Multiplicity and concentration results for a fractional Schr\"odinger-Poisson type equation with magnetic field

Abstract

This paper is devoted to the study of fractional Schr\"odinger-Poisson type equations with magnetic field of the type \begin{equation*} \varepsilon^{2s}(-\Delta)_{A/\varepsilon}^{s}u+V(x)u+\varepsilon^{-2t}(|x|^{2t-3}*|u|^{2})u=f(|u|^{2})u \quad \mbox{ in } \mathbb{R}^{3}, \end{equation*} where $\varepsilon>0$ is a parameter, $s,t\in (0, 1)$ are such that $2s+2t>3$, $A:\mathbb{R}^{3}\rightarrow \mathbb{R}^{3}$ is a smooth magnetic potential, $(-\Delta)^{s}_{A}$ is the fractional magnetic Laplacian, $V:\mathbb{R}^{3}\rightarrow \mathbb{R}$ is a continuous electric potential and $f:\mathbb{R}\rightarrow \mathbb{R}$ is a $C^{1}$ subcritical nonlinear term. Using variational methods, we obtain the existence, multiplicity and concentration of nontrivial solutions for $\varepsilon>0$ small enough.