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

Abstract

In this article, we are concerned with the following geometric equation \begin{equation}\label{MainEq} Δ^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.

15 pages