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

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 $\mathrm{\Omega }\subset {\mathbb{R}}^N$, $N\ge 3$, is an $\mathit{\text{exterior}}\mathit{\text{domain}}$, i.e., ${\mathrm{\Omega }}^c={\mathbb{R}}^N\setminus \mathrm{\Omega }$ is a bounded smooth domain where $\mathcal{B}u=u$ or $\mathcal{B}u=\frac{\mathrm{\partial }u}{\mathrm{\partial }\nu }$. 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.