On the Hardy-Schrödinger operator with a boundary singularity

Abstract

We investigate the Hardy-Schrödinger operator $L_γ=-Δ-\fracγ{|x|^2}$ on domains $Ω\subset\rn$, 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 $0\inΩ$, then $L_γ$ is positive if and only if $γ<\frac{(n-2)^2}{4}$, while if $0\in\partialΩ$ the operator $L_γ$ could be positive for larger value of $γ$, potentially reaching the maximal constant $\frac{n^2}{4}$ on convex domains. We prove optimal regularity and a Hopf-type Lemma for variational solutions of corresponding linear Dirichlet boundary value problems of the form $L_γ u=a(x)u$, but also for non-linear equations including $L_{_γ} u=\frac{|u|^{\crits-2}u}{|x|^s}$, where $γ<\frac{n^2}{4}$, $s\in [0,2)$ and $\crits:=\frac{2(n-s)}{n-2}$ 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 $γ=\frac{n^2-1}{4}$ turned out to be another critical threshold for the operator $L_γ$, and our analysis yields a corresponding notion of "Hardy singular boundary-mass" $m_γ(Ω)$ of a domain $Ω$ having $0\in \partial Ω$, which could be defined whenever $\frac{n^2-1}{4}<γ<\frac{n^2}{4}$.