Schrödinger operators with Leray-Hardy potential singular on the boundary

Huyuan Chen (1) , Laurent Veron (2)

1 Jiangxi Normal University
2 LMPT - Laboratoire de Mathématiques et Physique Théorique

We study the kernel function of the operator u → L µ u = −∆u + 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.

Mots clés: Hardy Potential; Harnack inequality; limit set; Radon Measure MSC2010: 35B44; 35J75

Domaines: Equations aux dérivées partielles [math.AP]

Huyuan Chen, Laurent Veron. Schrödinger operators with Leray-Hardy potential singular on the boundary. 2019. ⟨hal-02157156v2⟩