Multiplicity and concentration results for a magnetic Schr\"{o}dinger equation with exponential critical growth in $\mathbb{R}^{2}$

TL;DR

This paper investigates solutions to a magnetic nonlinear Schrödinger equation with exponential critical growth in two dimensions, demonstrating solution multiplicity and concentration as the parameter approaches zero using variational and topological methods.

Contribution

It introduces new multiplicity and concentration results for the magnetic Schrödinger equation with exponential critical growth, employing variational, penalization, and topological techniques.

Findings

Existence of multiple solutions for small epsilon.

Solutions concentrate around certain regions as epsilon approaches zero.

Application of variational and Ljusternick-Schnirelmann methods to this problem.

Abstract

In this paper we study the following nonlinear Schr\"{o}dinger equation with magnetic field \[ \Big(\frac{\varepsilon}{i}\nabla-A(x)\Big)^{2}u+V(x)u=f(| u|^{2})u,\quad x\in\mathbb{R}^{2}, \] where $\varepsilon>0$ is a parameter, $V:\mathbb{R}^{2}\rightarrow \mathbb{R}$ and $A: \mathbb{R}^{2}\rightarrow \mathbb{R}^{2}$ are continuous potentials and $f:\mathbb{R}\rightarrow \mathbb{R}$ has exponential critical growth. Under a local assumption on the potential $V$, by variational methods, penalization technique, and Ljusternick-Schnirelmann theory, we prove multiplicity and concentration of solutions for $\varepsilon$ small.