Existence and concentration of nontrivial solutions for a fractional magnetic Schr\"odinger-Poisson type equation

Abstract

We consider the following fractional Schr\"odinger-Poisson type equation with magnetic fields \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, $s\in (\frac{3}{4}, 1)$, $t\in (0,1)$, $(-\Delta)^{s}_{A}$ is the fractional magnetic Laplacian, $A:\mathbb{R}^{3}\rightarrow \mathbb{R}^{3}$ is a smooth magnetic potential, $V:\mathbb{R}^{3}\rightarrow \mathbb{R}$ is a positive continuous electric potential and $f:\mathbb{R}^{3}\rightarrow \mathbb{R}$ is a continuous function with subcritical growth. By using suitable variational methods, we show the existence of families of nontrivial solutions concentrating around local minima of the potential $V(x)$ as $\varepsilon\rightarrow 0$.