Correlation between two quasilinear elliptic problems with a source term involving the function or its gradient

Hamid, Haydar Abdel; Bidaut-Véron, Marie Françoise

doi:10.1016/j.crma.2008.10.002

Partial Differential Equations

Correlation between two quasilinear elliptic problems with a source term involving the function or its gradient
[Corrélation entre deux problèmes quasilinéaires elliptiques avec terme de source relatif à la fonction ou à son gradient]

Présenté par : Haïm Brezis

Hamid, Haydar Abdel ¹ ; Bidaut-Véron, Marie Françoise ¹

¹ Laboratoire de mathématiques et physique théorique, CNRS UMR 6083, faculté des sciences, 37200 Tours, France

Comptes Rendus. Mathématique, Tome 346 (2008) no. 23-24, pp. 1251-1256.

Résumé
Abstract

A l'aide d'un changement d'inconnue nous comparons deux problèmes elliptiques quasilinéaires avec conditions de Dirichlet dans un domaine borné Ω de $R^{N}$ . Le premier, de la forme $- Δ_{p} u = β (u) {| \nabla u |}^{p} + λ f (x)$ , où β est positif, comporte un terme de gradient à croissance critique. Le second, de la forme $- Δ_{p} v = λ f (x) (1 + g {(v))}^{p - 1}$ où g est croissante, contient un terme de source d'ordre 0. La comparaison donne des résultats nouveaux d'existence, nonexistence et multiplicité pour les deux problèmes.

Thanks to a change of unknown we compare two elliptic quasilinear problems with Dirichlet data in a bounded domain of $R^{N}$ . The first one, of the form $- Δ_{p} u = β (u) {| \nabla u |}^{p} + λ f (x)$ , where β is nonnegative, involves a gradient term with natural growth. The second one, of the form $- Δ_{p} v = λ f (x) (1 + g {(v))}^{p - 1}$ where g is nondecreasing, presents a source term of order 0. The correlation gives new results of existence, nonexistence and multiplicity for the two problems.

Reçu le : 2008-10-05
Publié le : 2008-10-31

DOI : 10.1016/j.crma.2008.10.002

Affiliations des auteurs :

Hamid, Haydar Abdel ¹ ; Bidaut-Véron, Marie Françoise ¹

¹ Laboratoire de mathématiques et physique théorique, CNRS UMR 6083, faculté des sciences, 37200 Tours, France

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     title = {Correlation between two quasilinear elliptic problems with a source term involving the function or its gradient},
     journal = {Comptes Rendus. Math\'ematique},
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Hamid, Haydar Abdel; Bidaut-Véron, Marie Françoise. Correlation between two quasilinear elliptic problems with a source term involving the function or its gradient. Comptes Rendus. Mathématique, Tome 346 (2008) no. 23-24, pp. 1251-1256. doi : 10.1016/j.crma.2008.10.002. http://www.numdam.org/articles/10.1016/j.crma.2008.10.002/

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