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

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.