Nodal Sets and Morse Indices of Solutions of Super-linear Elliptic PDEs

Abstract

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, andN−1 dimensional Hausdorff measure of the nodal set) of the solutions as the functions on the domainΩand Morse indices of the solutions as the critical points of the functional$\text{J(u)=∫}{}_{\mathrm{\Omega }}\text{[}\text{1}\text{2}\text{|∇u|}{}^2\text{−F(x,}\text{u)]}\text{dx}$onH¹₀(Ω).