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

Abstract

In this paper we study the following nonlinear Schrödinger 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 $\epsilon >0$ is a parameter, $V:{\mathbb{R}}^2\to \mathbb{R}$ and $A:{\mathbb{R}}^2\to {\mathbb{R}}^2$ are continuous potentials and $f:\mathbb{R}\to \mathbb{R}$ 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 $\epsilon$ small.

25 pages