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

Abstract

This paper is devoted to the magnetic nonlinear Schrödinger equation \[ \Big(\frac{\varepsilon}{i}\nabla-A(x)\Big)^{2}u+V(x)u=f(| u|^{2})u \text{ in } \mathbb{R}^{2}, \] where $\epsilon >0$ is a parameter, $V:{\mathbb{R}}^2\to \mathbb{R}$ and $A:{\mathbb{R}}^2\to {\mathbb{R}}^2$ are continuous functions and $f:\mathbb{R}\to \mathbb{R}$ 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 $\epsilon >0$ small.

35 pages