Infinitely many solutions for a class of nonlinear elliptic problems on fractals

Gabriele Bonanno; Giovanni Molica Bisci; Vicenţiu Rădulescu

doi:10.1016/j.crma.2012.01.027

Partial Differential Equations

Infinitely many solutions for a class of nonlinear elliptic problems on fractals

Presented by: 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 Dipartimento MECMAT, University of Reggio Calabria, Via Graziella, Feo di Vito, 89124 Reggio Calabria, Italy
³ Institute of Mathematics “Simion Stoilow” of the Romanian Academy, 014700 Bucharest, Romania
⁴ Department of Mathematics, University of Craiova, A.I. Cuza Street No. 13, 200585 Craiova, Romania

Comptes Rendus. Mathématique, Volume 350 (2012) no. 3-4, pp. 187-191.

Abstract
Résumé

We study the nonlinear problem $Δ u + a (x) u = λ g (x) f (u)$ in $V ∖ V_{0}$ , $u = 0$ on $V_{0}$ , where V is the Sierpiński gasket, $V_{0}$ is its intrinsic boundary, Δ denotes the weak Laplace operator, λ is a positive parameter, and f has an oscillatory behaviour either near the origin or at infinity. In both cases, we establish the existence of infinitely many solutions, which either converge to zero or have larger and larger energies.

On étudie le problème non linéaire $Δ u + a (x) u = λ g (x) f (u)$ dans $V ∖ V_{0}$ , $u = 0$ sur $V_{0}$ , où V est le joint de culasse de Sierpiński, $V_{0}$ est sa frontière intrinsèque, Δ dénote lʼopérateur de Laplace au sens faible, λ est un paramètre positif et f a un comportement oscillatoire autour de lʼorigine ou à lʼinfini. Dans les deux cas on établit lʼexistence dʼune infinité de solutions, qui ou bien convergent vers à zéro, ou bien ont des énergies de plus en plus grandes.

Received: 2011-12-26
Accepted: 2012-01-28
Published online: 2012-02-01

DOI: 10.1016/j.crma.2012.01.027

Author's affiliations:

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 Dipartimento MECMAT, University of Reggio Calabria, Via Graziella, Feo di Vito, 89124 Reggio Calabria, Italy
³ Institute of Mathematics “Simion Stoilow” of the Romanian Academy, 014700 Bucharest, Romania
⁴ Department of Mathematics, University of Craiova, A.I. Cuza Street No. 13, 200585 Craiova, Romania

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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 elliptic problems on fractals},
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     pages = {187--191},
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Gabriele Bonanno; Giovanni Molica Bisci; Vicenţiu Rădulescu. Infinitely many solutions for a class of nonlinear elliptic problems on fractals. Comptes Rendus. Mathématique, Volume 350 (2012) no. 3-4, pp. 187-191. doi : 10.1016/j.crma.2012.01.027. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2012.01.027/

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