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Schrödinger operators with Leray-Hardy potential singular on the boundary

Abstract : 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.
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https://hal.archives-ouvertes.fr/hal-02157156
Contributor : Laurent Veron <>
Submitted on : Tuesday, February 25, 2020 - 5:32:10 PM
Last modification on : Saturday, February 29, 2020 - 1:35:39 AM

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  • HAL Id : hal-02157156, version 3
  • ARXIV : 1906.07583

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Huyuan Chen, Laurent Veron. Schrödinger operators with Leray-Hardy potential singular on the boundary. Journal of Differential Equations, Elsevier, In press. ⟨hal-02157156v3⟩

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