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

This article concerns the existence of multi-bump positive solutions for the following logarithmic Schrödinger equation: { u + V ( x ) u = u log u 2 i n R N , u H 1 ( R N ) , where N ⩾ 1, ⋋ > 0 is a parameter and the nonnegative continuous function V : ℝ N → ℝ has potential well Ω:= int V −1 (0) which possesses k disjoint bounded components Ω= ∪ j = 1 k Ω j . Using the variational methods, we prove that if the parameter ⋋ > 0 is large enough, then the equation has at least 2 k − 1 multi-bump positive solutions.