Higher order Brezis-Nirenberg problem on hyperbolic spaces: Existence, nonexistence and symmetry of solutions

Abstract

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 Pk on bounded domains in the hyperbolic space Hn and as well as on the entire hyperbolic space Hn. Among techniques different from the higher order Brezis-Nirenberg problem in Euclidean spaces, one of our main novelties in the study of existence and symmetry 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 Pk−λ 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 avoid using the maximum principle to establish the symmetry of solutions to the higher order Brezis-Nirenberg problem by developing a moving plane method in integral form on hyperbolic spaces in the spirit of the work of [14].