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.
Categories: math.AP