Semiclassical states for a magnetic nonlinear Schr\"{o}dinger equation with exponential critical growth in $\mathbb{R}^{2}$

TL;DR

This paper investigates the existence and behavior of solutions to a magnetic nonlinear Schrödinger equation with exponential critical growth in two dimensions, using variational methods and topological tools.

Contribution

It establishes the existence, multiplicity, and concentration properties of solutions for small parameters, addressing the critical exponential growth challenge.

Findings

Existence of solutions for small epsilon

Multiple solutions under certain conditions

Solutions concentrate around specific regions

Abstract

This paper is devoted to the magnetic nonlinear Schr\"{o}dinger equation \[ \Big(\frac{\varepsilon}{i}\nabla-A(x)\Big)^{2}u+V(x)u=f(| u|^{2})u \text{ in } \mathbb{R}^{2}, \] where $\varepsilon>0$ is a parameter, $V:\mathbb{R}^{2}\rightarrow \mathbb{R}$ and $A: \mathbb{R}^{2}\rightarrow \mathbb{R}^{2}$ are continuous functions and $f:\mathbb{R}\rightarrow \mathbb{R}$ is a $C^{1}$ function having exponential critical growth. Under a global assumption on the potential $V$, we use variational methods and Ljusternick-Schnirelmann theory to prove existence, multiplicity, concentration, and decay of nontrivial solutions for $\varepsilon>0$ small.