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

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

doi:10.1016/j.crma.2012.01.027

Partial Differential Equations

Infinitely many solutions for a class of nonlinear elliptic problems on fractals
[Infinité de solutions pour une classe de problèmes elliptiques non linéaires sur des fractales]

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 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, Tome 350 (2012) no. 3-4, pp. 187-191.

Résumé
Abstract

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.

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.

Reçu le : 2011-12-26
Accepté le : 2012-01-28
Publié le : 2012-02-01

DOI : 10.1016/j.crma.2012.01.027

Affiliations des auteurs :

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

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Bonanno, Gabriele; Molica Bisci, Giovanni; Rădulescu, Vicenţiu. Infinitely many solutions for a class of nonlinear elliptic problems on fractals. Comptes Rendus. Mathématique, Tome 350 (2012) no. 3-4, pp. 187-191. doi : 10.1016/j.crma.2012.01.027. http://www.numdam.org/articles/10.1016/j.crma.2012.01.027/

Bibliographie
Cité par

[1] Bonanno, G.; Molica Bisci, G. Infinitely many solutions for a boundary value problem with discontinuous nonlinearities, Bound. Value Probl., Volume 2009 (2009), pp. 1-20

[2] G. Bonanno, G. Molica Bisci, V. Rădulescu, Variational analysis for a nonlinear elliptic problem on the Sierpiński gasket, ESAIM Control Optim. Calc. Var., , in press. | DOI

[3] Breckner, B.E.; Rădulescu, V.; Varga, Cs. Infinitely many solutions for the Dirichlet problem on the Sierpiński gasket, Anal. Appl., Volume 9 (2011), pp. 235-248

[4] Falconer, K.J. Fractal Geometry: Mathematical Foundations and Applications, John Wiley & Sons, 2003

[5] Falconer, K.J.; Hu, J. Nonlinear elliptical equations on the Sierpiński gasket, J. Math. Anal. Appl., Volume 240 (1999), pp. 552-573

[6] Fukushima, M.; Shima, T. On a spectral analysis for the Sierpiński gasket, Potential Anal., Volume 1 (1992), pp. 1-35

[7] Mandelbrot, B. Les objets fractals: forme, hasard et dimension, Flammarion, Paris, 1973

[8] Ricceri, B. A general variational principle and some of its applications, J. Comput. Appl. Math., Volume 113 (2000), pp. 401-410

[9] Sierpiński, W. Sur une courbe dont tout point est un point de ramification, C. R. Acad. Sci. Paris, Volume 160 (1915), pp. 302-305

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