Advances in Differential Equations

Monotonicity and symmetry results for $p$-Laplace equations and applications

Lucio Damascelli and Filomena Pacella

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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.

Article information

Adv. Differential Equations Volume 5, Number 7-9 (2000), 1179-1200.

First available in Project Euclid: 27 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35J65: Nonlinear boundary value problems for linear elliptic equations
Secondary: 35B05: Oscillation, zeros of solutions, mean value theorems, etc. 35B50: Maximum principles 35J70: Degenerate elliptic equations


Damascelli, Lucio; Pacella, Filomena. Monotonicity and symmetry results for $p$-Laplace equations and applications. Adv. Differential Equations 5 (2000), no. 7-9, 1179--1200.

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