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 $N\ge 1$, $\lambda >0$ is a parameter and the nonnegative continuous function $V:{\mathbb{R}}^N\to \mathbb{R}$ 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.

Accepted for Publication in SCIENCE CHINA Mathematics