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

In this paper we study the following nonlinear Schr"{o}dinger equation with magnetic field [ Big(rac{arepsilon}{i} abla-A(x)Big)^{2}u+V(x)u=f(| u|^{2})u,quad xinmathbb{R}^{2}, ] where $arepsilon>0$ is a parameter, $V:mathbb{R}^{2} ightarrow mathbb{R}$ and $A: mathbb{R}^{2} ightarrow mathbb{R}^{2}$ are continuous potentials and $f:mathbb{R}^{2} ightarrow mathbb{R}$ 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 $arepsilon$ small.