Solutions concentrating around the saddle points of the potential for Schrödinger equations with critical exponential growth

Abstract

In this paper, we deal with the following nonlinear Schrödinger equation $$-{\epsilon }^2\Delta u+V(x)u=f(u),u\in H^1({\mathbb{R}}^2),$$ where $f(t)$ has critical growth of Trudinger-Moser type. By using the variational techniques, we construct a positive solution $u_{\epsilon }$ concentrating around the saddle points of the potential $V(x)$ as $\epsilon \to 0$. Our results complete the analysis made in \cite{MR2900480} and \cite{MR3426106}, where the Schrödinger equation was studied in ${\mathbb{R}}^N$, $N\ge 3$ for sub-critical and critical case respectively in the sense of Sobolev embedding. Moreover, we relax the monotonicity condition on the nonlinear term $f(t)/t$ together with a compactness assumption on the potential $V(x)$, imposed in \cite{MR3503193}.