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 5, we consider positive solutions u of the biharmonic equation

Δ2u = u(n+4)(n4) on n {0},

with a nonremovable singularity at the origin. We show that |x|(n4)2u 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