Multi-bump positive solutions for a logarithmic Schrödinger equation with deepening potential well

Abstract

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