For $n \geq 5$, we consider positive solutions $u$ of the biharmonic equation \[ \Delta^2 u = u^\frac{ n+4}{ n-4} \qquad \text{ on}\ \mathbb R^n \setminus \{ 0\} \] with a non-removable singularity at the origin. We show that $|x|^{ \frac{ n-4}{ 2}} u$ is a periodic function of $\ln |x|$ 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.
Categories: math.AP math.CA math.DG