Semilinear PDEs on self-similar fractals

A Laplacian may be defined on self-similar fractal domains in terms of a su itable self-similar Dirichlet form, enabling discussion of elliptic PDEs on such domains. In this context it is shown that that semilinear equations s uch as Delta u + u(p) = 0, with zero Dirichlet boundary conditions, have no n-trivial non-negative solutions if 0 < nu less than or equal to 2 and p > 1, or if nu > 2 and 1 < p < (nu + 2)/(nu - 2), where nu is the "intrinsic d imension" or "spectral dimension" of the system. Thus the intrinsic dimensi on takes the role of the Euclidean dimension in the classical case in deter mining critical exponents of semilinear problems.