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

Abstract

We consider the Hardy-Schrödinger operator $ -Δ_{\mathbb{B}^n}-γ{V_2}$ on the Poincaré ball model of the Hyperbolic space ${\mathbb{B}^n}$ ($n \geq 3$). Here $V_2$ is a well chosen radially symmetric potential, which behaves like the Hardy potential around its singularity at $0$, i.e., $V_2(r)\sim \frac{1}{r^2}$. Just like in the Euclidean setting, the operator $ -Δ_{\mathbb{B}^n}-γ{V_2}$ is positive definite whenever $γ<\frac{(n-2)^2}{4}$, in which case we exhibit explicit solutions for the equation $$-Δ_{\mathbb{B}^n}u-γ{V_2}u=V_{2^*(s)}u^{2^*(s)-1}\quad{\text{ in }}\mathbb{B}^n,$$ where $0\leq s <2$, $2^*(s)=\frac{2(n-s)}{n-2}$, and $V_{2^*(s)}$ is a weight that behaves like $\frac{1}{r^s}$ around $0$. The same equation, on bounded domains $Ω$ of ${\mathbb{B}^n}$ containing $0$ but not touching the hyperbolic boundary, has positive solutions if $0 < γ\leq \frac{(n-2)^{2}}{4}-\frac{1}{4}$. However, if $\frac{(n-2)^{2}}{4}-\frac{1}{4}< γ< \frac{(n-2)^{2}}{4}$, the existence of solutions requires the positivity of the "hyperbolic Hardy mass" $m_{_{\mathbb{B}^n}}(Ω)$ of the domain, a notion that we introduce and analyse therein.