Classification of positive solutions to a nonlinear biharmonic equation with critical exponent

For $n\ge 5$, we consider positive solutions $u$ of the biharmonic equation $${\mathrm{\Delta }}^{2}u=u^{\frac{n+4}{n-4}}\text{on }{\mathbb{R}}^n\setminus \{0\}$$ with a non-removable singularity at the origin. We show that $\mathrm{\mid }x{\mathrm{\mid }}^{\frac{n-4}{2}}u$ is a periodic function of $\ln \mathrm{\mid }x\mathrm{\mid }$ and we classify all periodic functions obtained in this way. This result is relevant for the description of the asymptotic behavior near singularities and for the $Q$-curvature problem in conformal geometry.