The classification of solutions for 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. In this talk, we focus on the equation
\begin{equation*}
P_k u = f(u)
\end{equation*}
on hyperbolic spaces $\mathbb{H}^n$, where $P_k$ denotes the GJMS operators and $f : \mathbb{R} \to \mathbb{R}$ satisfies certain growth conditions. I will introduce a moving sphere approach on $\mathbb{H}^n$, to obtain the symmetry property as well as the classification result towards positive solutions. Our methods also rely on Helgason-Fourier analysis and Hardy-Littlewood-Sobolev inequalities on hyperbolic spaces together with a newly introduced Kelvin-type transform on $\mathbb{H}^n$.