Existence of a positive solution for a logarithmic Schrödinger equation with saddle-like potential

Abstract: In this article we use the variational method developed by Szulkin \cite{szulkin} to prove the existence of a positive solution for the following logarithmic Schr\"{o}dinger equation $$ \left{ \begin{array}{lc} -{\epsilon}^2\Delta u+ V(x)u=u \log u^2, & \mbox{in} \quad \mathbb{R}^{N}, \ %u(x)>0, & \mbox{in} \quad \mathbb{R}^{N} \ u \in H^1(\mathbb{R}^{N}), & \; \ \end{array} \right. $$ where $\epsilon >0, N \geq 1$ and $V$ is a saddle-like potential.