#### Vol. 12, No. 4, 2019

 Download this article For screen For printing
 Recent Issues
 The Journal About the Journal Editorial Board Subscriptions Editors’ Interests Scientific Advantages Submission Guidelines Submission Form Editorial Login Ethics Statement ISSN: 1948-206X (e-only) ISSN: 2157-5045 (print) Author Index To Appear Other MSP Journals
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
##### Milestones
Received: 2 November 2017
Revised: 28 May 2018
Accepted: 30 July 2018
Published: 20 October 2018
##### Authors
 Rupert L. Frank Mathematisches Institut Ludwig-Maximilians-Universität München München Germany Department of Mathematics Caltech Pasadena, CA United States Tobias König Mathematisches Institut Ludwig-Maximilians-Universität München München Germany