Semilinear PDEs on self-similar fractals

A Laplacian may be defined on self-similar fractal domains in terms of a suitable self-similar Dirichlet form, enabling discussion of elliptic PDEs on such domains. In this context it is shown that that semilinear equations such as Delta u + u(p) = 0, with zero Dirichlet boundary conditions, have non-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 dimension" or "spectral dimension" of the system. Thus the intrinsic dimension takes the role of the Euclidean dimension in the classical case in determining critical exponents of semilinear problems.