15, 17, 21, 24] and their references therein. In recent decades, system (1.1) and its related problems have attracted a lot of attention. When ε = 1, (1.5) Applying the finite dimensional reduction method, Wang and Zhou [22] constructed the infinitely many nonradial positive solutions of (1.5) if the potential functions P(x), Q(x) are radial and satisfy some algebraic decay at infinity. Also, if ε is small, for any positive integer k ≤ N + 1, Tang and Xie [20] proved that (1.1) has a k spikes solution concentrating at some strict local maximum of P(x) and Q(x) by using the finite dimensional reduction provided that |P(x) − P(y)| ≤ L 1 |x − y| θ1 and |Q(x) − Q(y)| ≤ L 2 |x − y| θ2 for some positive constants L 1 , L 2 , θ 1 , θ 2 .
Here we want to mention that, very recently, Luo, Peng and Yan [13] revisited the following Schrödinger equation with 2 < p < 2 * . Under the condition that V(x) obtains its local minimum or local maximum x 0 at a closed N − 1 dimensional hypersurface, they obtained the existence of a positive single peak solution for (1.6) concentrating at x 0 if x 0 is non-degenerate critical point of ΔV and also verified the local uniqueness of single peak solutions by using local Pohazaev type identity. Motivated by [13,20,22], we want to apply the finite-dimensional reduction to study the existence of positive single peak solutions for (1.1). Our purpose here is to prove that (1.1) has a single peak solution concentrating at some non-degenerate critical point of Δ(P + Q) on a closed N − 1 dimensional hypersurface.
− 1| ≤ δ 0 } for some small fixed δ 0 > 0. Then we can use the above conditions to the potentials P(x), Q(x).
Let us point out that if Γ is a local minimum (or local maximum) set of P(x) and Q(x), then for any x ∈ Γ, This implies that for any tangential vector ς of Γ at x, one has where Dς denotes the directional derivative at the direction ς.
Let U be the unique positive radial solution of the following problem It is well-known in [10] that U(x) is strictly decreasing and its s order derivative satisfies for |s| ≤ 1 and some constant C > 0. For x ε close to x 0 , if we denoteV For ε > 0 small, we will use (V ε,x ε , Wε,x ε ) to construct the single peak solutions concentrating at x 0 . First we give the following definitions. Definition 1.2. We say that (vε , wε) is a single peak solution of (1.1) concentrating at The main result of this paper is the following.
As in [13,20,22], we mainly apply the finite dimensional reduction method to prove our main result. Compared with [13], we have to overcome much difficulties in the reduction process which involves some technical and careful computations due to the χ (2) nonlinearity appears. Moreover, to our best knowledge, our result exhibits a new phenomenon for the coupled Schrödinger system with χ (2) nonlinearity.

Remark 1.5.
Combining the ideas from [9,13], where in [9] the coupled nonlinear Gross-Pitaevskii system was studied, we guess that the following conclusions may hold.
(1) On the basis of Theorem 1.4, further we can prove the local uniqueness of single peak solutions by using local Pohazaev type identity.
(2) Under the conditions of Theorem 1.4, if Δ(P + Q) has an isolated maximum or minimum point x 0 ∈ Γ, then for any integer k > 0, (1.1) has a k-peaks solution whose peaks cluster at x 0 .
The structure of this paper is organized as follows. In section 2, we do some preliminaries and then we carry out the finite dimensional reduction. We will prove our main result in section 3. In the sequel, for simplicity of notations we write f to mean the Lebesgue integral of f (x) in R N .

the finite dimensional reduction
In this section, we mainly give some preliminaries and do the finite dimensional reduction. Hereafter, for any function K(x) > 0, we define the Sobolev space endowed with the standard norm which is induced by the inner product Note that the variational functional corresponding to (1.1) is Then I ∈ C 2 (H, R) and its critical points are solutions of (1.1).
We can expand J ε(φ, ψ) as follows: and In order to find a critical point for J ε (φ, ψ), we need to discuss each terms in the expansion (2.2).
Proof. Recall that By the direct computations, we get Observe that by letting u ε(x) = u(εx), it is easy to check with u ε = ε 2 |∇u| 2 + u 2 1 2 for 2 ≤ p ≤ 2 * and some C > 0. Then we have This completes our proof.
Next we want to discuss the invertibility of L in E ε,xε . To this end, we denote (V , W) = (αU, βU)). Then Using the above result, we come to discuss the invertibility of L in E ε,xε .
Then, in view of (vn , wn) 2 ε n = ε N n , we get which implies that up to a subsequence, there exists v, w ∈ H 1 (R N ) such that as n → +∞, Now we claim that v = w = 0. Considering (2.5), we find

Proof of our main result
In this section, we assume that x 0 ∈ Γ is a non-degenerate critical point of Δ(P + Q) and we will construct a single peak solution (v ε , wε) of (1.1) concentrating at x 0 . From Proposition 2.5, we can get the following result.
For y close to x 0 , y ∈ Γ t for some t close to 1. In the following, we denote by ν the unit normal vector of Γ t at y and we use ς i , i = 1, · · · , N − 1, to denote the principal direction of Γ t at y. Then, at y, one has Dς i P(y) = 0, |∇P(y)| = |Dν P(y)| and D ς i Q(y) = 0, |∇Q(y)| = |Dν Q(y)|.
First, we prove the following results.
As a result, the result follows.