Non-linear elliptical equations on the Sierpinski gasket

This paper investigates properties of certain nonlinear PDEs on fractal set s. With an appropriately defined Laplacian, we obtain a number of results o n the existence of non-trivial solutions of the semilinear elliptic equatio n 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 an d the saddle point theorem to study such equations for different classes of a and f. A strong Sobolev-type inequality leads to properties that contras t with those for classical domains. (C) 1999 Academic Press.