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 \[ \left(\frac{\varepsilon}{i}\nabla-A(x)\right)^{2}u+V(x)u=f(|u|^{2})u\,\ \text{in}\,\ \mathbb{R}^{2}, \] where $\varepsilon>0$ 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 $\varepsilon>0$. 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.