Multiplicity and concentration results for a fractional Schrödinger-Poisson type equation with magnetic field (1807.06861v2)
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.