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

Abstract

For $n\ge 5$, we consider positive solutions $u$ of the biharmonic equation

$${\mathrm{\Delta }}^{2}u=u^{\left(n+4\right)∕\left(n-4\right)}\text{ on }{ℝ}^n\setminus \left\{0\right\},$$

with a nonremovable singularity at the origin. We show that $|x{|}^{\left(n-4\right)∕2}u$ is a periodic function of $ln\left|x\right|$ and we classify all periodic functions obtained in this way. This result is relevant for the description of the asymptotic behavior of local solutions near singularities and for the $Q$-curvature problem in conformal geometry.