The Method of Moving Spheres on the Hyperbolic Space and the Classification of Solutions and the prescribed Q-curvature problem

TL;DR

This work develops a moving sphere method on the hyperbolic space $\mathbb{H}^n$ to study the higher-order equation $P_k u = f(u)$ with GJMS operators, leveraging a novel Kelvin transform, Helgason-Fourier analysis, and Hardy-Littlewood-Sobolev inequalities. It provides a sharp symmetry/classification result: any positive solution is either constant or radially symmetric about some point with an explicit form $u(x)=\frac{\alpha}{(\cosh^2(\frac{r}{2})+\beta)^{\frac{n-2k}{2}}}$, with the nonlinearity necessarily of critical growth when nontrivial. The paper then applies these insights to the higher order prescribed $Q$-curvature problem on $\mathbb{H}^n$, proving existence of radial positive solutions for constant $Q$ and showing that, for nonconstant $\tilde Q$, the radial solution forces $\tilde Q$ to be constant. Overall, the results extend Euclidean symmetry techniques to the noncompact hyperbolic setting and provide new tools for conformal geometry problems on $\mathbb{H}^n$ with connections to the Euclidean case.

Abstract

The classification of solutions to semilinear partial differential equations, as well as the classification of critical points of the corresponding functionals, have wide applications in the study of partial differential equations and differential geometry. The classical moving plane method and the method of moving sphere on the Euclidean space $\mathbb{R}^n$ provide an effective approach to capture the symmetry of solutions. As far as we know, the moving sphere method has yet to be developed on the hyperbolic space $\mathbb{H}^n$. In the present paper, we focus on the following equation \begin{equation*} P_k u = f(u) \end{equation*} on hyperbolic spaces $\mathbb{H}^n$, where $P_k$ denotes the GJMS operators on $\mathbb{H}^n$ and $f : \mathbb{R} \to \mathbb{R}$ satisfies certain growth conditions. We develop a moving sphere approach on $\mathbb{H}^n$ to obtain the symmetry propertyas well as the classification of positive solutions to the above equation. Our methods also rely on the Helgason-Fourier analysis and Hardy-Littlewood-Sobolev inequalities on hyperbolic space together with a Kelvin transform we introduce on the hyperbolic space in this paper. We also present applications to the higher order prescribed $Q$-curvature problem on the hyperbolic space.

Theorems & Definitions (27)

• Theorem A
• Theorem 1.1
• Corollary 1.2
• Theorem B
• Theorem 1.3
• Theorem C
• Theorem D
• Theorem 1.4
• Theorem 1.5
• Corollary 1.6