Concentrating solutions for a class of nonlinear fractional Schrödinger equations in ${\mathbb{R}}^N$

Abstract

We deal with the existence of positive solutions for the following fractional Schrödinger equation:

where $\epsilon >0$ is a parameter, $s\in (0,1)$, $N\ge 2$, $(-\Delta {)}^s$ is the fractional Laplacian operator, and $V:{\mathbb{R}}^N\to \mathbb{R}$ is a positive continuous function. Under the assumptions that the nonlinearity $f$ is either asymptotically linear or superlinear at infinity, we prove the existence of a family of positive solutions which concentrates at a local minimum of $V$ as $\epsilon$ tends to zero.