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In this paper we will prove the existence of a positive solution for a classof 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{exteriordomain}, i.e., $\Omega^c=\mathbb{R}^N \setminus \Omega$ is a bounded smoothdomain 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 ofproblems.

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