Nonlinear fractional magnetic Schrödinger equation: Existence and multiplicity

In this paper we focus our attention on the following nonlinear fractional Schrödinger equation with magnetic field egin{equation*} arepsilon^{2s} (-Delta)^{s}_{A/arepsilon} u + V(x) u= f(|u|^{2})u mbox{ in } mathbb{R}^{N}, end{equation*} where $arepsilon>0$ is a parameter, $sin (0,1)$, $Ngeq 3$, $(-Delta)^{s}_{A}$ is the fractional magnetic Laplacian, $V:mathbb{R}^{N} ightarrow mathbb{R}$ and $A:mathbb{R}^{N} ightarrow mathbb{R}^{N}$ are continuous potentials and $f: mathbb{R}^{N} ightarrow mathbb{R}$ is a subcritical nonlinearity. By applying variational methods and Ljusternick–Schnirelmann theory, we prove existence and multiplicity of solutions for ε small.