Concentration phenomena for a fractional Choquard equation with magnetic field

TL;DR

This paper studies a fractional magnetic Choquard equation, proving the existence and concentration of solutions as a parameter approaches zero, using variational methods in a nonlocal magnetic setting.

Contribution

It introduces a new analysis of fractional magnetic Choquard equations with potential wells, establishing solution existence and concentration phenomena for small parameters.

Findings

Existence of solutions for small epsilon

Solutions concentrate near potential minima

Application of variational methods to nonlocal magnetic equations

Abstract

We consider the following nonlinear fractional Choquard equation $$ \varepsilon^{2s}(-\Delta)^{s}_{A/\varepsilon} u + V(x)u = \varepsilon^{\mu-N}\left(\frac{1}{|x|^{\mu}}*F(|u|^{2})\right)f(|u|^{2})u \mbox{ in } \mathbb{R}^{N}, $$ where $\varepsilon>0$ is a parameter, $s\in (0, 1)$, $0<\mu<2s$, $N\geq 3$, $(-\Delta)^{s}_{A}$ is the fractional magnetic Laplacian, $A:\mathbb{R}^{N}\rightarrow \mathbb{R}^{N}$ is a smooth magnetic potential, $V:\mathbb{R}^{N}\rightarrow \mathbb{R}$ is a positive potential with a local minimum and $f$ is a continuous nonlinearity with subcritical growth. By using variational methods we prove the existence and concentration of nontrivial solutions for $\varepsilon>0$ small enough.