Schrödinger operators with Leray-Hardy potential singular on the boundary

2019/06/18 by Chen, Huyuan, Veron, Laurent
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1906.07583

Abstract

We study the kernel function of the operator u → L μ u = --Δu + μ |x| 2 u in a bounded smooth domain Ω ⊂ R N + such that 0 ∈ ∂Ω, where μ ≥ -- N 2 4 is a constant. We show the existence of a Poisson kernel vanishing at 0 and a singular kernel with a singularity at 0. We prove the existence and uniqueness of weak solutions of L μ u = 0 in Ω with boundary data ν + kδ 0 , where ν is a Radon measure on ∂Ω 0, k ∈ R and show that this boundary data corresponds in a unique way to the boundary trace of positive solution of L μ u = 0 in Ω.

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