Nonlinear fractional magnetic Schr\"odinger equation: existence and multiplicity

In this paper we focus our attention on the following nonlinear fractional Schr"odinger equation with magnetic field \begin{equation*} \varepsilon^{2s}(-\Delta)_{A/\varepsilon}^{s}u+V(x)u=f(|u|^{2})u \quad \mbox{ in } $\mathbb{R}$^{N}, \end{equation*} where $\epsilon >0$ is a parameter, $s\in (0,1)$, $N\ge 3$, $(-\mathrm{\Delta }{)}_A^s$ is the fractional magnetic Laplacian, $V:{\mathbb{R}}^N\to \mathbb{R}$ and $A:{\mathbb{R}}^N\to {\mathbb{R}}^N$ are continuous potentials and $f:{\mathbb{R}}^N\to \mathbb{R}$ is a subcritical nonlinearity. By applying variational methods and Ljusternick-Schnirelmann theory, we prove existence and multiplicity of solutions for $\epsilon$ small.