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

In this article we use the variational method developed by Szulkin (Ann Inst H Poincaré Anal Non Linéire 3:77–109, 1986) to prove the existence of a positive solution for the following logarithmic Schrödinger equation - ϵ 2 Δ u + V ( x ) u = u log u 2 , in R N , u ∈ H 1 ( R N ) , where ϵ > 0 , N ≥ 1 and V is a saddle-like potential.