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

Published online by Cambridge University Press:  15 August 2002

Françoise Demengel
Françoise Demengel*
Affiliation:
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.
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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 $L^{N\over N-1}(\Omega)$.

Keywords

BV functionsbest constant for Sobolev embeddings.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 4 , 1999 , pp. 667 - 686
Copyright
© EDP Sciences, SMAI, 1999

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