The Hardy-Schrödinger operator on the Poincaré Ball: compactness and multiplicity

Nassif Ghoussoub (1) , Saikat Mazumdar (1) , Frédéric Robert (2)

1 UBC Mathematics - Department of Mathematics [Vancouver]
2 IECL - Institut Élie Cartan de Lorraine

Let $\Omega$ be a compact smooth domain containing zero in the Poincar\'e ball model of the Hyperbolic space $\mathbb{B}_n$ ($n \geq 3$) and let $-\Delta_{\mathbb{B}_n}$ be the Laplace-Beltrami operator on $\mathbb{B}_n$, associated with the metric $g_{\mathbb{B}_n}= \frac{4}{(1-|x|^{2})^2}g_{_{\hbox{Eucl}}}$. 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^*(s)}|u|^{2^*(s)-2}u &\hbox{ in }\Omega \\ \hfill u &=0 & \hbox{ on } \partial \Omega, \end{array} \right. \end{eqnarray} where $0\leq \gamma \leq \frac{(n-2)^2}{4}$, $0< s <2$, $2^*(s):=\frac{2(n-s)}{n-2}$ is the corresponding critical Sobolev exponent, $V_{2}$ (resp., $V_{2^*(s)}$) 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 that we perform on approximate solutions of (E) with bounded but arbitrary high energies. When these approximate solutions are positive, our analysis leads to improvements of results in [6] regarding positive ground state solutions for (E), as we show that they exist whenever $n \geq 4$, $0 \leq \gamma \leq \frac{(n-2)^2}{4}-1$ and $ \lambda > 0$. The latter result also holds true for $n\geq 3$ 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 $n\geq 5$, $0\leq \gamma<\frac{(n-2)^2}{4}-4$ and $ \lambda > \frac{n-2}{n-4} \left(\frac{n(n-4)}{4}-\gamma \right)$.

Domains: Analysis of PDEs [math.AP]

Nassif Ghoussoub, Saikat Mazumdar, Frédéric Robert. The Hardy-Schrödinger operator on the Poincaré Ball: compactness and multiplicity. 2021. ⟨hal-01770803v2⟩