On the multiplicity and concentration of positive solutions for a $p$-fractional Choquard equation in $\mathbb{R}^{N}$

Abstract

In this paper we deal with the following fractional Choquard equation \begin{equation*} \left\{ \begin{array}{ll} \varepsilon^{sp}(-\Delta)^{s}_{p} u + V(x)|u|^{p-2}u = \varepsilon^{\mu-N}\left(\frac{1}{|x|^{\mu}}*F(u)\right)f(u) \mbox{ in } \mathbb{R}^{N},\\ u\in W^{s,p}(\R^{N}), \quad u>0 \mbox{ in } \mathbb{R}^{N}, \end{array} \right. \end{equation*} where $\varepsilon>0$ is a small parameter, $s\in (0, 1)$, $p\in (1, \infty)$, $N>sp$, $(-\Delta)^{s}_{p}$ is the fractional $p$-Laplacian, $V$ is a positive continuous potential, $0<\mu<sp$, and $f$ is a continuous superlinear function with subcritical growth. Using minimax arguments and the Ljusternik-Schnirelmann category theory, we obtain the existence, multiplicity and concentration of positive solutions for $\varepsilon>0$ small enough.