Stable solitary waves for a class of nonlinear Schrödinger system with quadratic interaction

We consider the existence and orbital stability of bound state solitary waves and ground state solitary waves for a class of nonlinear Schrödinger system with quadratic interaction in Rn (n = 2, 3). The existence of bound state and ground state solitary waves are studied by variational arguments and Concentration-compactness Lemma. In additional, we also prove the orbital stability of bound state and ground state solitary waves.


Introduction
In this paper, we consider the following system of nonlinear Schrödinger equations where u and v are complex-valued wave fields, m and M are positive constants, λ and µ are complex constants, and u is the complex conjugate of u.Such systems have interesting applications in several branches of physics, such as in the study of interactions of waves with different polarizations [1,11].The Cauchy problem for System 1.1 has been studied from the point of view of small data scattering [6,7].In 2013, Hayashi, Ozawa and Tanaka [8] studied the well-posedness of Cauchy problem for System 1.1 with large data.In particular, System 1.1 is regarded as a non-relativistic limit of the system of nonlinear Klein-Gordon equations Corresponding author.Email: zgqw2001@usst.edu.cnunder the mass resonance condition M = 2m, where c is the speed of light.Assume λ = cµ, c > 0, λ = 0 and µ = 0, we introduce new functions ( u, v) defined by x, t , and System (1.1) satisfies Using the ansatz ( u(x, t), v(x, t)) = (e iωt φ(x), e i2ωt ψ(x)), φ(x), ψ(x) ≡ 0 with ω > 0, System (1.3) becomes where κ = m M .Let L p (R n ) denote the usual Lebesgue space with the norm |u| p = ( R n |u| p dx) 1 2 , and H 1 r (R n ) := {u ∈ H 1 (R n ); u is radially symmetric}.Recently, as 2 ≤ n ≤ 5, Hayashi, Ozawa and Tanaka [8] obtained the existence of radially symmetric ground states for System (1.4) by using rearrangement method, Pohozaev identity and the Sobolev compact embedding In this paper, firstly, we prove the existence of bound states for System (1.4) by using the Concentration-compactness Lemma and direct methods in the critical points theory.Secondly, we discuss the general case for System (1.4), i.e., where (λ 1 , λ 2 ) ∈ R 2 .By using the Concentration-compactness Lemma, variational arguments and rearrangement result of Shibata [13], we obtain the existence of ground states for System (1.5).In particular, if λ 1 = 1 2 λ 2 > 0, then System (1.5) can be reduced to System (1.4) and the existence of ground states for System (1.4) is obtained in [8].Furthermore, we also prove the orbital stability of bound states and ground states.
Remark 1.1.In contrast to results in [8], we obtain the existence of bound states in the whole space ) is only continuous, we apply the Concentration-compactness Lemma and variational arguments to obtain the existence of bound states.

Preliminaries and main results
In this section, we state our main results in this paper.Now, we define the functionals I, J and Q : It is obvious that I, J and 2 > 0} for some N > 0, and the minimizing problem Besides, for every N > 0, let P N denote the set of bound states of System (1.4), that is, which generates the solitary waves of System (1.1).
Theorem 2.1.Let n = 2, 3. Then we have: (1) For all N > 0, there exists (φ (2) If (φ N , ψ N ) is a solution of the minimizing problem (2.2), then there exists a Lagrange multiplier where σ N is given by (3) The set is a closed graph in (0, +∞) × (0, +∞).In particular, if Σ is a function, then it is continuous and there exists N 0 > 0 such that Next, we define the set for any α, β > 0, and the minimizing problem Besides, for any α, β > 0, let which denotes the set of ground states of System (1.5).
Now, we recall the rearrangement results of Shibata [13] as presented in [9].Let u be a Borel measureable function on R n .Then u is said to vanish at infinity and {u, v} by {u, v} (x

Bound states
Let {(φ n , ψ n )} n≥1 be a minimizing sequence for the minimizing problem (2.1), that is, the sequence for all n, and the functional I is bounded below on M N .
Proof.By the Gagliardo-Nirenberg inequality, we have Hence, we have Since n = 2, 3, we have n 6 + n 12 < 1.Thus, I is coercive and in particular Lemma 3.2.For any N > 0, I N < 0 and I N is continuous with respect to N.
In order to prove that I N is a continuous function, we assume N n = N + o(1).From the definition of I N n , for any ε > 0, there exists (φ n , ψ n ) ∈ M N n such that we have that (u n , v n ) ∈ M N and Combining (3.1) and (3.2), we obtain Reversing the argument, we obtain similarly that Therefore, since ε > 0 is arbitrary, we deduce that and we can write We can choose a, b > 0 such that 2b − na = 1, b > 2a and it follows from (3.3) that Since (φ(x), ψ(x)) ∈ M N ⇔ (φ θ (x), ψ θ (x)) ∈ M θN , ∀θ, N > 0, it follows that Thus, Lemma 3.4.For any N > 0 and λ ∈ (0, N), we have I N < I λ + I N−λ .
Proof.Thanks to the following well-known inequality: ∀a, b, A, B > 0, where the equalities hold if and only if a A = b B , we get Without loss of generality, we assume −I λ λ is larger than −I N−λ N−λ , then By Lemma 3.3, we have Proof of Theorem 2.1.Our proof is divided into five steps: Step 1.The minimizing problem (2.2) has a solution.By Lemma 3.1, the sequence for some R > 0, the φ n → 0, ψ n → 0 in L p (R n ) for 2 < p < 2 * , see [11,12].This is incompatible with the fact that I N < 0, see Lemma 3.2.Thus, the vanishing of minimizing sequence {(φ n , ψ n )} does not exist.Besides, Lemma 3.4 prevents their dichotomy.According to Concentration-compactness Lemma, only concentration exists, and we get a solution (φ N , ψ N ) of the minimizing problem (2.