  We are interested in the qualitative properties of positive entire solutions
$u \in C^4 (\mathbb{R}^n \backslash \{0\})$ of the equation \begin{equation}
\label{0.0} \Delta^2 u=u^{\frac{n+4}{n-4}} \;\;\mbox{in $\mathbb{R}^n
\backslash \{0\}$ and 0 is a non-removable singularity of $u(x)$}.
\end{equation} It is known from [Theorem 4.2] that any positive entire solution
$u$ of \eqref{0.0} is radially symmetric with respect to $x=0$, i.e.
$u(x)=u(|x|)$, and equation \eqref{0.0} also admits a special positive entire
solution $u_s (x)=\Big(\frac{n^2 (n-4)^2}{16} \Big)^{\frac{n-4}{8}}
|x|^{-\frac{n-4}{2}}$. We first show that $u-u_s$ changes signs infinitely many
times in $(0, \infty)$ for any positive singular entire solution $u \not \equiv
u_s$ in $\mathbb{R}^N \backslash \{0\}$ of \eqref{0.0}. Moreover, equation
\eqref{0.0} admits a positive entire singular solution $u(x) \; (=u(|x|)$ such
that the scalar curvature of the conformal metric with conformal factor
$u^{\frac{4}{n-4}}$ is positive and $v(t):=e^{\frac{n-4}{2} t} u(e^t)$ is
$2T$-periodic with suitably large $T$. It is still open that
$v(t):=e^{\frac{n-4}{2} t} u(e^t)$ is periodic for any positive entire solution
$u(x)$ of \eqref{0.0}.
