Existence and uniqueness results for nonlinear elliptic problems with a lower order term and measure datum

M.Francesca Betta; Anna Mercaldo; François Murat; M.Michaela Porzio

doi:10.1016/S1631-073X(02)02338-5

Existence and uniqueness results for nonlinear elliptic problems with a lower order term and measure datum
[Existence et unicité de solutions renormalisées d'équations elliptiques non linéaires avec des termes d'ordre inférieur et données mesures]

M.Francesca Betta ¹ ; Anna Mercaldo ² ; François Murat ³ ; M.Michaela Porzio ⁴

¹ Dipartimento di Matematica, Seconda Università di Napoli, via Vivaldi 43, 81100 Caserta, Italy
² Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università degli Studi di Napoli “Federico II”, via Cintia, 80126 Napoli, Italy
³ Laboratoire Jacques-Louis Lions, Université Paris VI, boı̂te courrier 187, 75252 Paris cedex 05, France
⁴ Facoltà di Scienze Matematiche, Fisiche e Naturali, Università degli Studi del Sannio, via Port'Arsa 11, 82100 Benevento, Italy

Comptes Rendus. Mathématique, Volume 334 (2002) no. 9, pp. 757-762.

Résumé
Abstract

Dans cette Note nous considérons une classe de problèmes non linéaires et non coercifs dont le prototype est

- △_{p} u + b (x) {| \nabla u |}^{λ} = μ dans Ω, u = 0 sur \partial Ω,

où

Ω

est un ouvert borné de

ℝ^{N}

(N⩾2), △_p est le p-Laplacien (1<p<N) ou une variante de cet opérateur, μ est une mesure de Radon bornée ou une function de

L^{1} (Ω)

, λ⩾0 et b appartient à l'espace de Lorentz

L^{N, 1} (Ω)

ou à l'espace de Lebesgue

L^{\infty} (Ω)

. Nous démontrons l'existence et l'unicité de solutions renormalisées de ce problème.

In this Note we consider a class of noncoercive nonlinear problems whose prototype is

- △_{p} u + b (x) {| \nabla u |}^{λ} = μ in Ω, u = 0 on \partial Ω,

where

Ω

is a bounded open subset of

ℝ^{N}

(N⩾2), △_p is the so called p-Laplace operator (1<p<N) or a variant of it, μ is a Radon measure with bounded variation on

Ω

or a function in

L^{1} (Ω)

, λ⩾0 and b belongs to the Lorentz space

L^{N, 1} (Ω)

or to the Lebesgue space

L^{\infty} (Ω)

. We prove existence and uniqueness of renormalized solutions.

Reçu le : 2001-11-27
Révisé le : 2002-02-28
Publié le : 2002-01-01

DOI : 10.1016/S1631-073X(02)02338-5

Affiliations des auteurs :

M.Francesca Betta ¹ ; Anna Mercaldo ² ; François Murat ³ ; M.Michaela Porzio ⁴

¹ Dipartimento di Matematica, Seconda Università di Napoli, via Vivaldi 43, 81100 Caserta, Italy
² Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università degli Studi di Napoli “Federico II”, via Cintia, 80126 Napoli, Italy
³ Laboratoire Jacques-Louis Lions, Université Paris VI, boı̂te courrier 187, 75252 Paris cedex 05, France
⁴ Facoltà di Scienze Matematiche, Fisiche e Naturali, Università degli Studi del Sannio, via Port'Arsa 11, 82100 Benevento, Italy

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     title = {Existence and uniqueness results for nonlinear elliptic problems with a lower order term and measure datum},
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M.Francesca Betta; Anna Mercaldo; François Murat; M.Michaela Porzio. Existence and uniqueness results for nonlinear elliptic problems with a lower order term and measure datum. Comptes Rendus. Mathématique, Volume 334 (2002) no. 9, pp. 757-762. doi : 10.1016/S1631-073X(02)02338-5. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(02)02338-5/

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