Existence of a positive solution for a class of Schr\"odinger logarithmic equations on exterior domains

TL;DR

This paper proves the existence of positive solutions for a class of Schrödinger logarithmic equations on exterior domains using a novel variational approach that handles the challenges posed by the logarithmic nonlinearity.

Contribution

It introduces a new method enabling the application of standard $C^1$-variational techniques to Schrödinger equations with logarithmic nonlinearities on exterior domains.

Findings

Established existence of positive solutions in exterior domains.

Applied a new variational approach suitable for logarithmic nonlinearities.

Extended the applicability of variational methods to this class of problems.

Abstract

In this paper we will prove the existence of a positive solution for a class of Schr\"odinger logarithmic equation of the form \begin{equation} \left\{\begin{aligned} -\Delta u &+ u =Q(x)u\log u^2,\;\;\mbox{in}\;\;\Omega,\nonumber &\mathcal{B}u=0 \,\,\, \mbox{on} \,\,\, \partial \Omega , \end{aligned} \right. \end{equation} where $\Omega \subset \mathbb{R}^N$, $N \geq 3$, 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.