Multiple positive solutions for a Schr\"{o}dinger logarithmic equation

Abstract

This article concerns with the existence of multiple positive solutions 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 continuous function with a global minimum. Using variational method, we prove that for small enough $\epsilon>0$, the "shape" of the graph of the function $V$ affects the number of nontrivial solutions.