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

2019/08/24 by Alves, Claudianor O., Ji, Chao
#35A15 #35B09 #35J10 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1908.09153

Abstract

This article concerns the existence of multi-bump positive solutions for the following logarithmic Schrödinger equation \ -Δu+ λV(x)u=u log u², · amp; in ℝ^N,
u ∈ H¹(ℝ^N), . where N ≥ 1, λ>0 is a parameter and the nonnegative continuous function V: ℝ^N→ ℝ has a potential well Ω: =int V^-1(0) which possesses k disjoint bounded components Ω=\bigcup_j=1^kΩ_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.

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