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}.
Categories: math.AP