  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 $\R^N \; (N \geq 3)$}, \;\; 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 ${\underline u}_a (r)$ and a
family of non-minimal positive entire radial solutions $u_a (r)$ such that $u_a
(0)={\underline u}_a (0)=a$ and $u_a (r) \geq {\underline u}_a (r)$ for $r \in
(0, \infty)$. Moreover, the asymptotic behaviors of ${\underline u}_a (r)$ and
$u_a (r)$ at $r=\infty$ are also known. We will see in this paper that the
asymptotic behaviors similar to those of ${\underline u}_a (r)$ and $u_a (r)$
at $r=\infty$ 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.

Analysis of PDEs