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

TL;DR

This paper investigates the existence, nonexistence, and symmetry of solutions to higher order Brezis-Nirenberg problems involving GJMS operators on hyperbolic spaces, employing advanced Fourier analysis and Green's function techniques.

Contribution

It introduces novel methods using Helgason-Fourier analysis and Green's functions to analyze solutions without relying on maximum principles.

Findings

Established conditions for existence and nonexistence of solutions.

Demonstrated symmetry properties of solutions in hyperbolic spaces.

Developed integral representations of solutions using Green's functions.

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 $P_k$ 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.