Abstract:

In this paper, we are concerned with the following nonlinear Schrödinger system with quadratic nonlinearities

$\begin{align*} \begin{cases} -\epsilon^2\Delta u_1+V_1(x)u_1=\alpha u_1 u_{2} \ \text{ in } \mathbb{R}^N,\\ -\epsilon^2\Delta u_2+V_2(x)u_2=\frac{\alpha}{2}u_1^2+\beta u_2^{2} \ \text{ in } \mathbb{R}^N, \end{cases} \end{align*}$

where$2\leq N<6$, $\epsilon>0$is a small parameter,$\alpha>0$and$\alpha >\beta,$ the functions $V_i$ are positive, $V_i$, $|\nabla V_i| \in L^\infty(\mathbb{R}^N).$ As $\epsilon$ goes to zero, applying the finite dimensional reduction method, we construct synchronized solution which concentrates at the non-degenerate critical point of a new function constructed by the potential functions $V_{i}(x)(i=1,2).$ Moreover, by the contradiction argument combining some local Pohozeav identities and the blow-up technique we prove the uniqueness of this single-peak solution. Our results extend the results in [Gross M. Ann Inst H Poincar'e C Anal Non Lin'eaire, 2002] about a single Schrödinger equation to our system.

Key words: Schr?dinger system, synchronized solutions, uniqueness, ?nite dimensional reduction, Pohzeav identity.