Multiplicity and concentration of solutions for fractional Schrödinger systems via penalization method

Abstract

The aim of this paper is to investigate the existence, multiplicity and concentration of positive solutions for the following nonlocal system of fractional Schrödinger equations

where $\epsilon >0$ is a parameter, $s\in (0,1)$, $N>2s$, $(-\Delta {)}^s$ is the fractional Laplacian, $V:{\mathbb{R}}^N\to \mathbb{R}$ and $W:{\mathbb{R}}^N\to \mathbb{R}$ are positive continuous potentials, $Q$ is a homogeneous $C^2$-function with subcritical growth. In order to relate the number of solutions with the topology of the set where the potentials $V$ and $W$ attain their minimum values, we apply penalization techniques, Nehari manifold arguments and Ljusternik–Schnirelmann theory.