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 Schr\"{o}dinger equations with the fractional Laplacian \[ \left\{ \begin{aligned} (-\Delta)^\alpha u_1 & = \lambda_1u_1+f_1(u_1)+\partial_1F(u_1,u_2)\ \ \mathrm{in}\ \mathbb{R}^N, \\ (-\Delta)^\alpha u_2 & = \lambda_2u_2+f_2(u_2)+\partial_2F(u_1,u_2)\ \ \mathrm{in}\ \mathbb{R}^N, \end{aligned} \right. \] where $u_1, u_2:\mathbb{R}^N\to \mathbb{C},\ N\geq 2,$ 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 \[ \int_{\mathbb{R}^N}|u_1|^2\ dx = \sigma_1\ \ \textrm{and}\ \ \int_{\mathbb{R}^N}|u_2|^2\ dx=\sigma_2 \] 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.