Nodal sets and morse indices of solutions of super-linear elliptic PDEs

In this paper, we develop a Sturm-Liouville type theory for the nodal sets and Morse indices of solutions of super-linear elliptic PDEs with Dirichlet boundary condition. It shows that there are some relationships between analytic properties (e.g., L-P-norm, vanishing order of the nodal point, and N - 1 dimensional Hausdorff measure of the nodal set) of the solutions as the functions on the domain Omega and Morse indices of the solutions as the critical points of the functional J(u)= integral(Omega) [1/2 \del u\(2) - F(x, u)] dx on H-0(1)(Omega). (C) 1998 Academic Press.