Solutions concentrating around the saddle points of the potential for Schr\"{o}dinger equations with critical exponential growth

TL;DR

This paper studies a nonlinear Schrödinger equation with critical exponential growth in two dimensions, constructing solutions that concentrate around saddle points of the potential as a small parameter tends to zero, extending previous results.

Contribution

It introduces new variational methods to handle critical exponential growth in 2D and relaxes previous conditions on the nonlinear term and potential.

Findings

Constructed positive solutions concentrating at saddle points

Extended analysis to 2D with critical exponential growth

Relaxed conditions on the nonlinear term and potential

Abstract

In this paper, we deal with the following nonlinear Schr\"odinger 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 $\epsilon\rightarrow 0$. Our results complete the analysis made in \cite{MR2900480} and \cite{MR3426106}, where the Schr\"odinger equation was studied in $\mathbb R^N$, $N\geq 3$ 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}.