Nonhomogeneous boundary value problems in anisotropic Sobolev spaces

Mihai Mihăilescu; Patrizia Pucci; Vicenţiu Rădulescu

doi:10.1016/j.crma.2007.10.012

Partial Differential Equations

Nonhomogeneous boundary value problems in anisotropic Sobolev spaces

Presented by: Philippe G. Ciarlet

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

Comptes Rendus. Mathématique, Volume 345 (2007) no. 10, pp. 561-566.

Abstract
Résumé

We study the nonlinear boundary value problem $- \sum_{i = 1}^{N} {({| u_{x_{i}} |}^{p_{i} (x) - 2} u_{x_{i}})}_{x_{i}} = λ {| u |}^{q (x) - 2} u$ in Ω, $u = 0$ on ∂Ω, where $Ω \subset R^{N}$ $(N ⩾ 3)$ is a bounded domain with smooth boundary, λ is a positive real number, and the continuous functions $p_{i}$ and q satisfy $2 ⩽ p_{i} (x) < N$ and $q (x) > 1$ for any $x \in \bar{Ω}$ and any $i \in {1, \dots, N}$ . By analyzing the growth of the functions $p_{i}$ and q we prove in this Note several existence results in Sobolev spaces with variable exponents.

On étudie le problème non linéaire $- \sum_{i = 1}^{N} {({| u_{x_{i}} |}^{p_{i} (x) - 2} u_{x_{i}})}_{x_{i}} = λ {| u |}^{q (x) - 2} u$ dans Ω, $u = 0$ sur ∂Ω, où $Ω \subset R^{N}$ $(N ⩾ 3)$ est un domaine borné et régulier, λ est un nombre réel positif et $p_{i}$ et q sont des fonctions continues telles que $2 ⩽ p_{i} (x) < N$ et $q (x) > 1$ pour tout $x \in \bar{Ω}$ et chaque $i \in {1, \dots, N}$ . En étudiant la croissance des fonctions $p_{i}$ et q on obtient dans cette Note plusieurs résultats d'existence dans des espaces de Sobolev aux exposants variables.

Received: 2007-08-22
Accepted: 2007-09-21
Published online: 2007-11-07

DOI: 10.1016/j.crma.2007.10.012

Author's affiliations:

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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     author = {Mihai Mih\u{a}ilescu and Patrizia Pucci and Vicen\c{t}iu R\u{a}dulescu},
     title = {Nonhomogeneous boundary value problems in anisotropic {Sobolev} spaces},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {561--566},
     publisher = {Elsevier},
     volume = {345},
     number = {10},
     year = {2007},
     doi = {10.1016/j.crma.2007.10.012},
     language = {en},
}

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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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