[Existence of multiple solutions for a Schröndiger logarithmic equation

This paper concerns the existence of multiple solutions for a Schrödinger logarithmic equation of the form \begin{equation} \left{\begin{aligned} -\varepsilon^2Δu + V(x)u & =u\log u^2,;;\mbox{in};;$\mathbb{R}$^{N},\nonumber u \in H^{1}($\mathbb{R}$^{N}), \end{aligned} \right.\leqno{(P_\varepsilon)} \end{equation} where $V:{\mathbb{R}}^N⟶\mathbb{R}$ is a continuous function that satisfies some technical conditions and $\epsilon$ is a positive parameter. We will establish the multiplicity of solution for $(P_{\epsilon })$ by using the notion of Lusternik-Schnirelmann category, by introducing a new function space where the energy functional is $C^1$.