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2000 Monotonicity and symmetry results for $p$-Laplace equations and applications
Lucio Damascelli, Filomena Pacella
Adv. Differential Equations 5(7-9): 1179-1200 (2000). DOI: 10.57262/ade/1356651297


In this paper we prove monotonicity and symmetry properties of positive solutions of the equation $ - div (|Du|^{p-2}Du)=f(u)$, $1 <p <2$, in a smooth bounded domain ${\Omega} $ satisfying the boundary condition $u=0$ on ${\partial} {\Omega} $. We assume $f$ locally Lipschitz continuous only in $(0, \infty) $ and either $f \geq 0 $ in $[0, \infty] $ or $f$ satisfying a growth condition near zero. In particular we can treat the case of $f(s) = s^{{\alpha}} -c\, s^q $, ${\alpha} >0 $, $ c \geq 0 $, $ q \geq p-1 $. As a consequence we get an extension to the $p$--Laplacian case of a symmetry theorem of Serrin for an overdetermined problem in bounded domains. Finally we apply the results obtained to the problem of finding the best constants for the classical isoperimetric inequality and for some Sobolev embeddings.


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Lucio Damascelli. Filomena Pacella. "Monotonicity and symmetry results for $p$-Laplace equations and applications." Adv. Differential Equations 5 (7-9) 1179 - 1200, 2000.


Published: 2000
First available in Project Euclid: 27 December 2012

zbMATH: 1002.35045
MathSciNet: MR1776351
Digital Object Identifier: 10.57262/ade/1356651297

Primary: 35J65
Secondary: 35B05 , 35B50 , 35J70

Rights: Copyright © 2000 Khayyam Publishing, Inc.


Vol.5 • No. 7-9 • 2000
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