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

• Published in: Advanced nonlinear studies Vol. 18; no. 4; pp. 671 - 689
• Main Authors: Chan Hardy, Ghoussoub Nassif, Mazumdar Saikat, Shakerian Shaya, de Oliveira Faria Luiz Fernando
• Format: Journal Article
• Language: English
• Published: De Gruyter 01-11-2018
• Subjects: 35j20; 35j75; 53c25; hardy–schrödinger operator; hardy–sobolev inequalities; hyperbolic mass; poincaré ball
• ISSN: 1536-1365; 2169-0375
• DOI: 10.1515/ans-2018-2025

We consider the Hardy–Schrödinger operator Lγ:=-Δ𝔹n-γ⁢V2{L_{\gamma}:=-\Delta_{\mathbb{B}^{n}}-\gamma{V_{2}}} on the Poincaré ball model of the hyperbolic space 𝔹n{\mathbb{B}^{n}} (n≥3{n\geq 3}). Here V2{V_{2}} is a radially symmetric potential, which behaves like the Hardy potential around its singularity at 0, i.e., V2⁢(r)∼1r2{V_{2}(r)\sim\frac{1}{r^{2}}}. As in the Euclidean setting, Lγ{L_{\gamma}} is positive definite whenever γ<(n-2)24{\gamma<\frac{(n-2)^{2}}{4}}, in which case we exhibit explicit solutions for the critical equation Lγ⁢u=V2*⁢(s)⁢u2*⁢(s)-1{L_{\gamma}u=V_{2^{*}(s)}u^{2^{*}(s)-1}} in 𝔹n,{\mathbb{B}^{n},} where 0≤s<2{0\leq s<2}, 2*⁢(s)=2⁢(n-s)n-2{2^{*}(s)=\frac{2(n-s)}{n-2}}, and V2*⁢(s){V_{2^{*}(s)}} is a weight that behaves like 1rs{\frac{1}{r^{s}}} around 0. In dimensions n≥5{n\geq 5}, the equation Lγ⁢u-λ⁢u=V2*⁢(s)⁢u2*⁢(s)-1{L_{\gamma}u-\lambda u=V_{2^{*}(s)}u^{2^{*}(s)-1}} in a domain Ω of 𝔹n{\mathbb{B}^{n}} away from the boundary but containing 0 has a ground state solution, whenever 0<γ≤n⁢(n-4)4{0<\gamma\leq\frac{n(n-4)}{4}}, and λ>n-2n-4⁢(n⁢(n-4)4-γ){\lambda>\frac{n-2}{n-4}(\frac{n(n-4)}{4}-\gamma)}. 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.