On the Hardy-Schrödinger operator with a boundary singularity

Abstract

We investigate the Hardy-Schrödinger operator $L_{\gamma }=-\Delta -\frac{\gamma }{|x{|}^2}$ 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 $\Omega$. For one, if $0\in \Omega$, then $L_{\gamma }$ is positive if and only if $\gamma <\frac{(n-2{)}^2}{4}$, while if $0\in \partial \Omega$ the operator $L_{\gamma }$ could be positive for larger value of $\gamma$, 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 $L_{\gamma }u=a(x)u$, but also for non-linear equations including $L_{_γ} u=\frac{|u|^{\crits-2}u}{|x|^s}$, where $\gamma <\frac{n^2}{4}$, $s\in [0,2)$ 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 $\gamma =\frac{n^2-1}{4}$ turned out to be another critical threshold for the operator $L_{\gamma }$, and our analysis yields a corresponding notion of "Hardy singular boundary-mass" $m_{\gamma }(\Omega )$ of a domain $\Omega$ having $0\in \partial \Omega$, which could be defined whenever $\frac{n^2-1}{4}<\gamma <\frac{n^2}{4}$.

81 pages, Updated version - if any - can be downloaded at http://www.birs.ca/~nassif/

References in corpus (1)

• Optimal Hardy Weight for Second-Order Elliptic Operator: An Answer to a Problem of Agmon

Cited by in corpus (2)

• Multiplicity and stability of the Pohozaev obstruction for Hardy-Schrödinger equations with boundary singularity
• The Hardy--Schrödinger Operator on the Poincaré Ball: Compactness and Multiplicity