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 Schr\"{o}dinger equations with fractional laplacian $$ \left\{ \begin{array}{lr} (-\Delta)^{s_{1}} u+V_{1}(x)u=f_{1}(u)+\lambda(x)v, & x\in\mathbb{R}^{N}, (-\Delta)^{s_{2}} v+V_{2}(x)v=f_{2}(v)+\lambda(x)u, & x\in\mathbb{R}^{N}, \end{array} \right. $$ where $(-\Delta)^{s}$ denotes de fractional Laplacian, $s_{1},s_{2}\in(0,1)$ and $N\geq2$. The coupling function $\lambda:\mathbb{R}^{N} \rightarrow \mathbb{R}$ is related with the potentials by $|\lambda(x)|\leq \delta\sqrt{V_{1}(x)V_{2}(x)}$, 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.