Comptes Rendus
Partial Differential Equations
Nonhomogeneous boundary value problems in anisotropic Sobolev spaces
[Problèmes aux limites non homogènes en espaces de Sobolev anisotropiques]
Comptes Rendus. Mathématique, Volume 345 (2007) no. 10, pp. 561-566.

On étudie le problème non linéaire i=1N(|uxi|pi(x)2uxi)xi=λ|u|q(x)2u dans Ω, u=0 sur ∂Ω, où ΩRN (N3) est un domaine borné et régulier, λ est un nombre réel positif et pi et q sont des fonctions continues telles que 2pi(x)<N et q(x)>1 pour tout xΩ¯ et chaque i{1,,N}. En étudiant la croissance des fonctions pi et q on obtient dans cette Note plusieurs résultats d'existence dans des espaces de Sobolev aux exposants variables.

We study the nonlinear boundary value problem i=1N(|uxi|pi(x)2uxi)xi=λ|u|q(x)2u in Ω, u=0 on ∂Ω, where ΩRN (N3) is a bounded domain with smooth boundary, λ is a positive real number, and the continuous functions pi and q satisfy 2pi(x)<N and q(x)>1 for any xΩ¯ and any i{1,,N}. By analyzing the growth of the functions pi and q we prove in this Note several existence results in Sobolev spaces with variable exponents.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2007.10.012

Mihai Mihăilescu 1, 2 ; Patrizia Pucci 3 ; Vicenţiu Rădulescu 1, 4

1 University of Craiova, Department of Mathematics, 200585 Craiova, Romania
2 Department of Mathematics, Central European University, 1051 Budapest, Hungary
3 Università degli Studi di Perugia, Dipartimento di Matematica e Informatica, 06123 Perugia, Italy
4 Institute of Mathematics “Simion Stoilow” of the Romanian Academy, 014700 Bucharest, Romania
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Mihai Mihăilescu; Patrizia Pucci; Vicenţiu Rădulescu. Nonhomogeneous boundary value problems in anisotropic Sobolev spaces. Comptes Rendus. Mathématique, Volume 345 (2007) no. 10, pp. 561-566. doi : 10.1016/j.crma.2007.10.012. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.10.012/

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