Radial symmetry of positive entire solutions of a fourth order elliptic equation with a singular nonlinearity

The necessary and sufficient conditions for a regular positive entire solution $u$ of the biharmonic equation: \begin{equation} \label{0.1} -\Delta^2 u=u^{-p} ;; \mbox{in ${\mathbb{R}}^N(N\ge 3)$}, ;; p>1 \end{equation} to be a radially symmetric solution are obtained via the moving plane method (MPM) of a system of equations. It is well-known that for any $a>0$, \eqref{0.1} admits a unique minimal positive entire radial solution ${\underset{‾}{u}}_a(r)$ and a family of non-minimal positive entire radial solutions $u_a(r)$ such that $u_a (0)={\underline u}_a (0)=a$and$u_a ® \geq {\underline u}_a ®$for$r \in (0, \infty)$\mathrm{.}Moreover,theasymptoticbehaviorsof${\underline u}_a ®$and$u_a ®$at$r=\infty$ are also known. We will see in this paper that the asymptotic behaviors similar to those of ${\underset{‾}{u}}_a(r)$ and $u_a(r)$ at $r=\mathrm{\infty }$ can determine the radial symmetry of a general regular positive entire solution $u$ of \eqref{0.1}. The precisely asymptotic behaviors of $u (x)$and$-\Delta u (x)$at$|x|=\infty$ need to be established such that the moving-plane procedure can be started. We provide the necessary and sufficient conditions not only for a regular positive entire solution $u$ of \eqref{0.1} to be the minimal entire radial solution, but also for $u$ to be a non-minimal entire radial solution.