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

Tien-Tai Nguyen

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arXiv · 1909.09238

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

Tien-Tai Nguyen

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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 $1 3$ thanks to the method of moving plane.

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Tien-Tai Nguyen. 2019-09-19. A necessary and sufficient condition for radial property of positive entire solutions of $\Delta^2 u+u^{-q}=0$ in $\mathbf{R}^3$. https://arxiv.org/abs/1909.09238

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