Non-linear Elliptical Equations on the Sierpifiski Gasket

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 Au + a ( x ) u = f ( x , u ) , with zero Dirichlet boundary conditions, where u is defined on the Sierpifiski 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.