Regularity of the extremal solution for a biharmonic problem with general nonlinearity

Guillaume Warnault

doi:10.3934/cpaa.2009.8.1709

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2009, Volume 8, Issue 5: 1709-1723. Doi: 10.3934/cpaa.2009.8.1709

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Regularity of the extremal solution for a biharmonic problem with general nonlinearity

Guillaume Warnault^1,

1.
Laboratoire Amiénois de Mathématiques Fondamentale et Appliquée, Faculté de Mathématiques et d' Informatique, 33, rue Saint-Leu 80039 Amiens Cedex 1

Received: June 30, 2008

Revised: December 31, 2008

Published: August 2009

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Abstract

We consider the class of radial solutions of semilinear equations $\Delta^2 u=\lambda f(u)$ in the unit ball of $\mathbb R^N$. It is the class of stable solutions which includes minimal solutions and extremal solution. We establish the regularity of this extremal solution for $N\leq 9$. Our regularity results do not depend on the specific nonlinearity $f$.

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Mathematics Subject Classification: Primary: 31B30; Secondary: 74B0.

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