Analysis & PDE
- Anal. PDE
- Volume 10, Number 5 (2017), 1017-1079.
Hardy-singular boundary mass and Sobolev-critical variational problems
We investigate the Hardy–Schrödinger operator on smooth domains whose boundaries contain the singularity . We prove a Hopf-type result and optimal regularity for variational solutions of corresponding linear and nonlinear Dirichlet boundary value problems, including the equation , where , and 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 turns out to be a critical threshold for the operator . When , a notion of Hardy singular boundary mass associated to the operator can be assigned to any conformally bounded domain such that . 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 , and by Chern and Lin for the case . More specifically, we show that extremals exist when if the mean curvature of at is negative. On the other hand, if , extremals then exist whenever the Hardy singular boundary mass of the domain is positive.
Anal. PDE, Volume 10, Number 5 (2017), 1017-1079.
Received: 4 January 2016
Revised: 23 February 2017
Accepted: 3 April 2017
First available in Project Euclid: 16 November 2017
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 35J35: Variational methods for higher-order elliptic equations 35J60: Nonlinear elliptic equations 58J05: Elliptic equations on manifolds, general theory [See also 35-XX] 35B44: Blow-up
Ghoussoub, Nassif; Robert, Frédéric. Hardy-singular boundary mass and Sobolev-critical variational problems. Anal. PDE 10 (2017), no. 5, 1017--1079. doi:10.2140/apde.2017.10.1017. https://projecteuclid.org/euclid.apde/1510843492