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

Abstract

Let $\Omega$ be a compact smooth domain containing zero in the Poincaré ball model of the Hyperbolic space ${\mathbb{B}}^n$ ($n\ge 3$) and let $-{\Delta }_{{\mathbb{B}}^n}$ be the Laplace-Beltrami operator on ${\mathbb{B}}^n$, associated with the metric $g_{{\mathbb{B}}^n}=\frac{4}{(1-|x{|}^2{)}^2}{g}_{{}_{Eucl}}$. We consider issues of non-existence, existence, and multiplicity of variational solutions for the borderline Dirichlet problem, \begin{eqnarray*} (E)~ \left\{ \begin{array}{lll} -Δ_{\mathbb{B}^{n}}u-γ{V_2}u -λu&=V_{2^\star(s)}|u|^{2^\star(s)-2}u &\hbox{ in }Ω\\ \hfill u &=0 & \hbox{ on } \partial Ω, \end{array} \right. \end{eqnarray*} where $0\le \gamma \le \frac{(n-2{)}^2}{4}$, $0<s<2$, $2^{\star }(s):=\frac{2(n-s)}{n-2}$ 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 $\lambda >\frac{n-2}{n-4}\left(\frac{n(n-4)}{4}-\gamma \right)$.