Semilinear PDEs on self-similar fractals
The author considers a Laplacian on self-similar fractal domains defined in terms of a suitable self-similar Dirichlet form, see \textit{U. Mosco} [AMS/IP Stud. Adv. Math. 8, 117-155 (1998; Zbl 0914.31007)]. It is shown that the semilinear equations such as \(\Delta u +u^p=0\) with zero Dirichlet boundary conditions have nontrivial nonnegative solution for any \(p>1\) if \(0<\nu\leq 2\), and for \(1<p<(\nu+2)/(\nu-2)\) if \(\nu>2\), where \(\nu\) is the intrinsic dimension of the fractal domain.
scientific article; zbMATH DE number 1382933