  We consider positive solutions $u$ of the semilinear biharmonic equation
$\Delta^2 u = |x|^{-\frac{n+4}{2}} g(|x|^\frac{n-4}{2} u)$ in $\mathbb R^n
\setminus \{0\}$ with non-removable singularities at the origin. Under natural
assumptions on the nonlinearity $g$, we show that $|x|^\frac{n-4}{2} u$ is a
periodic function of $\ln |x|$ and we classify all such solutions.
