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
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  • 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

Abstract

We consider the Hardy–Schrödinger operator Lγ:=-Δ𝔹n-γV2 on the Poincaré ball model of the hyperbolic space 𝔹n (n3). 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 Lγu=V2*(s)u2*(s)-1 in 𝔹n, where 0s<2, 2*(s)=2(n-s)n-2, and V2*(s) is a weight that behaves like 1rs around 0. In dimensions n5, the equation Lγu-λu=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<γn(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.

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