On a singular eigenvalue problem and its applications in computing the Morse index of solutions to semilinear PDE's (preprint)

We investigate nodal radial solutions to semilinear problems of type {−Δu=f(|x|,u) in Ω,u=0 on ∂Ω, where Ω is a bounded radially symmetric domain of RN (N≥2) and f is a real function. We characterize both the Morse index and the degeneracy in terms of a singular one dimensional eigenvalue problem, and describe the symmetries of the eigenfunctions. Next we use this characterization to give a lower bound for the Morse index; in such a way we give an alternative proof of an already known estimate for the autonomous problem and we furnish a new estimate for H\'enon type problems with f(|x|,u)=|x|αf(u). Concerning the real H\'enon problem, f(|x|,u)=|x|α|u|p−1u, we prove radial nondegeneracy and show that the radial Morse index is equal to the number of nodal zones.