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 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.