Advances in Differential Equations

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

Lucio Damascelli and Filomena Pacella

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Abstract

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

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

Dates
First available in Project Euclid: 27 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.ade/1356651297

Mathematical Reviews number (MathSciNet)
MR1776351

Zentralblatt MATH identifier
1002.35045

Subjects
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

Citation

Damascelli, Lucio; Pacella, Filomena. Monotonicity and symmetry results for $p$-Laplace equations and applications. Adv. Differential Equations 5 (2000), no. 7-9, 1179--1200. https://projecteuclid.org/euclid.ade/1356651297.


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