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: $$\left\{ {\begin{array}{*{20}{l}} { - \Delta u + \lambda V(x)u = u\log {u^2}\;\;\;in\;\;\;{\mathbb{R}^N},} \\ {u \;\in {H^1}({R^N}),} \end{array}} \right.$$ where N ⩾ 1, ⋋ > 0 is a parameter and the nonnegative continuous function V: ℝN → ℝ has potential well Ω:= int V−1(0) which possesses k disjoint bounded components $${\rm{\Omega}}\,{\rm{=}}\, \cup _{j = 1}^k{{\rm{\Omega}}_j}$$ . Using the variational methods, we prove that if the parameter ⋋ > 0 is large enough, then the equation has at least 2k − 1 multi-bump positive solutions.