Multi-bump positive solutions for a logarithmic Schr\"{o}dinger equation with deepening potential well

Abstract

This article concerns the existence of multi-bump positive solutions for the following logarithmic Schr\"{o}dinger equation $$ \left\{ \begin{array}{lc} -\Delta u+ \lambda V(x)u=u \log u^2, & \mbox{in} \quad \mathbb{R}^{N}, \\ u \in H^1(\mathbb{R}^{N}), \\ \end{array} \right. $$ where $N \geq 1$, $\lambda>0$ is a parameter and the nonnegative continuous function $V: \mathbb{R}^{N}\rightarrow \mathbb{R}$ has a potential well $\Omega: =\text{int}\, V^{-1}(0)$ which possesses $k$ disjoint bounded components $\Omega=\bigcup_{j=1}^{k}\Omega_{j}$. Using the variational methods, we prove that if the parameter $\lambda>0$ is large enough, then the equation has at least $2^{k}-1$ multi-bump positive solutions.