Conformally invariant equations with negative critical exponents on the three dimensional hyperbolic space

TL;DR

This paper proves symmetry and monotonicity of positive solutions to a fourth-order conformally invariant PDE with negative critical exponent on hyperbolic space, highlighting differences from Euclidean cases.

Contribution

It establishes a symmetry result for solutions to a fourth-order PDE with negative critical growth on hyperbolic space, overcoming challenges posed by the Green's function behavior.

Findings

Solutions are radial and decreasing around some point in hyperbolic space.

Solutions with exponential growth at infinity are characterized.

The method adapts symmetry techniques to hyperbolic geometry.

Abstract

We establish a symmetry result for positive entire solutions with a prescribed growth rate to the following fourth order equation on the 3-dimensional hyperbolic space $\mathbb{H}^3$: \[ P_2 u = - u^{-7}, \] where $P_2$ denotes the fourth-order Paneitz operator. We prove that any positive solution $u$ on $\mathbb{H}^3$ exhibiting exponential growth at infinity must, up to hyperbolic isometries, be radial and strictly decreasing with respect to some point $P \in \mathbb{H}^3$. Fourth order equations with negative critical growth on 3-dimensional Euclidean space $\mathbb{R}^3$ has been studied by Choi and Xu in \cite{CX09 }, and subsequently by McKenna and Reichel \cite{MR03} and Xu \cite{Xu05}. Unlike the Euclidean case, the behavior of the Green's function of $P_2$ is substantially different, which prevents us from using the moving plane (sphere) method directly.