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

TL;DR

This paper classifies positive solutions with singularities of a critical biharmonic equation in higher dimensions, revealing their periodic behavior and implications for geometric analysis.

Contribution

It provides a complete classification of singular solutions to a critical biharmonic equation, linking their asymptotic behavior to periodic functions of logarithmic radius.

Findings

Solutions exhibit a specific periodicity in logarithmic scale.

All such solutions can be explicitly classified.

Results have implications for conformal geometry and Q-curvature.

Abstract

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.