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

TL;DR

This paper investigates fractional Schr"odinger-Poisson equations with magnetic fields, establishing existence, multiplicity, and concentration of solutions using variational methods for small parameters.

Contribution

It introduces new results on the existence and multiplicity of solutions for a fractional magnetic Schr"odinger-Poisson equation with concentration phenomena.

Findings

Existence of solutions for small 

Multiple solutions under certain conditions

Solutions concentrate as  approaches zero

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.