The Method of Moving spheres on Hyperbolic Space and Symmetry of Solutions to a Class of PDEs

Abstract

The classification of solutions of 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 moving sphere method on $\mathbb{R}^n$ provide an effective approach to capturing 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 for integral equations on $\mathbb{H}^n$, to obtain the symmetry property as well as a characterization result towards positive solutions. Our methods also rely on Helgason-Fourier analysis and Hardy-Littlewood-Sobolev inequalities on hyperbolic spaces together with a Kelvin transform we introduce on $\mathbb{H}^n$ in this paper.Comment: Some references are added and typos fixe