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Comptes Rendus Mathématique
Volume 344, n° 8
pages 487-492 (avril 2007)
Doi : 10.1016/j.crma.2007.02.007
Received : 5 April 2005 ;  accepted : 30 January 2007
Uniqueness results for pseudomonotone problems with  
Résultats dʼunicité pour des problèmes pseudomonotones avec  
 

Juan Casado-Díaz a , François Murat b , Alessio Porretta c
a Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, c/Tarfia s/n, 41012 Sevilla, Spain 
b Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, boîte courrier 187, 75252 Paris cedex 05, France 
c Dipartimento di Matematica, Università di Roma Tor Vergata, Via della ricerca scientifica 1, 00133 Roma, Italy 

Abstract

We consider a pseudomonotone operator, the model of which is   with   and   a Lipschitz continuous function in s which hold satisfies  . We show that the comparison principle (and therefore the uniqueness for the Dirichlet problem) in two particular cases, namely the one-dimensional case, and the case where at least one of the right-hand sides does not change sign. To the best of our knowledge these results are new for  . Full detailed proofs are given in the present Note. The results continue to hold when is unbounded. To cite this article: J. Casado-Díaz et al., C. R. Acad. Sci. Paris, Ser. I 344 (2007).

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Résumé

Nous considérons un opérateur pseudomonotone du type  , avec   et   une fonction Lipschitzienne en s qui vérifie  . Nous démontrons que cet opérateur satisfait le principe de comparaison (et donc quʼon a unicité pour le problème de Dirichlet) dans deux cas particuliers : en dimension 1, et dans le cas où au moins lʼun des deux seconds membres ne change pas de signe. A notre connaissance, ces résultats sont nouveaux quand  . Les démonstrations complètes sont données dans cette Note. Les résultats restent valides quand est non borné. Pour citer cet article : J. Casado-Díaz et al., C. R. Acad. Sci. Paris, Ser. I 344 (2007).

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