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Journal für die reine und angewandte Mathematik

Managing Editor: Weissauer, Rainer

Ed. by Colding, Tobias / Huybrechts, Daniel / Hwang, Jun-Muk / Williamson, Geordie

IMPACT FACTOR 2017: 1.686

CiteScore 2018: 1.14

SCImago Journal Rank (SJR) 2018: 2.554
Source Normalized Impact per Paper (SNIP) 2018: 1.411

Mathematical Citation Quotient (MCQ) 2017: 1.49

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Volume 2008, Issue 620


Radial symmetry of positive solutions to nonlinear polyharmonic Dirichlet problems

Elvise Berchio / Filippo Gazzola
  • Dipartimento di Matematica del Politecnico, Piazza L. da Vinci 33, 20133 Milano, Italy. e-mail:
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/ Tobias Weth
Published Online: 2008-07-16 | DOI: https://doi.org/10.1515/CRELLE.2008.052


We extend the symmetry result of Gidas-Ni-Nirenberg [B. Gidas, W. M. Ni, L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979), 209–243.] to semilinear polyharmonic Dirichlet problems in the unit ball. In the proof we develop a new variant of the method of moving planes relying on fine estimates for the Green function of the polyharmonic operator. We also consider minimizers for subcritical higher order Sobolev embeddings. For embeddings into weighted spaces with a radially symmetric weight function, we show that the minimizers are at least axially symmetric. This result is sharp since we exhibit examples of minimizers which do not have full radial symmetry.

About the article

Received: 2006-12-12

Revised: 2007-04-18

Published Online: 2008-07-16

Published in Print: 2008-07-01

Citation Information: Journal für die reine und angewandte Mathematik (Crelles Journal), Volume 2008, Issue 620, Pages 165–183, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102, DOI: https://doi.org/10.1515/CRELLE.2008.052.

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