On the Hardy-Schrödinger operator with a boundary singularity
Résumé
We investigate the Hardy-Schrödinger operator Lγ = −∆ − γ |x| 2 on domains Ω ⊂ R n , whose boundary contain the singularity 0. The situation is quite different from the well-studied case when 0 is in the interior of Ω. For one, if 0 ∈ Ω, then Lγ is positive if and only if γ < (n−2) 2 4 , while if 0 ∈ ∂Ω the operator Lγ could be positive for larger value of γ, potentially reaching the maximal constant n 2 4 on convex domains. We prove optimal regularity and a Hopf-type Lemma for variational solu-tions of corresponding linear Dirichlet boundary value problems of the form Lγ u = a(x)u, but also for non-linear equations including L γ u = |u| 2 (s)−2 u |x| s , where γ < n 2 4 , s ∈ [0, 2) and 2 (s) := 2(n−s) n−2 is the critical Hardy-Sobolev exponent. We also provide a Harnack inequality and a complete description of the profile of all positive solutions –variational or not– of the corresponding linear equation on the punctured domain. The value γ = n 2 −1 4 turned out to be another critical threshold for the operator Lγ , and our analysis yields a corresponding notion of "Hardy singular boundary-mass" mγ (Ω) of a domain Ω having 0 ∈ ∂Ω, which could be defined whenever n 2 −1 4 < γ < n 2 4 . As a byproduct, we give a complete answer to problems of existence of extremals for Hardy-Sobolev inequalities of the form C Ω u 2 (s) |x| s dx 2 2 (s) ≤ Ω |∇u| 2 dx − γ Ω u 2 |x| 2 dx for all u ∈ D 1,2 (Ω), whenever γ < n 2 4 , and in particular, for those of Caffarelli-Kohn-Nirenberg. These results extend previous contributions by the authors in the case γ = 0, and by Chern-Lin for the case γ < (n−2) 2 4 . Namely, if 0 ≤ γ ≤ n 2 −1 4 , then the negativity of the mean curvature of ∂Ω at 0 is sufficient for the existence of extremals. This is however not sufficient for n 2 −1 4 < γ < n 2 4 , which then requires the positivity of the Hardy singular boundary-mass of the domain under consideration.
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