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The Hardy–Schrödinger operator on the Poincaré ball: Compactness, multiplicity, and stability of the Pohozaev obstruction
Journal
Journal of Differential Equations
ISSN
00220396
Date Issued
2022-05-25
Author(s)
Ghoussoub, Nassif
Mazumdar, Saikat
Robert, Frédéric
Abstract
Let Ω be a smooth relatively compact domain containing zero in the Poincaré ball model of the Hyperbolic space Bn (n≥3) and let −ΔBn be the Laplace-Beltrami operator on Bn. We consider problems of non-existence, existence, and multiplicity of variational solutions for the borderline Dirichlet problem, {−ΔBnu−γV2u−λu=V2⋆(s)|u|2⋆(s)−2u in Ωu=0 on ∂Ω, where [Formula presented], 0<s<2, [Formula presented] being the corresponding critical Sobolev exponent, while V2 (resp., V2⋆(s)) is a Hardy-type potential (resp., Hardy-Sobolev weight) that is invariant under hyperbolic scaling and which behaves like [Formula presented] (resp. [Formula presented]) at the origin. The bulk of this paper is a sharp blow-up analysis that we perform on certain approximate solutions of (1) with bounded but arbitrary high energies. When these approximate solutions are positive, our analysis leads to improvements of results in [5] regarding positive ground state solutions for (1), as we show that they exist whenever n≥4, [Formula presented] and λ>0. The latter result also holds true for n≥3 and [Formula presented] provided the domain has a positive “hyperbolic mass”. On the other hand, the same analysis yields that if [Formula presented] and the mass is non vanishing, then there is a surprising stability of regimes where no variational positive solution exists. As for higher energy solutions to (1), we show that there are infinitely many of them provided n≥5, [Formula presented] and [Formula presented].
Volume
320