Variational analysis for a nonlinear elliptic problem on the Sierpiński gasket

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

ESAIM: Control, Optimisation and Calculus of Variations, Tome 18 (2012) no. 4, pp. 941-953.

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

Classification : 35J20, 28A80, 35J25, 35J60, 47J30, 49J52
Mots clés : Sierpiński gasket, nonlinear elliptic equation, Dirichlet form, weak laplacian

[1] S. Alexander, Some properties of the spectrum of the Sierpiński gasket in a magnetic field. Phys. Rev. B 29 (1984) 5504-5508. | MR

[2] G. Bonanno and R. Livrea, Multiplicity theorems for the Dirichlet problem involving the p-Laplacian. Nonlinear Anal. 54 (2003) 1-7. | MR | Zbl

[3] G. Bonanno and G. Molica Bisci, Infinitely many solutions for a boundary value problem with discontinuous nonlinearities. Bound. Value Probl. 2009 (2009) 1-20. | EuDML | MR | Zbl

[4] G. Bonanno and G. Molica Bisci, Infinitely many solutions for a Dirichlet problem involving the p-Laplacian. Proc. R. Soc. Edinb. Sect. A 140 (2010) 737-752. | MR | Zbl

[5] G. Bonanno, G. Molica Bisci and D. O'Regan, Infinitely many weak solutions for a class of quasilinear elliptic systems. Math. Comput. Model. 52 (2010) 152-160. | MR | Zbl

[6] B.E. Breckner, D. Repovš and Cs. Varga, On the existence of three solutions for the Dirichlet problem on the Sierpiński gasket. Nonlinear Anal. 73 (2010) 2980-2990. | MR | Zbl

[7] B.E. Breckner, V. Rădulescu and Cs. Varga, Infinitely many solutions for the Dirichlet problem on the Sierpiński gasket. Analysis and Applications 9 (2011) 235-248. | Zbl

[8] G. D'Aguì and G. Molica Bisci, Infinitely many solutions for perturbed hemivariational inequalities. Bound. Value Probl. 2011 (2011) 1-19. | EuDML | Zbl

[9] G. D'Aguì and G. Molica Bisci, Existence results for an Elliptic Dirichlet problem, Le Matematiche LXVI, Fasc. I (2011) 133-141. | MR | Zbl

[10] K.J. Falconer, Semilinear PDEs on self-similar fractals. Commun. Math. Phys. 206 (1999) 235-245. | MR | Zbl

[11] K.J. Falconer, Fractal Geometry : Mathematical Foundations and Applications, 2nd edition. John Wiley & Sons (2003). | MR | Zbl

[12] K.J. Falconer and J. Hu, Nonlinear elliptical equations on the Sierpiński gasket. J. Math. Anal. Appl. 240 (1999) 552-573. | MR | Zbl

[13] M. Fukushima and T. Shima, On a spectral analysis for the Sierpiński gasket. Potential Anal. 1 (1992) 1-35. | MR | Zbl

[14] 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. | MR | Zbl

[15] J. Hu, Multiple solutions for a class of nonlinear elliptic equations on the Sierpiński gasket. Sci. China Ser. A 47 (2004) 772-786. | Zbl

[16] C. Hua and H. Zhenya, Semilinear elliptic equations on fractal sets. Acta Mathematica Scientica 29 B (2009) 232-242. | MR | Zbl

[17] A. Kristály and G. Moroşanu, New competition phenomena in Dirichlet problems. J. Math. Pures Appl. 94 (2010) 555-570. | Zbl

[18] 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). | Zbl

[19] J. Kigami, Analysis on Fractals. Cambridge University Press, Cambridge (2001). | MR | Zbl

[20] S. Kusuoka, A diffusion process on a fractal. Probabilistic Methods in Mathematical Physics, Katata/Kyoto (1985) 251-274; Academic Press, Boston, MA (1987). | MR | Zbl

[21] B.B. Mandelbrot, How long is the coast of Britain? Statistical self-similarity and fractional dimension, Science 156 (1967) 636-638.

[22] B.B. Mandelbrot, Fractals : Form, Chance and Dimension. W.H. Freeman & Co., San Francisco (1977). | MR | Zbl

[23] B.B. Mandelbrot, The Fractal Geometry of Nature. W.H. Freeman & Co., San Francisco (1982). | MR | Zbl

[24] P. Omari and F. Zanolin, Infinitely many solutions of a quasilinear elliptic problem with an oscillatory potential. Comm. Partial Differential Equations 21 (1996) 721-733. | MR | Zbl

[25] 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. | MR | Zbl

[26] R. Rammal, A spectrum of harmonic excitations on fractals. J. Phys. Lett. 45 (1984) 191-206. | MR

[27] R. Rammal and G. Toulouse, Random walks on fractal structures and percolation clusters. J. Phys. Lett. 44 (1983) L13-L22.

[28] B. Ricceri, A general variational principle and some of its applications. J. Comput. Appl. Math. 113 (2000) 401-410. | MR | Zbl

[29] W. Sierpiński, Sur une courbe dont tout point est un point de ramification. Comptes Rendus (Paris) 160 (1915) 302-305. | JFM

[30] R.S. Strichartz, Analysis on fractals. Notices Amer. Math. Soc. 46 (1999) 1199-1208. | MR | Zbl

[31] R.S. Strichartz, Solvability for differential equations on fractals. J. Anal. Math. 96 (2005) 247-267. | MR | Zbl

[32] R.S. Strichartz, Differential Equations on Fractals, A Tutorial. Princeton University Press, Princeton, NJ (2006). | MR | Zbl