Nonlinear fractional magnetic Schr\"odinger equation: existence and multiplicity

TL;DR

This paper investigates a nonlinear fractional magnetic Schrödinger equation, establishing the existence and multiplicity of solutions for small parameters using variational methods and topological tools.

Contribution

It introduces new results on the existence and multiplicity of solutions for a fractional magnetic Schrödinger equation with subcritical nonlinearity.

Findings

Proves existence of solutions for small epsilon

Establishes multiple solutions using topological methods

Employs variational techniques and Ljusternick-Schnirelmann theory

Abstract

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