Multi-peak solutions for a magnetic nonlinear Schrödinger equation with exponential critical growth in ${\mathbb{R}}^2$

In this paper, we study the following magnetic nonlinear Schrödinger equation $${(\frac{\epsilon }{i}\mathrm{\nabla }-A(x))}^{2}u+V(x)u=f(\mathrm{\mid }u{\mathrm{\mid }}^2)u\text{ in}{\mathbb{R}}^2,$$ where $\epsilon >0$ is a parameter, $A\in C({\mathbb{R}}^2,{\mathbb{R}}^2)$ is a magnetic potential, $V\in C({\mathbb{R}}^2,\mathbb{R})$ is a multi-well potential, and $f\in C^1(\mathbb{R},\mathbb{R})$ exhibits exponential critical growth. By variational methods and a penalization technique, we prove the existence and multiplicity of nontrivial solutions for sufficiently small $\epsilon >0$. If, in addition, $A\in C^1({\mathbb{R}}^2,{\mathbb{R}}^2)$, we show that these solutions have multiple peaks and we establish their concentration behavior and exponential decay.