Mass and Extremals Associated with the Hardy-Schr\"odinger Operator on Hyperbolic Space

• Main Authors: Chan, Hardy, Ghoussoub, Nassif, Mazumdar, Saikat, Shakerian, Shaya, Faria, Luiz Fernando de Oliveira
• Format: Journal Article
• Language: English
• Published: 03-10-2017
• Subjects: Mathematics - Analysis of PDEs
• DOI: 10.48550/arxiv.1710.01271

We consider the Hardy-Schr\"odinger operator $
-\Delta_{\mathbb{B}^n}-\gamma{V_2}$ on the Poincar\'e 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 $ -\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
$$-\Delta_{\mathbb{B}^n}u-\gamma{V_2}u=V_{2^*(s)}u^{2^*(s)-1}\quad{\text{ in
}}\mathbb{B}^n,$$ 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 $\Omega$ of ${\mathbb{B}^n}$ containing $0$ but
not touching the hyperbolic boundary, has positive solutions if $0 < \gamma
\leq \frac{(n-2)^{2}}{4}-\frac{1}{4}$. 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.