A necessary and sufficient condition for radial property of positive entire solutions of $\Delta^2 u+u^{-q}=0$ in $\mathbf{R}^3$

TL;DR

This paper establishes a necessary and sufficient condition for the radial symmetry of positive entire solutions to a biharmonic equation with a negative power nonlinearity in three-dimensional space, extending previous results to the case where q>3.

Contribution

It extends the characterization of radial symmetry for solutions of a biharmonic equation to the case q>3 using the method of moving plane.

Findings

Characterization of radial symmetry for q>3

Extension of previous results for 1<q<3

Application of the method of moving plane

Abstract

In this article, we are concerned with the following geometric equation \begin{equation}\label{MainEq} \Delta^2 u = -u^{-q} \qquad \text{in } \mathbf{R}^3 \end{equation} for $q>0$. Recently in \cite{GWZ18}, Guo, Wei and Zhou have established the relationship between the radial symmetry and the exact growth rate at infinity of a positive entire solution of that equation as $1<q<3$. The aim of this paper is to obtain the similar result in the case $q>3$ thanks to the method of moving plane.