Source Title:
Advanced Nonlinear Studies
|
Volume/Issue:
Volume 18: Issue 4

Mass and Extremals Associated with the Hardy–Schrödinger Operator on Hyperbolic Space

Hardy Chan 1 , Nassif Ghoussoub 2 , Saikat Mazumdar 1 , Shaya Shakerian 1 , and Luiz Fernando de Oliveira Faria 3
  • 1 Department of Mathematics, The University of British Columbia, BC, V6T 1Z2, Vancouver, Canada
  • 2 Department of Mathematics, The University of British Columbia, BC, V6T 1Z2, Vancouver, Canada
  • 3 Departamento de Matemática, Universidade Federal de Juiz de Fora, Juiz de Fora, Brazil
Hardy Chan, Nassif Ghoussoub
  • Corresponding author
  • Department of Mathematics, The University of British Columbia, Vancouver, BC, V6T 1Z2, Canada
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, Saikat Mazumdar, Shaya Shakerian and Luiz Fernando de Oliveira Faria
DOI:
https://doi.org/10.1515/ans-2018-2025
|
Published online:
07 Jul 2018

Abstract

We consider the Hardy–Schrödinger operator Lγ:=-Δ𝔹n-γ⁢V2 on the Poincaré ball model of the hyperbolic space 𝔹n (n≥3). Here V2 is a radially symmetric potential, which behaves like the Hardy potential around its singularity at 0, i.e., V2⁢(r)∼1r2. As in the Euclidean setting, Lγ is positive definite whenever γ<(n-2)24, in which case we exhibit explicit solutions for the critical equation Lγ⁢u=V2*⁢(s)⁢u2*⁢(s)-1 in 𝔹n, where 0≤s<2, 2*⁢(s)=2⁢(n-s)n-2, and V2*⁢(s) is a weight that behaves like 1rs around 0. In dimensions n≥5, the equation Lγ⁢u-λ⁢u=V2*⁢(s)⁢u2*⁢(s)-1 in a domain Ω of 𝔹n away from the boundary but containing 0 has a ground state solution, whenever 0<γ≤n⁢(n-4)4, and λ>n-2n-4⁢(n⁢(n-4)4-γ). On the other hand, in dimensions 3 and 4, the existence of solutions depends on whether the domain has a positive “hyperbolic mass” a notion that we introduce and analyze therein.

  • [1]

    Adimurthi and A. Sekar, Role of the fundamental solution in Hardy–Sobolev-type inequalities, Proc. Roy. Soc. Edinburgh Sect. A 136 (2006), no. 6, 1111–1130.

    • Crossref
    • Export Citation
  • [2]

    K. Akutagawa and H. Kumura, Geometric relative Hardy inequalities and the discrete spectrum of Schrödinger operators on manifolds, Calc. Var. Partial Differential Equations 48 (2013), no. 1–2, 67–88.

    • Crossref
    • Export Citation
  • [3]

    W. Beckner, On Hardy–Sobolev embedding, preprint (2009), https://arxiv.org/abs/0907.3932v1.

  • [4]

    M. Bhakta and K. Sandeep, Poincaré–Sobolev equations in the hyperbolic space, Calc. Var. Partial Differential Equations 44 (2012), no. 1–2, 247–269.

    • Crossref
    • Export Citation
  • [5]

    M. Bonforte, F. Gazzola, G. Grillo and J. L. Vázquez, Classification of radial solutions to the Emden–Fowler equation on the hyperbolic space, Calc. Var. Partial Differential Equations 46 (2013), no. 1–2, 375–401.

    • Crossref
    • Export Citation
  • [6]

    L. Caffarelli, R. V. Kohn and L. Nirenberg, First order interpolation inequalities with weights, Compos. Math. 53 (1984), no. 3, 259–275.

  • [7]

    G. Carron, Inégalités de Hardy sur les variétés riemanniennes non-compactes, J. Math. Pures Appl. (9) 76 (1997), no. 10, 883–891.

    • Crossref
    • Export Citation
  • [8]

    L. D’Ambrosio and S. Dipierro, Hardy inequalities on Riemannian manifolds and applications, Ann. Inst. H. Poincaré Anal. Non Linéaire 31 (2014), no. 3, 449–475.

    • Crossref
    • Export Citation
  • [9]

    B. Devyver, M. Fraas and Y. Pinchover, Optimal Hardy weight for second-order elliptic operator: An answer to a problem of Agmon, J. Funct. Anal. 266 (2014), no. 7, 4422–4489.

    • Crossref
    • Export Citation
  • [10]

    D. Ganguly and K. Sandeep, Sign changing solutions of the Brezis–Nirenberg problem in the hyperbolic space, Calc. Var. Partial Differential Equations 50 (2014), no. 1–2, 69–91.

    • Crossref
    • Export Citation
  • [11]

    N. Ghoussoub and F. Robert, Sobolev inequalities for the Hardy–Schrödinger operator: Extremals and critical dimensions, Bull. Math. Sci. 6 (2016), no. 1, 89–144.

    • Crossref
    • Export Citation
  • [12]

    N. Ghoussoub and F. Robert, The Hardy–Schrödinger operator with interior singularity: The remaining cases, Calc. Var. Partial Differential Equations 56 (2017), no. 5, Article ID 149.

  • [13]

    I. Kombe and M. Ozaydin, Improved Hardy and Rellich inequalities on Riemannian manifolds, Trans. Amer. Math. Soc. 361 (2009), no. 12, 6191–6203.

    • Crossref
    • Export Citation
  • [14]

    P. Li and J. Wang, Weighted Poincaré inequality and rigidity of complete manifolds, Ann. Sci. Éc. Norm. Supér. (4) 39 (2006), no. 6, 921–982.

    • Crossref
    • Export Citation
  • [15]

    G. Mancini and K. Sandeep, On a semilinear equation in ℍ n {\mathbb{H}^{n}}, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 7 (2008), no. 4, 635–671.

  • [16]

    Y. Pinchover and K. Tintarev, A ground state alternative for singular Schrödinger operators, J. Funct. Anal. 230 (2006), no. 1, 65–77.

    • Crossref
    • Export Citation
  • [17]

    P. Quitnner and P. Souplet, Superlinear Parabolic Problems. Blow-up, Global Existence and Steady States, Birkhäuser Adv. Texts Basler Lehrbücher, Birkhäuser, Basel, 2007.

  • [18]

    K. Sandeep and C. Tintarev, A subset of Caffarelli–Kohn–Nirenberg inequalities in the hyperbolic space ℍ N {\mathbb{H}^{N}}, Ann. Mat. Pura Appl. (4) 196 (2017), no. 6, 2005–2021.

    • Crossref
    • Export Citation
  • [19]

    J. L. Vazquez, Fundamental solution and long time behaviour of the porous medium equation in hyperbolic space, J. Math. Pures Appl. (9) 104 (2015), no. 3, 454–484.

    • Crossref
    • Export Citation
  • [20]

    Q. Yang, D. Su and Y. Kong, Hardy inequalities on Riemannian manifolds with negative curvature, Commun. Contemp. Math. 16 (2014), no. 2, Article ID 1350043.

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Article information

Received:
2018-04-27
Revised:
2018-06-11
Accepted:
2018-06-11
Published Online:
2018-07-07
Published in Print:
2018-11-01

Funding Source:
Natural Sciences and Engineering Research Council of Canada
Award identifier / Grant number:
22R8-4808

This work was done while Hardy Chan was completing his PhD thesis, and Saikat Mazumdar and Shaya Shakerian were holding postdoctoral positions at the University of British Columbia, under the supervision of Nassif Ghoussoub. All were partially supported by a grant from the Natural Sciences and Engineering Research Council of Canada (22R8-4808). During his visit to UBC, Luiz Fernando de Oliveira Faria was partially supported by CAPES/Brazil (Proc. 6129/2015-03).

Citation Information: 
Advanced Nonlinear Studies, Volume 18, Issue 4, Pages 671–689, eISSN 2169-0375, ISSN 1536-1365, DOI: https://doi.org/10.1515/ans-2018-2025.

© 2018 Walter de Gruyter GmbH, Berlin/Boston.

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Journal + Issues

Advanced Nonlinear Studies

Online ISSN:
2169-0375
First published:
01 Aug 2001
Language:
English
Publisher:
De Gruyter

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