Multiplicity and concentration results for a fractional Choquard equation via penalization method

This paper is devoted to the study of the following fractional Choquard equation $${\epsilon }^{2s}(-\mathrm{\Delta }{)}^{s}u+V(x)u={\epsilon }^{\mu -N}(\frac{1}{\mathrm{\mid }x{\mathrm{\mid }}^{\mu }}\ast F(u))f(u)in{\mathbb{R}}^N,$$ where $\epsilon >0$ is a parameter, $s\in (0,1)$, $N>2s$, $(-\mathrm{\Delta }{)}^s$ is the fractional Laplacian, $V$ is a positive continuous potential with local minimum, $0<\mu <2s$, and $f$ is a superlinear continuous function with subcritical growth. By using the penalization method and the Ljusternik-Schnirelmann theory, we investigate the multiplicity and concentration of positive solutions for the above problem.