The Hardy--Schrödinger Operator on the Poincaré Ball: Compactness and Multiplicity

Nassif Ghoussoub, Saikat Mazumdar, Frédéric Robert

Published 2018-04-17Version 1

Let $\Omega$ be a compact smooth domain in the Poincar\'e ball model of the Hyperbolic space $\mathbb{B}^n$, $n \geq 5$. Let $0< s <2$ and write $2^{\star}(s):=\frac{2(n-s)}{n-2}$ for the corresponding critical Sobolev exponent. We show that if $\gamma<\frac{(n-2)^2}{4}-4$ and $ \lambda > \frac{n-2}{n-4} \left(\frac{n(n-4)}{4}-\gamma \right)$, then the Dirichlet boundary value problem: \begin{eqnarray*} \left\{ \begin{array}{lll} -\Delta_{\mathbb{B}^n}u-\gamma{V_2}u -\lambda u&=V_{2^{\star}(s)}|u|^{2^{\star}(s)-2}u &\hbox{ in }\Omega \hfill u &=0 & \hbox{ on } \partial \Omega, \end{array} \right. \end{eqnarray*} has infinitely many solutions. Here $-\Delta_{\mathbb{B}^n}$ is the Laplace-Beltrami operator on $\mathbb{B}^n$, $V_{2}$ is a Hardy-type potential that behaves like $\frac{1}{r^{2}}$ at the origin, while $V_{2^{\star}(s)}$ is the Hardy-Sobolev weight, which behaves like $\frac{1}{r^{s}}$ at the origin. The solutions belong to $ C^{2}(\overline{\Omega}\setminus\{0\})$, while around $0$ they behave like \begin{align*} u(x) \sim \frac{c(n,\gamma)}{|x|^{\frac{n-2}{2}-\sqrt{\frac{(n-2)^{2}}{4}-\gamma}}}, \end{align*} where $c(n,\gamma)$ is some constant in $\mathbb{R}.$