Nonlinear elliptical equations on the Sierpinski gasket

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59 Citations (Scopus)

Abstract

This paper investigates properties of certain nonlinear PDEs on fractal sets. With an appropriately defined Laplacian, we obtain a number of results on the existence of non-trivial solutions of the semilinear elliptic equation Delta u + a(x)u = f(x, u), with zero Dirichlet boundary conditions, where u is defined on the Sierpinski gasket. We use the mountain pass theorem and the saddle point theorem to study such equations for different classes of a and f. A strong Sobolev-type inequality leads to properties that contrast with those for classical domains. (C) 1999 Academic Press.

Original languageEnglish
Pages (from-to)552-573
Number of pages22
JournalJournal of Mathematical Analysis and Applications
Volume240
Publication statusPublished - 15 Dec 1999

Keywords

  • Sierpinski gasket
  • Laplacian operator
  • weak solution
  • mountain pass theorem
  • saddle point theorem
  • Sobolev-type inequality
  • FRACTALS

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