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. \(\begin{aligned} \left\{ \begin{aligned} -\Delta u&+ u =Q(x)u\log u^2,\;\;\hbox {in}\;\;\Omega , \\&{\mathcal {B}}u=0 \,\,\, \hbox {on} \,\,\, \partial \Omega , \end{aligned} \right. \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill -\mathrm{\Delta }u& +u=Q(x)ulog{u}^2,\text{in}\mathrm{\Omega },\hfill \\ \hfill & \mathcal{B}u=0\text{on}\partial \mathrm{\Omega },\hfill \end{array}\right)\end{array}$$ where \(\Omega \subset {\mathbb {R}}^N\) $\mathrm{\Omega }\subset {\mathbb{R}}^N$ , \(N \ge 3\) $N\ge 3$ , is an exterior domain, i.e., \(\Omega ^c={\mathbb {R}}^N {\setminus } \Omega \) ${\mathrm{\Omega }}^c={\mathbb{R}}^N\backslash \mathrm{\Omega }$ is a bounded smooth domain where \({\mathcal {B}}u=u\) $\mathcal{B}u=u$ or \({\mathcal {B}}u=\frac{\partial u}{\partial \nu }\) $\mathcal{B}u=\frac{\partial u}{\partial \nu }$ . We have used new approach that allows us to apply the usual \(C^1\) $C^1$ -variational methods to get a nontrivial solutions for these classes of problems.