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

Bonanno, Gabriele; Molica Bisci, Giovanni; Rădulescu, Vicenţiu

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

Bonanno, Gabriele ¹ ; Molica Bisci, Giovanni ² ; Rădulescu, Vicenţiu ^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, Tome 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 :

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

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     title = {Infinitely many solutions for a class of nonlinear eigenvalue problem in {Orlicz{\textendash}Sobolev} spaces},
     journal = {Comptes Rendus. Math\'ematique},
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Bonanno, Gabriele; Molica Bisci, Giovanni; Rădulescu, Vicenţiu. Infinitely many solutions for a class of nonlinear eigenvalue problem in Orlicz–Sobolev spaces. Comptes Rendus. Mathématique, Tome 349 (2011) no. 5-6, pp. 263-268. doi : 10.1016/j.crma.2011.02.009. http://www.numdam.org/articles/10.1016/j.crma.2011.02.009/

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