Comptes Rendus
Partial Differential Equations
Uniqueness results for pseudomonotone problems with p>2
[Résultats d'unicité pour des problèmes pseudomonotones avec p>2]
Comptes Rendus. Mathématique, Volume 344 (2007) no. 8, pp. 487-492.

We consider a pseudomonotone operator, the model of which is div(b(x,u)|u|p2u) with 1<p<+ and b(x,s) a Lipschitz continuous function in s which hold satisfies 0<αb(x,s)β<+. 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 p>2. Full detailed proofs are given in the present Note. The results continue to hold when Ω is unbounded.

Nous considérons un opérateur pseudomonotone du type div(b(x,u)|u|p2u), avec 1<p<+ et b(x,s) une fonction Lipschitzienne en s qui vérifie 0<αb(x,s)β<+. Nous démontrons que cet opérateur satisfait le principe de comparaison (et donc qu'on a unicité pour le problème de Dirichlet) dans deux cas particuliers : en dimension 1, et dans le cas où au moins l'un des deux seconds membres ne change pas de signe. A notre connaissance, ces résultats sont nouveaux quand p>2. Les démonstrations complètes sont données dans cette Note. Les résultats restent valides quand Ω est non borné.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2007.02.007

Juan Casado-Díaz 1 ; François Murat 2 ; Alessio Porretta 3

1 Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, c/Tarfia s/n, 41012 Sevilla, Spain
2 Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, boîte courrier 187, 75252 Paris cedex 05, France
3 Dipartimento di Matematica, Università di Roma Tor Vergata, Via della ricerca scientifica 1, 00133 Roma, Italy
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Juan Casado-Díaz; François Murat; Alessio Porretta. Uniqueness results for pseudomonotone problems with $ p>2$. Comptes Rendus. Mathématique, Volume 344 (2007) no. 8, pp. 487-492. doi : 10.1016/j.crma.2007.02.007. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.02.007/

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