On Some Nonlinear Partial Differential Equations Involving the “1”-Laplacian and Critical Sobolev Exponent

Françoise Demengel

doi:10.1051/cocv:1999126

All issues

Volume 4 (1999)

ESAIM: COCV, 4 (1999) 667-686

Abstract

Free Access

Issue		ESAIM: COCV Volume 4, 1999


Page(s)		667 - 686
DOI		https://doi.org/10.1051/cocv:1999126
Published online		15 August 2002

ESAIM: COCV 4 (1999) 667-686

On Some Nonlinear Partial Differential Equations Involving the “1”-Laplacian and Critical Sobolev Exponent

Françoise Demengel

Université de Cergy-Pontoise, Département de Mathématiques, Site de Saint-Martin, 2 avenue Adolphe Chauvain, 95302 Cergy-Pontoise Cedex, France; Francoise.DEMENGEL@math.u-cergy.fr.

Received: 8 July 1998
Revised: 7 September 1999

Abstract

Let Ω be a smooth bounded domain in ${\bf R}^n$ , n > 1, let a and f be continuous functions on $\bar\Omega$ , $1^\star = {n\over n-1}$ . We are concerned here with the existence of solution in $BV(\Omega)$ , positive or not, to the problem: 

$\left\{ \begin{array}{rl} -{\rm div}\ \sigma+a(x) sign\ u &= f|u|^{1^\star-2} u\cr \sigma.\nabla u &= |\nabla u|\ {\rm in}\ \Omega\cr u\ {\rm is\ not \ identically\ zero}, &-\sigma.n (u) = |u|\ {\rm on }\ \partial \Omega.\end{array}\right.$

This problem is closely related to the extremal functions for the problem of the best constant of $W^{1,1}(\Omega)$ into $L^{N\over N-1}(\Omega)$ .

Résumé

On s'intéresse aux solutions dans $BV(\Omega)$ , non identiquement nulles au problème

$-{\rm div}(\sigma+a(x) sign\ u = f|u|^{1^\star-2}u$

$\sigma.\nabla u= |\nabla u|$

$-\sigma.\vec nu=|u|\ {\rm sur }\ \partial \Omega$

(où Ω est un ouvert borné régulier de ${\bf R}^N$ .)

Mathematics Subject Classification: 35J70

Key words: BV functions / best constant for Sobolev embeddings.

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