Nonhomogeneous boundary value problems in Orlicz–Sobolev spaces

Mihai Mihăilescu; Vicenţiu Rădulescu

doi:10.1016/j.crma.2006.11.020

Partial Differential Equations

Nonhomogeneous boundary value problems in Orlicz–Sobolev spaces
[Problèmes aux limites non homogènes dans les espaces d'Orlicz–Sobolev]

Présenté par : Philippe G. Ciarlet

Mihai Mihăilescu ¹ ; Vicenţiu Rădulescu ¹

¹ University of Craiova, Department of Mathematics, Street A.I. Cuza No. 13, 200585 Craiova, Romania

Comptes Rendus. Mathématique, Volume 344 (2007) no. 1, pp. 15-20.

Résumé
Abstract

On étudie le problème de Dirichlet non linéaire $- div (\log (1 + {| \nabla u |}^{q}) {| \nabla u |}^{p - 2} \nabla u) = - λ {| u |}^{p - 2} u + {| u |}^{r - 2} u$ dans Ω, $u = 0$ sur ∂Ω, où Ω est un domaine borné, régulier et p, q, r sont des nombres réels tels que $p, q > 1$ , $p + q < \min {N, r}$ , $r < (N p - N + p) / (N - p)$ . Le résultat principal de cette Note montre que pour tout $λ > 0$ ce problème admet une infinité de solutions dans l'espace d'Orlicz–Sobolev $W_{0}^{1} L_{Φ} (Ω)$ , où $Φ (t) = \int_{0}^{t} \log (1 + {| s |}^{q}) \cdot {| s |}^{p - 2} s d s$ .

We study the nonlinear Dirichlet problem $- div (\log (1 + {| \nabla u |}^{q}) {| \nabla u |}^{p - 2} \nabla u) = - λ {| u |}^{p - 2} u + {| u |}^{r - 2} u$ in Ω, $u = 0$ on ∂Ω, where Ω is a bounded domain in $R^{N}$ with smooth boundary, while p, q and r are real numbers satisfying $p, q > 1$ , $p + q < \min {N, r}$ , $r < (N p - N + p) / (N - p)$ . The main result of this Note establishes that for any $λ > 0$ this boundary value problem has infinitely many solutions in the Orlicz–Sobolev space $W_{0}^{1} L_{Φ} (Ω)$ , where $Φ (t) = \int_{0}^{t} \log (1 + {| s |}^{q}) \cdot {| s |}^{p - 2} s d s$ .

Reçu le : 2006-10-23
Accepté le : 2006-11-07
Publié le : 2006-12-18

DOI : 10.1016/j.crma.2006.11.020

Affiliations des auteurs :

Mihai Mihăilescu ¹ ; Vicenţiu Rădulescu ¹

¹ University of Craiova, Department of Mathematics, Street A.I. Cuza No. 13, 200585 Craiova, Romania

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     title = {Nonhomogeneous boundary value problems in {Orlicz{\textendash}Sobolev} spaces},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {15--20},
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     language = {en},
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Mihai Mihăilescu; Vicenţiu Rădulescu. Nonhomogeneous boundary value problems in Orlicz–Sobolev spaces. Comptes Rendus. Mathématique, Volume 344 (2007) no. 1, pp. 15-20. doi : 10.1016/j.crma.2006.11.020. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2006.11.020/

Bibliographie
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[9] M. Mihăilescu, V. Rădulescu, On a nonhomogeneous quasilinear eigenvalue problem in Sobolev spaces with variable exponent, Proc. Amer. Math. Soc., in press

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