Existence and concentration of solution for a class of fractional elliptic equation in $\mathbb{R}^N$ via penalization method

Abstract

In this paper, we study the existence and concentration of positive solution for the following class of fractional elliptic equation $$ \epsilon^{2s} (-\Delta)^{s}{u}+V(z)u=f(u)\,\,\, \mbox{in} \,\,\, \mathbb{R}^{N}, $$ where $\epsilon$ is a positive parameter, $f$ has a subcritical growth, $V$ possesses a local minimum, $N > 2s,$ $s \in (0,1),$ and $ (-\Delta)^{s}u$ is the fractional laplacian.