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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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

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

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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

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


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.

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  • A. Attar, S. Merchán, and I. Peral, “A remark on the existence properties of a semilinear heat equation involving a Hardy–Leray potential”, J. Evol. Equ. 15:1 (2015), 239–250.
  • T. Aubin, “Problèmes isopérimétriques et espaces de Sobolev”, J. Differential Geometry 11:4 (1976), 573–598.
  • T. Bartsch, S. Peng, and Z. Zhang, “Existence and non-existence of solutions to elliptic equations related to the Caffarelli–Kohn–Nirenberg inequalities”, Calc. Var. Partial Differential Equations 30:1 (2007), 113–136.
  • L. Caffarelli, R. Kohn, and L. Nirenberg, “First order interpolation inequalities with weights”, Compositio Math. 53:3 (1984), 259–275.
  • J.-L. Chern and C.-S. Lin, “The symmetry of least-energy solutions for semilinear elliptic equations”, J. Differential Equations 187:2 (2003), 240–268.
  • J.-L. Chern and C.-S. Lin, “Minimizers of Caffarelli–Kohn–Nirenberg inequalities with the singularity on the boundary”, Arch. Ration. Mech. Anal. 197:2 (2010), 401–432.
  • C. Cowan, “Optimal Hardy inequalities for general elliptic operators with improvements”, Commun. Pure Appl. Anal. 9:1 (2010), 109–140.
  • J. Dávila and I. Peral, “Nonlinear elliptic problems with a singular weight on the boundary”, Calc. Var. Partial Differential Equations 41:3-4 (2011), 567–586.
  • O. Druet, “Elliptic equations with critical Sobolev exponents in dimension 3”, Ann. Inst. H. Poincaré Anal. Non Linéaire 19:2 (2002), 125–142.
  • O. Druet, “Optimal Sobolev inequalities and extremal functions: the three-dimensional case”, Indiana Univ. Math. J. 51:1 (2002), 69–88.
  • M. M. Fall, “On the Hardy–Poincaré inequality with boundary singularities”, Commun. Contemp. Math. 14:3 (2012), art. id. 1250019.
  • M. M. Fall and R. Musina, “Hardy–Poincaré inequalities with boundary singularities”, Proc. Roy. Soc. Edinburgh Sect. A 142:4 (2012), 769–786.
  • N. Ghoussoub and X. S. Kang, “Hardy–Sobolev critical elliptic equations with boundary singularities”, Ann. Inst. H. Poincaré Anal. Non Linéaire 21:6 (2004), 767–793.
  • N. Ghoussoub and A. Moradifam, Functional inequalities: new perspectives and new applications, Mathematical Surveys and Monographs 187, American Mathematical Society, Providence, RI, 2013.
  • N. Ghoussoub and F. Robert, “Concentration estimates for Emden–Fowler equations with boundary singularities and critical growth”, IMRP Int. Math. Res. Pap. (2006), art. id. 21867.
  • N. Ghoussoub and F. Robert, “The effect of curvature on the best constant in the Hardy–Sobolev inequalities”, Geom. Funct. Anal. 16:6 (2006), 1201–1245.
  • N. Ghoussoub and F. Robert, “Sobolev inequalities for the Hardy–Schrödinger operator: extremals and critical dimensions”, Bull. Math. Sci. 6:1 (2016), 89–144.
  • D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, revised 2nd ed., Grundlehren der Math. Wissenschaften 224, Springer, Berlin, 1998.
  • A. Gmira and L. Véron, “Boundary singularities of solutions of some nonlinear elliptic equations”, Duke Math. J. 64:2 (1991), 271–324.
  • E. Hebey, Introduction à l'analyse non linéaire sur les variétés, Diderot Editeur, Paris, 1997.
  • H. Jaber, “Hardy–Sobolev equations on compact Riemannian manifolds”, Nonlinear Anal. 103 (2014), 39–54.
  • C.-S. Lin and H. Wadade, “Minimizing problems for the Hardy–Sobolev type inequality with the singularity on the boundary”, Tohoku Math. J. $(2)$ 64:1 (2012), 79–103.
  • Y. Pinchover, “On positive Liouville theorems and asymptotic behavior of solutions of Fuchsian type elliptic operators”, Ann. Inst. H. Poincaré Anal. Non Linéaire 11:3 (1994), 313–341.
  • Y. Pinchover and K. Tintarev, “Existence of minimizers for Schrödinger operators under domain perturbations with application to Hardy's inequality”, Indiana Univ. Math. J. 54:4 (2005), 1061–1074.
  • F. Robert, “Existence et asymptotiques optimales des fonctions de Green des opérateurs elliptiques d'ordre deux”, unpublished notes, 2010, hook \posturlhook.
  • R. Schoen, “Conformal deformation of a Riemannian metric to constant scalar curvature”, J. Differential Geom. 20:2 (1984), 479–495.
  • R. Schoen and S.-T. Yau, “Conformally flat manifolds, Kleinian groups and scalar curvature”, Invent. Math. 92:1 (1988), 47–71.