Vol. 12, No. 4, 2019

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

Rupert L. Frank and Tobias König

Vol. 12 (2019), No. 4, 1101–1113
DOI: 10.2140/apde.2019.12.1101
Abstract

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

with a nonremovable singularity at the origin. We show that $|x{|}^{\left(n-4\right)∕2}u$ 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 of local solutions near singularities and for the $Q$-curvature problem in conformal geometry.

Keywords
fourth-order equation, critical exponent, classification, periodic solutions
Mathematical Subject Classification 2010
Primary: 34C25, 35B09, 35J30, 53A30