[Existence of multiple solutions for a Schr\"ondiger logarithmic equation

Abstract

This paper concerns the existence of multiple solutions for a Schr\"odinger logarithmic equation of the form \begin{equation} \left\{\begin{aligned} -\varepsilon^2\Delta u + 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 $V:\mathbb{R}^N\longrightarrow \mathbb{R}$ is a continuous function that satisfies some technical conditions and $\varepsilon$ is a positive parameter. We will establish the multiplicity of solution for $(P_\varepsilon)$ by using the notion of Lusternik-Schnirelmann category, by introducing a new function space where the energy functional is $C^1$.