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

TL;DR

This paper proves the existence of solutions to a fractional magnetic Schr"odinger-Poisson equation that concentrate near potential minima as a small parameter approaches zero, using variational methods.

Contribution

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

Findings

Solutions exist and concentrate near local minima of V(x) as epsilon approaches zero.

The methods handle fractional magnetic operators with subcritical nonlinearities.

The results extend previous work to include magnetic fields and fractional operators.

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 $\varepsilon\rightarrow 0$.