  The main purpose of this paper is to establish the existence, nonexistence
and symmetry of nontrivial solutions to the higher order Brezis-Nirenberg
problems associated with the GJMS operators $P_k$ on bounded domains in the
hyperbolic space $\mathbb{H}^n$ and as well as on the entire hyperbolic space
$\mathbb{H}^n$. Among other techniques, one of our main novelties is to use
crucially the Helgason-Fourier analysis on hyperbolic spaces and the higher
order Hardy-Sobolev-Maz'ya inequalities and careful study of delicate
properties of Green's functions of $P_k-\lambda$ on hyperbolic spaces which are
of independent interests in dealing with such problems. Such Green's functions
allow us to obtain the integral representations of solutions and thus to avoid
using the maximum principle to establish the symmetry of solutions.
