Existence and stability of standing waves for nonlinear Schrodinger systems involving the fractional Laplacian

Abstract

In the present paper we consider the coupled system of nonlinear Schrödinger equations with the fractional Laplacian \[ \left\{ \begin{aligned} (-Δ)^αu_1 & = λ_1u_1+f_1(u_1)+\partial_1F(u_1,u_2)\ \ \mathrm{in}\ \mathbb{R}^N, \\ (-Δ)^αu_2 & = λ_2u_2+f_2(u_2)+\partial_2F(u_1,u_2)\ \ \mathrm{in}\ \mathbb{R}^N, \end{aligned} \right. \] where $u_1,u_2:{\mathbb{R}}^N\to \mathbb{C},N\ge 2,$ and $0<\alpha <1.$ By studying an appropriate family of constrained minimization problems, we obtain the existence of solutions in the space $H^{\alpha }({\mathbb{R}}^N)\times H^{\alpha }({\mathbb{R}}^N)$ satisfying \[ \int_{\mathbb{R}^N}|u_1|^2\ dx = σ_1\ \ \textrm{and}\ \ \int_{\mathbb{R}^N}|u_2|^2\ dx=σ_2 \] for given ${\sigma }_j>0.$ The numbers ${\lambda }_1$ and ${\lambda }_2$ in the system appear as Lagrange multiplier. The method is based on the concentration compactness arguments, but introduces a new way to verify some of the properties of the variational problem that are required in order for the concentration compactness method to work. We consider the case when $f_j(s)={\mu }_j|s{|}^{p_j-2}s$ and $F(s,t)=\beta |s{|}^{r_1}|t{|}^{r_2}$ with ${\mu }_j>0,\beta >0,$ and the values $r_i>1,2<p_j,r_1+r_2<2+\frac{4\alpha }{N}.$ The method also enables us to prove the stability result of standing wave solutions associated with the set of global minimizers.

22 pages