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The classification of solutions to semilinear partial differential equations,as well as the classification of critical points of the correspondingfunctionals, have wide applications in the study of partial differentialequations and differential geometry. The classical moving plane method and themethod of moving sphere on the Euclidean space $\mathbb{R}^n$ provide aneffective approach to capture the symmetry of solutions. As far as we know, themoving 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 sphereapproach on $\mathbb{H}^n$ to obtain the symmetry propertyas well as theclassification of positive solutions to the above equation. Our methods alsorely on the Helgason-Fourier analysis and Hardy-Littlewood-Sobolev inequalitieson hyperbolic space together with a Kelvin transform we introduce on thehyperbolic space in this paper. We also present applications to the higherorder prescribed $Q$-curvature problem on the hyperbolic space.

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