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Uniqueness of renormalized solutions to nonlinear elliptic equations with a lower order term and right-hand side in L1(Ω)

Published online by Cambridge University Press:  15 August 2002

M. F. Betta ,
A. Mercaldo ,
F. Murat  and
M. M. Porzio
M. F. Betta
Affiliation:
Dipartimento di Matematica, Seconda Università di Napoli, via Vivaldi 43, 81100 Caserta, Italy; mariafrancesca.betta@unina2.it.
A. Mercaldo
Affiliation:
Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università degli Studi di Napoli “Federico II", Complesso Monte S. Angelo, via Cintia, 80126 Napoli, Italy; mercaldo@unina.it.
F. Murat
Affiliation:
Laboratoire Jacques-Louis Lions, Université Paris VI, Boîte courrier 187, 75252 Paris Cedex 05, France; murat@ann.jussieu.fr.
M. M. Porzio
Affiliation:
Facoltà di Scienze Matematiche, Fisiche e Naturali, Università degli Studi del Sannio, via Port'Arsa 11, 82100 Benevento, Italy; porzio@unisannio.it.
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Abstract

In this paper we prove uniqueness results for the renormalized solution, if it exists, of a class of non coercive nonlinear problems whose prototype is
$$ \left\{ - \hbox{div}( a(x)(1+|\nabla u|^{2})^{\frac{p-2}{2}}\nabla u) +b(x)(1+|\nabla u|^{2})^{\frac{\lambda}{2}} =f \hbox{in} \quad \Omega, u=0 \hbox{on} \quad \partial\Omega, \right. $$
where Ω is a bounded open subset of ${\mathbb{R}}^N$, N > 2, 2-1/N < p < N , a belongs to L∞(Ω), $a(x) \ge \alpha_0>0$, f is a function in L1(Ω), b is a function in $L^r(\Omega)$ and 0 ≤ λ < λ *(N,p,r), for some r and λ *(N,p,r).

Keywords

Uniquenessnonlinear elliptic equationsnoncoercive problemsdata in L1.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 8: A tribute to JL Lions , 2002 , pp. 239 - 272
Copyright
© EDP Sciences, SMAI, 2002

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References

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