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

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.

24 pages