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

In the present paper we consider the coupled system of nonlinear Schr"{o}dinger equations with the fractional Laplacian $$\{\begin{array}{rl}{\displaystyle (-\mathrm{\Delta }{)}^{\alpha }{u}_1}& {\displaystyle ={\lambda }_1{u}_1+f_1(u_1)+{\mathrm{\partial }}_{1}F(u_1,u_2)\mathrm{i}\mathrm{n}{\mathbb{R}}^N,}\\ {\displaystyle (-\mathrm{\Delta }{)}^{\alpha }{u}_2}& {\displaystyle ={\lambda }_2{u}_2+f_2(u_2)+{\mathrm{\partial }}_{2}F(u_1,u_2)\mathrm{i}\mathrm{n}{\mathbb{R}}^N,}\end{array}$$ 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 $${\int }_{{\mathbb{R}}^N}\mathrm{\mid }{u}_1{\mathrm{\mid }}^{2}dx={\sigma }_1\text{ and }{\int }_{{\mathbb{R}}^N}\mathrm{\mid }{u}_2{\mathrm{\mid }}^{2}dx={\sigma }_2$$ 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 $f_j(s)={\mu }_j\mathrm{\mid }s{\mathrm{\mid }}^{p_j-2}s$ and $F(s,t)=\beta \mathrm{\mid }s{\mathrm{\mid }}^{r_1}\mathrm{\mid }t{\mathrm{\mid }}^{r_2}$ with $\mu_j>0, \beta>0,$andthevalues$r_i>1, 2<p_j, r_1+r_2<2+\frac{4\alpha}{N}.$ The method also enables us to prove the stability result of standing wave solutions associated with the set of global minimizers.