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.
Categories: math.AP