[Paper Review] Mass and Extremals Associated with the Hardy-Schrödinger Operator on Hyperbolic Space

Hardy Chan, Nassif Ghoussoub+3 more|arXiv (Cornell University)|Oct 3, 2017

Geometric Analysis and Curvature Flows1 references3 citations

TL;DR

This paper studies the Hardy–Schrödinger operator $ L_{ ilde{\gamma}} = -\Delta_{\mathbb{B}^n} - \gamma V_2 $ on the Poincaré ball model of hyperbolic space $ \mathbb{B}^n $, $ n \geq 3 $, where $ V_2 \sim r^{-2} $ near the origin. It establishes existence of ground state solutions to the critical equation $ L_{\gamma}u = V_{2^*(s)} u^{2^*(s)-1} $ via a hyperbolic scaling transformation and introduces a notion of 'hyperbolic mass' that determines existence in dimensions 3 and 4, with explicit coercivity and positivity conditions for $ \gamma $ and $ \lambda $.

ABSTRACT

We consider the Hardy-Schrödinger operator $ -Δ_{\mathbb{B}^n}-γ{V_2}$ on the Poincaré 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 $ -Δ_{\mathbb{B}^n}-γ{V_2}$ is positive definite whenever $γ

Motivation & Objective

• To analyze the spectral and variational properties of the Hardy–Schrödinger operator $ L_{\gamma} = -\Delta_{\mathbb{B}^n} - \gamma V_2 $ on the Poincaré ball model of hyperbolic space $ \mathbb{B}^n $, $ n \geq 3 $.
• To establish existence of ground state solutions to the critical equation $ L_{\gamma}u = V_{2^*(s)} u^{2^*(s)-1} $ in domains $ \Omega \subset \mathbb{B}^n $ containing the origin.
• To introduce and analyze the concept of 'hyperbolic mass' as a necessary and sufficient condition for existence of solutions in dimensions $ n = 3 $ and $ n = 4 $, where standard coercivity fails.
• To extend the classical Euclidean scaling invariance to the hyperbolic setting using the fundamental solution of the hyperbolic Laplacian, enabling construction of explicit solutions and energy estimates.

Proposed method

• The authors define a hyperbolic scaling transformation $ u_\lambda(r) = \lambda^{-1/2} u(G^{-1}(\lambda G(r))) $, where $ G(r) = \int_r^1 f(t) dt $, $ f(r) = (1 - r^2)^{n-2} / r^{n-1} $, which preserves the $ H^1 $-norm and weighted $ L^p $-norms for radial functions.
• They relate the hyperbolic operator $ L_{\gamma} $ to a corresponding Euclidean operator $ L_{\gamma,h} $ via the change of variables $ v = \varphi u $, where $ \varphi $ is a conformal factor, enabling transfer of coercivity and positivity properties.
• The critical equation is transformed into a form $ -\Delta v - (\gamma |x|^{-2} + h_{\gamma,\lambda}(x))v = b(x) |x|^{-s} |v|^{2^*(s)-2} v $ in $ \mathbb{R}^n $, with $ b(x) $ smooth and positive near 0, allowing application of known existence theorems.
• The behavior of the potential $ h_{\gamma,\lambda}(x) $ near the origin is analyzed: for $ n \geq 5 $, it behaves like a constant; for $ n = 3 $, like $ 4\gamma / r $; for $ n = 4 $, like $ 8\gamma \log(1/|x|) $, which determines the coercivity threshold.
• The concept of 'hyperbolic mass' $ m^{H}_{\gamma,\lambda}(\Omega) $ is introduced as a renormalized limit of the energy functional, and its positivity is shown to be necessary and sufficient for existence of positive solutions in $ n = 3,4 $.
• The paper proves that $ L_{\gamma} $ is coercive if and only if the corresponding Euclidean operator $ L_{\gamma,h} $ is coercive, linking hyperbolic and Euclidean analysis through conformal invariance.

Experimental results

Research questions

• RQ1 Under what conditions does the Hardy–Schrödinger operator $ L_{\gamma} = -\Delta_{\mathbb{B}^n} - \gamma V_2 $ on $ \mathbb{B}^n $ admit a ground state solution to the critical equation $ L_{\gamma}u = V_{2^*(s)} u^{2^*(s)-1} $ in a domain $ \Omega \subset \mathbb{B}^n $ containing the origin?
• RQ2 How does the hyperbolic scaling transformation $ u_\lambda(r) = \lambda^{-1/2} u(G^{-1}(\lambda G(r))) $ preserve the energy and weighted norms, and how does it enable explicit construction of solutions?
• RQ3 What is the role of the 'hyperbolic mass' $ m^{H}_{\gamma,\lambda}(\Omega) $ in determining the existence of positive solutions in dimensions $ n = 3 $ and $ n = 4 $, where standard coercivity fails?
• RQ4 For which values of $ \gamma $ and $ \lambda $ is the operator $ L_{\gamma} - \lambda $ coercive on $ \mathbb{B}^n $, and how does this relate to the coercivity of the corresponding Euclidean operator?
• RQ5 How does the behavior of the potential $ h_{\gamma,\lambda}(x) $ near the origin—specifically $ \sim 4\gamma / r $ for $ n=3 $, $ \sim 8\gamma \log(1/|x|) $ for $ n=4 $, and a constant for $ n \geq 5 $—affect the existence and regularity of solutions?

Key findings

• For $ n \geq 5 $, the critical equation $ L_{\gamma}u = V_{2^*(s)} u^{2^*(s)-1} $ admits a ground state solution in any domain $ \Omega \subset \mathbb{B}^n $ containing the origin, provided $ 0 < \gamma \leq \frac{n(n-4)}{4} $ and $ \lambda > \frac{n-2}{n-4}\left(\frac{n(n-4)}{4} - \gamma\right) $, ensuring coercivity of $ L_{\gamma} - \lambda $.
• In dimension $ n = 3 $, a positive solution exists if and only if the hyperbolic mass $ m^{H}_{\gamma,\lambda}(\Omega) $ is positive, and this requires $ \gamma > 0 $, with $ h_{\gamma,\lambda}(x) \sim 4\gamma / r $ near the origin.
• In dimension $ n = 4 $, a positive solution exists if and only if the hyperbolic mass $ m^{H}_{\gamma,\lambda}(\Omega) $ is positive, and this requires $ \gamma > 0 $, with $ h_{\gamma,\lambda}(x) \sim 8\gamma \log(1/|x|) $ near the origin, though $ \gamma < 0 $ is required for coercivity in the standard case, making the mass condition essential.
• The hyperbolic mass $ m^{H}_{\gamma,\lambda}(\Omega) $ is a positive multiple of the mass of the corresponding Euclidean operator, so their signs are identical, and it is invariant under the hyperbolic scaling transformation.
• The operator $ L_{\gamma} $ is positive definite if and only if $ \gamma < \frac{(n-2)^2}{4} $, which is the sharp Hardy constant on $ \mathbb{B}^n $, analogous to the Euclidean case.
• The scaling invariance of the energy and weighted norms under $ u_\lambda $ ensures that the critical equation's solutions are preserved under this transformation, enabling the construction of explicit extremals via conformal transformation to the Euclidean setting.