Existence of bound and ground states for fractional coupled systems in ${\mathbb{R}}^N$

Abstract

In this work we consider the following class of nonlocal linearly coupled systems involving Schrödinger equations with fractional laplacian $$\{\begin{array}{lrc}(-\Delta {)}^{s_1}u+V_1(x)u=f_1(u)+\lambda (x)v,& x\in {\mathbb{R}}^N,(-\Delta {)}^{s_2}v+V_2(x)v=f_2(v)+\lambda (x)u,& x\in {\mathbb{R}}^N,\end{array}$$ where $(-\Delta {)}^s$ denotes de fractional Laplacian, $s_1,s_2\in (0,1)$ and $N\ge 2$. The coupling function $\lambda :{\mathbb{R}}^N\to \mathbb{R}$ is related with the potentials by $|\lambda (x)|\le \delta \sqrt{V_1(x)V_2(x)}$, for some $\delta \in (0,1)$. We deal with periodic and asymptotically periodic bounded potentials. On the nonlinear terms $f_1$ and $f_2$, we assume "superlinear" at infinity and at the origin. We use a variational approach to obtain the existence of bound and ground states without assuming the well known Ambrosetti-Rabinowitz condition at infinity. Moreover, we give a description of the ground states when the coupling function goes to zero.