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

TL;DR

This paper proves the existence of bound and ground states for a class of fractional Schrödinger systems with nonlocal coupling, using variational methods without the Ambrosetti-Rabinowitz condition, and describes their behavior as coupling diminishes.

Contribution

It introduces a variational approach to establish bound and ground states for fractional coupled systems without relying on the Ambrosetti-Rabinowitz condition.

Findings

Existence of bound and ground states for fractional coupled systems.

Results hold for periodic and asymptotically periodic potentials.

Description of ground states as coupling function approaches zero.

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.