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

We investigate nodal radial solutions to semilinear problems of type −Δu=f(|x|,u)inΩ,u=0on∂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, which is studied in full detail. The presented approach also describes the symmetries of the eigenfunctions. This characterization enables to give a lower bound for the Morse index in a forthcoming work.