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Variational analysis for a nonlinear elliptic problem on theSierpiński gasket

Published online by Cambridge University Press:  16 January 2012

Gabriele Bonanno ,
Giovanni Molica Bisci  and
Vicenţiu Rădulescu
Gabriele Bonanno
Affiliation:
Department of Science for Engineering and Architecture (Mathematics Section) Engineering Faculty, University of Messina, 98166 Messina, Italy. [email protected]
Giovanni Molica Bisci
Affiliation:
Dipartimento MECMAT, University of Reggio Calabria, Via Graziella, Feo di Vito, 89124 Reggio Calabria, Italy; [email protected]
Vicenţiu Rădulescu
Affiliation:
Institute of Mathematics “Simion Stoilow” of the Romanian Academy, P.O. Box 1-764, 014700 Bucharest, Romania Department of Mathematics, University of Craiova, Street A.I. Cuza No. 13, 200585 Craiova, Romania; [email protected]
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Abstract

Under an appropriate oscillating behaviour either at zero or at infinity of the nonlinearterm, the existence of a sequence of weak solutions for an eigenvalue Dirichlet problem onthe Sierpiński gasket is proved. Our approach is based on variational methods and on someanalytic and geometrical properties of the Sierpiński fractal. The abstract results areillustrated by explicit examples.

Keywords

Sierpiński gasketnonlinear elliptic equationDirichlet formweak Laplacian
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 18 , Issue 4 , October 2012 , pp. 941 - 953
Copyright
© EDP Sciences, SMAI, 2012

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References

Alexander, S., Some properties of the spectrum of the Sierpiński gasket in a magnetic field. Phys. Rev. B 29 (1984) 5504–5508. Google Scholar
Bonanno, G. and Livrea, R., Multiplicity theorems for the Dirichlet problem involving the p-Laplacian. Nonlinear Anal. 54 (2003) 1–7. Google Scholar
Bonanno, G. and Molica Bisci, G., Infinitely many solutions for a boundary value problem with discontinuous nonlinearities. Bound. Value Probl. 2009 (2009) 1–20. Google Scholar
Bonanno, G. and Molica Bisci, G., Infinitely many solutions for a Dirichlet problem involving the p-Laplacian. Proc. R. Soc. Edinb. Sect. A 140 (2010) 737–752. Google Scholar
Bonanno, G., Bisci, G. Molica and O’Regan, D., Infinitely many weak solutions for a class of quasilinear elliptic systems. Math. Comput. Model. 52 (2010) 152–160. Google Scholar
Breckner, B.E., Repovš, D. and Varga, Cs., On the existence of three solutions for the Dirichlet problem on the Sierpiński gasket. Nonlinear Anal. 73 (2010) 2980–2990. Google Scholar
Breckner, B.E., Rădulescu, V. and Varga, Cs., Infinitely many solutions for the Dirichlet problem on the Sierpiński gasket. Analysis and Applications 9 (2011) 235–248. Google Scholar
D’Aguì, G. and Molica Bisci, G., Infinitely many solutions for perturbed hemivariational inequalities. Bound. Value Probl. 2011 (2011) 1–19. Google Scholar
G. D’Aguì and G. Molica Bisci, Existence results for an Elliptic Dirichlet problem, Le Matematiche LXVI, Fasc. I (2011) 133–141.
Falconer, K.J., Semilinear PDEs on self-similar fractals. Commun. Math. Phys. 206 (1999) 235–245. Google Scholar
K.J. Falconer, Fractal Geometry : Mathematical Foundations and Applications, 2nd edition. John Wiley & Sons (2003).
Falconer, K.J. and Hu, J., Nonlinear elliptical equations on the Sierpiński gasket. J. Math. Anal. Appl. 240 (1999) 552–573. Google Scholar
Fukushima, M. and Shima, T., On a spectral analysis for the Sierpiński gasket. Potential Anal. 1 (1992) 1–35. Google Scholar
S. Goldstein, Random walks and diffusions on fractals, in Percolation Theory and Ergodic Theory of Infinite Particle Systems, IMA Math. Appl. 8, edited by H. Kesten. Springer, New York (1987) 121–129.
Hu, J., Multiple solutions for a class of nonlinear elliptic equations on the Sierpiński gasket. Sci. China Ser. A 47 (2004) 772–786. Google Scholar
Hua, C. and Zhenya, H., Semilinear elliptic equations on fractal sets. Acta Mathematica Scientica 29 B (2009) 232–242. Google Scholar
Kristály, A. and Moroşanu, G., New competition phenomena in Dirichlet problems. J. Math. Pures Appl. 94 (2010) 555–570. Google Scholar
A. Kristály, V. Rădulescu and C. Varga, Variational Principles in Mathematical Physics, Geometry, and Economics : Qualitative Analysis of Nonlinear Equations and Unilateral Problems. Cambridge University Press, Cambridge (2010).
J. Kigami, Analysis on Fractals. Cambridge University Press, Cambridge (2001).
S. Kusuoka, A diffusion process on a fractal. Probabilistic Methods in Mathematical Physics, Katata/Kyoto (1985) 251–274; Academic Press, Boston, MA (1987).
Mandelbrot, B.B., How long is the coast of Britain? Statistical self-similarity and fractional dimension, Science 156 (1967) 636–638. Google Scholar
B.B. Mandelbrot, Fractals : Form, Chance and Dimension. W.H. Freeman & Co., San Francisco (1977).
B.B. Mandelbrot, The Fractal Geometry of Nature. W.H. Freeman & Co., San Francisco (1982).
Omari, P. and Zanolin, F., Infinitely many solutions of a quasilinear elliptic problem with an oscillatory potential. Comm. Partial Differential Equations 21 (1996) 721–733. Google Scholar
P. Omari and F. Zanolin, An elliptic problem with arbitrarily small positive solutions, Proceedings of the Conference on Nonlinear Differential Equations (Coral Gables, FL, 1999). Electron. J. Differ. Equ. Conf. 5. Southwest Texas State Univ., San Marcos, TX (2000) 301–308.
Rammal, R., A spectrum of harmonic excitations on fractals. J. Phys. Lett. 45 (1984) 191–206. Google Scholar
Rammal, R. and Toulouse, G., Random walks on fractal structures and percolation clusters. J. Phys. Lett. 44 (1983) L13–L22. Google Scholar
Ricceri, B., A general variational principle and some of its applications. J. Comput. Appl. Math. 113 (2000) 401–410. Google Scholar
Sierpiński, W., Sur une courbe dont tout point est un point de ramification. Comptes Rendus (Paris) 160 (1915) 302–305. Google Scholar
Strichartz, R.S., Analysis on fractals. Notices Amer. Math. Soc. 46 (1999) 1199–1208. Google Scholar
Strichartz, R.S., Solvability for differential equations on fractals. J. Anal. Math. 96 (2005) 247–267. Google Scholar
R.S. Strichartz, Differential Equations on Fractals, A Tutorial. Princeton University Press, Princeton, NJ (2006).