Infinitely many solutions for a class of nonlinear eigenvalue problem in Orlicz–Sobolev spaces

Gabriele Bonanno; Giovanni Molica Bisci; Vicenţiu Rădulescu

doi:10.1016/j.crma.2011.02.009

Partial Differential Equations

Infinitely many solutions for a class of nonlinear eigenvalue problem in Orlicz–Sobolev spaces
[Infinité de solutions pour une classe de problèmes non linéaires de valeurs propres dans les espaces dʼOrlicz–Sobolev]

Présenté par : Philippe G. Ciarlet

Gabriele Bonanno ¹ ; Giovanni Molica Bisci ² ; Vicenţiu Rădulescu ^3,⁴

¹ Department of Science for Engineering and Architecture (Mathematics Section), Engineering Faculty, University of Messina, 98166 Messina, Italy
² Department P.A.U., Architecture Faculty, University of Reggio Calabria, 89100 Reggio Calabria, Italy
³ Institute of Mathematics “Simion Stoilow” of the Romanian Academy, 014700 Bucharest, Romania
⁴ Department of Mathematics, University of Craiova, Street A.I. Cuza No. 13, 200585 Craiova, Romania

Comptes Rendus. Mathématique, Volume 349 (2011) no. 5-6, pp. 263-268.

Résumé
Abstract

On étudie le problème de Neumann $- div (α (| \nabla u |) \nabla u) + α (| u |) u = λ f (x, u)$ dans Ω, $\partial u / \partial ν = 0$ sur ∂Ω, où Ω est un domaine borné régulier de $R^{N}$ , λ est un paramètre positif, f est une fonction continue et α est une application définie sur $(0, \infty)$ . Le résultat principal de cette Note montre que pour tout λ dans un certain intervalle ouvert, ce problème admet une infinité de solutions qui convergent vers zéro dans lʼespace dʼOrlicz–Sobolev $W^{1} L_{Φ} (Ω)$ .

We study the Neumann problem $- div (α (| \nabla u |) \nabla u) + α (| u |) u = λ f (x, u)$ in Ω, $\partial u / \partial ν = 0$ on ∂Ω, where Ω is a smooth bounded domain in $R^{N}$ , λ is a positive parameter, f is a continuous function, and α is a real-valued mapping defined on $(0, \infty)$ . The main result in this Note establishes that for all λ in a prescribed open interval, this problem has infinitely many solutions that converge to zero in the Orlicz–Sobolev space $W^{1} L_{Φ} (Ω)$ .

Reçu le : 2011-01-16
Accepté le : 2011-02-05
Publié le : 2011-02-24

DOI : 10.1016/j.crma.2011.02.009

Affiliations des auteurs :

Gabriele Bonanno ¹ ; Giovanni Molica Bisci ² ; Vicenţiu Rădulescu ^{3, 4}

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     author = {Gabriele Bonanno and Giovanni Molica Bisci and Vicen\c{t}iu R\u{a}dulescu},
     title = {Infinitely many solutions for a class of nonlinear eigenvalue problem in {Orlicz{\textendash}Sobolev} spaces},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {263--268},
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     number = {5-6},
     year = {2011},
     doi = {10.1016/j.crma.2011.02.009},
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Gabriele Bonanno; Giovanni Molica Bisci; Vicenţiu Rădulescu. Infinitely many solutions for a class of nonlinear eigenvalue problem in Orlicz–Sobolev spaces. Comptes Rendus. Mathématique, Volume 349 (2011) no. 5-6, pp. 263-268. doi : 10.1016/j.crma.2011.02.009. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2011.02.009/

Bibliographie
Cité par

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[4] Ph. Clément; M. García-Huidobro; R. Manásevich; K. Schmitt Mountain pass type solutions for quasilinear elliptic equations, Calc. Var., Volume 11 (2000), pp. 33-62

[5] Ph. Clément; B. de Pagter; G. Sweers; F. de Thélin Existence of solutions to a semilinear elliptic system through Orlicz–Sobolev spaces, Mediterr. J. Math., Volume 1 (2004), pp. 241-267

[6] M. Garciá-Huidobro; V.K. Le; R. Manásevich; K. Schmitt On principal eigenvalues for quasilinear elliptic differential operators: an Orlicz–Sobolev space setting, Nonlinear Differential Equations Appl. (NoDEA), Volume 6 (1999), pp. 207-225

[7] A. Kristály; M. Mihăilescu; V. Rădulescu Two non-trivial solutions for a non-homogeneous Neumann problem: an Orlicz–Sobolev space setting, Proc. Roy. Soc. Edinburgh Sect. A, Volume 139 (2009), pp. 367-379

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