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

Abstract

We consider the Hardy-Schrödinger operator $-{\Delta }_{{\mathbb{B}}^n}-\gamma V_2$ on the Poincaré ball model of the Hyperbolic space ${\mathbb{B}}^n$ ($n\ge 3$). 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 $-{\Delta }_{{\mathbb{B}}^n}-\gamma V_2$ is positive definite whenever $\gamma <\frac{(n-2{)}^2}{4}$, in which case we exhibit explicit solutions for the equation $$-{\Delta }_{{\mathbb{B}}^n}u-\gamma V_{2}u=V_{2^{\ast }(s)}{u}^{2^{\ast }(s)-1}\text{ in }{\mathbb{B}}^n,$$ where $0\le s<2$, $2^{\ast }(s)=\frac{2(n-s)}{n-2}$, and $V_{2^{\ast }(s)}$ is a weight that behaves like $\frac{1}{r^s}$ around $0$. The same equation, on bounded domains $\Omega$ of ${\mathbb{B}}^n$ containing $0$ but not touching the hyperbolic boundary, has positive solutions if $0<\gamma \le \frac{(n-2{)}^2}{4}-\frac{1}{4}$. However, if $\frac{(n-2{)}^2}{4}-\frac{1}{4}<\gamma <\frac{(n-2{)}^2}{4}$, the existence of solutions requires the positivity of the "hyperbolic Hardy mass" $m_{{}_{{\mathbb{B}}^n}}(\Omega )$ of the domain, a notion that we introduce and analyse therein.

21 pages, The previous version of this article contained a consequential error in one of the estimates. This is a corrected and revised version. Updated version - if any - can be downloaded at http://www.birs.ca/~nassif/