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 $C^1$.
Categories: math.AP