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

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.

arXiv comment32 pages. The paper has been improved substantially in this new version. We established the classification of the solutions to the higher order equations associated to the GJMS operators on the hyperbolic space by removing the growth condition assumption on the solution at the infinity. (See the statement of Theorem 1.1)