Hardy-singular boundary mass and Sobolev-critical variational problems

Abstract

We investigate the Hardy–Schrödinger operator $L_{\gamma }=-\Delta -\gamma ∕|x{|}^2$ on smooth domains $\Omega \subset {ℝ}^n$ whose boundaries contain the singularity $0$. We prove a Hopf-type result and optimal regularity for variational solutions of corresponding linear and nonlinear Dirichlet boundary value problems, including the equation $L_{{}_{\gamma }}u=u^{2^{\star }\left(s\right)-1}∕|x{|}^s$, where $\gamma <\frac{1}{4}{n}^2$, $s\in \left[0,2\right)$ and $2^{\star }\left(s\right):=2\left(n-s\right)∕\left(n-2\right)$ is the critical Hardy–Sobolev exponent. We also give a complete description of the profile of all positive solutions — variational or not — of the corresponding linear equation on the punctured domain. The value $\gamma =\frac{1}{4}\left(n^2-1\right)$ turns out to be a critical threshold for the operator $L_{\gamma }$. When $\frac{1}{4}\left(n^2-1\right)<\gamma <\frac{1}{4}{n}^2$, a notion of Hardy singular boundary mass $m_{\gamma }\left(\Omega \right)$ associated to the operator $L_{\gamma }$ can be assigned to any conformally bounded domain $\Omega$ such that $0\in \partial \Omega$. As a byproduct, we give a complete answer to problems of existence of extremals for Hardy–Sobolev inequalities, and consequently for those of Caffarelli, Kohn and Nirenberg. These results extend previous contributions by the authors in the case $\gamma =0$, and by Chern and Lin for the case $\gamma <\frac{1}{4}{\left(n-2\right)}^2$. More specifically, we show that extremals exist when $0\le \gamma \le \frac{1}{4}\left(n^2-1\right)$ if the mean curvature of $\partial \Omega$ at $0$ is negative. On the other hand, if $\frac{1}{4}\left(n^2-1\right)<\gamma <\frac{1}{4}{n}^2$, extremals then exist whenever the Hardy singular boundary mass $m_{\gamma }\left(\Omega \right)$ of the domain is positive.