Existence of a positive solution for a class of Schrödinger logarithmic equations on exterior domains

Abstract

In this paper we will prove the existence of a positive solution for a class of Schrödinger logarithmic equation of the form \begin{equation} \left\{\begin{aligned} -Δu &+ u =Q(x)u\log u^2,\;\;\mbox{in}\;\;Ω,\nonumber &\mathcal{B}u=0 \,\,\, \mbox{on} \,\,\, \partial Ω, \end{aligned} \right. \end{equation} where $\Omega \subset {\mathbb{R}}^N$, $N\ge 3$, is an \textit{exterior domain}, i.e., ${\Omega }^c={\mathbb{R}}^N\setminus \Omega$ is a bounded smooth domain where $\mathcal{B}u=u$ or $\mathcal{B}u=\frac{\partial u}{\partial \nu }$. We have used new approach that allows us to apply the usual $C^1$-variational methods to get a nontrivial solutions for these classes of problems.