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-Mazya inequalities and careful study of delicate properties of Greens functions of $P_k-lambda$ on hyperbolic spaces which are of independent interests in dealing with such problems. Such Greens functions allow us to obtain the integral representations of solutions and thus to avoid using the maximum principle to establish the symmetry of solutions.
Let $mathbb{H}^{n}=mathbb{C}^{n}timesmathbb{R}$ be the $n$-dimensional Heisenberg group, $Q=2n+2$ be the homogeneous dimension of $mathbb{H}^{n}$. We extend the well-known concentration-compactness principle on finite domains in the Euclidean spaces
of P. L. Lions to the setting of the Heisenberg group $mathbb{H}^{n}$. Furthermore, we also obtain the corresponding concentration-compactness principle for the Sobolev space $HW^{1,Q}left( mathbb{H}^{n}right) $ on the entire Heisenberg group $mathbb{H}^{n}$. Our results improve the sharp Trudinger-Moser inequality on domains of finite measure in $mathbb{H}^{n}$ by Cohn and the second author [8] and the corresponding one on the whole space $mathbb{H}^n$ by Lam and the second author [21]. All the proofs of the concentration-compactness principles in the literature even in the Euclidean spaces use the rearrangement argument and the Polya-Szeg{o} inequality. Due to the absence of the Polya-Szeg{o} inequality on the Heisenberg group, we will develop a different argument. Our approach is surprisingly simple and general and can be easily applied to other settings where symmetrization argument does not work. As an application of the concentration-compactness principle, we establish the existence of ground state solutions for a class of $Q$- Laplacian subelliptic equations on $mathbb{H}^{n}$ with nonlinear terms $f$ of maximal exponential growth $expleft( alpha t^{frac{Q}{Q-1}}right) $ as $trightarrow+infty$.
