The Hardy--Schr\"odinger Operator on the Poincar\'e Ball: Compactness and Multiplicity

Let $\mathrm{\Omega }$ be a compact smooth domain containing zero in the Poincar'e ball model of the Hyperbolic space ${\mathbb{B}}^n$ ($n\ge 3$) and let $-{\mathrm{\Delta }}_{{\mathbb{B}}^n}$ be the Laplace-Beltrami operator on ${\mathbb{B}}^n$, associated with the metric $g_{$\mathbb{B}$^{n}}= \frac{4}{(1-|x|^{2})^2}g_{{\hbox{Eucl}}}$. We consider issues of non-existence, existence, and multiplicity of variational solutions for the borderline Dirichlet problem, \begin{eqnarray*} (E)~ \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*} where $0\leq \gamma \leq \frac{(n-2)^2}{4}$,$0< s <2$,${2^\star(s)}:=\frac{2(n-s)}{n-2}$ is the corresponding critical Sobolev exponent, $V_2$ (resp., $V_{2^{\star }(s)}$) is a Hardy-type potential (resp., Hardy-Sobolev weight) that is invariant under hyperbolic scaling and which behaves like $\frac{1}{r^2}$ (resp., $\frac{1}{r^s}$) at the origin. The bulk of this paper is a sharp blow-up analysis on approximate solutions of $(E)$ with bounded but arbitrary high energies. Our analysis leads to existence of positive ground state solutions for $(E)$, whenever $n\ge 4$, $0\le \gamma \le \frac{(n-2{)}^2}{4}-1$ and $\lambda >0$. The latter result also holds true for $n\ge 3$ and $\gamma > \frac{(n-2)^2}{4}-1$ provided the domain has a positive “hyperbolic mass”. On the other hand, the same analysis yields that if $\gamma >\frac{(n-2{)}^2}{4}-1$ and the mass is non vanishing, then there is a surprising stability of regimes where no variational positive solution exists. As for higher energy solutions to $(E)$, we show that there are infinitely many of them provided $n\ge 5$, $0\le \gamma <\frac{(n-2{)}^2}{4}-4$ and $ \lambda > \frac{n-2}{n-4} \left(\frac{n(n-4)}{4}-\gamma \right)$.