Analysis & PDE

  • Anal. PDE
  • Volume 10, Number 5 (2017), 1017-1079.

Hardy-singular boundary mass and Sobolev-critical variational problems

Nassif Ghoussoub and Frédéric Robert

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Abstract

We investigate the Hardy–Schrödinger operator Lγ = Δ γ|x|2 on smooth domains Ω 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 Lγu = u2(s)1 |x|s , where γ < 1 4n2 , s [0,2) and 2(s) := 2(n s)(n 2) 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 γ = 1 4(n2 1) turns out to be a critical threshold for the operator Lγ. When 1 4(n2 1) < γ < 1 4n2 , a notion of Hardy singular boundary mass mγ(Ω) associated to the operator Lγ can be assigned to any conformally bounded domain Ω such that 0 Ω. 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 γ = 0, and by Chern and Lin for the case γ < 1 4(n 2)2 . More specifically, we show that extremals exist when 0 γ 1 4(n2 1) if the mean curvature of Ω at 0 is negative. On the other hand, if 1 4(n2 1) < γ < 1 4n2 , extremals then exist whenever the Hardy singular boundary mass mγ(Ω) of the domain is positive.

Article information

Source
Anal. PDE, Volume 10, Number 5 (2017), 1017-1079.

Dates
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
https://projecteuclid.org/euclid.apde/1510843492

Digital Object Identifier
doi:10.2140/apde.2017.10.1017

Mathematical Reviews number (MathSciNet)
MR3668583

Zentralblatt MATH identifier
1379.35077

Subjects
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

Keywords
Hardy–Schrödinger operator Hardy-singular boundary mass Hardy–Sobolev inequalities mean curvature

Citation

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


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