Existence and concentrate behavior of Schrödinger equations with critical exponential growth in $\mathbb{R}^N$

Abstract

We consider the nonlinear Schrödinger equation \begin{equation*} -\Delta u + (1+\mu g(x))u = f(u) \quad \text{in } \mathbb{R}^N, \end{equation*} where $N \ge 3$, $\mu \ge 0$; the function $g \ge 0$ has a potential well and $f$ has critical growth. By using variational methods, the existence and concentration behavior of the ground state solution are obtained.