Comptes Rendus
no. 3-4
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

Gabriele Bonanno1 ; Giovanni Molica Bisci2 ; Vicenţiu Rădulescu3, 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.

On étudie le problème non linéaire Δu+a(x)u=λg(x)f(u) dans V∖V0, u=0 sur V0, où V est le joint de culasse de Sierpiński, V0 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 Δu+a(x)u=λg(x)f(u) in V∖V0, u=0 on V0, where V is the Sierpiński gasket, V0 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.

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DOI : 10.1016/j.crma.2012.01.027
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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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/

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

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

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

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

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

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

[9] W. Sierpiński 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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