  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.
