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

Debdip Ganguly, Jungang Li, Guozhen Lu, Jianxiong Wang

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.

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 : where denotes the fourth-order Paneitz operator. We prove that any positive solution on exhibiting exponential growth at infinity must, up to hyperbolic isometries, be radial and strictly decreasing with respect to some point . Fourth order equations with negative critical growth on 3-dimensional Euclidean space 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 is substantially different, which prevents us from using the moving plane (sphere) method directly.
Paper Structure (11 sections, 14 theorems, 106 equations)

This paper contains 11 sections, 14 theorems, 106 equations.

Table of Contents

  1. Introduction
  2. Preliminaries: Geometric and functional Analytic tools
  3. Models of hyperbolic spaces
  4. Foliations of hyperbolic spaces
  5. GJMS operator and its fundamental solution
  6. Green function properties, integral representation and Proof of Theorem \ref{['integral_rep']}
  7. Radial symmetry, and Proof of Theorem \ref{['thm:main']} using the moving plane method
  8. The moving plane argument
  9. Remarks on the Integral equation on the hyperbolic space
  10. Asymptotic behaviors of the Green's function of on
  11. Nonexistence of solutions

Key Result

Theorem 1.1

Let $v$ be a positive $W^{2,2}$ weak solution of eq:biharmonic, satisfying $\inf |v| \geq \epsilon > 0$ for some $\epsilon$. Then $v$ admits the following integral representation: where $B_1$ denotes the unit ball in $\mathbb{R}^3$ centered at the origin, and the constants are given by

Theorems & Definitions (22)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Lemma 3.1
  • Lemma 3.2
  • proof
  • proof
  • Lemma 4.1
  • Lemma 4.2
  • ...and 12 more