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 $$\{\begin{array}{ll}-\mathrm{\Delta }u=f(\mathrm{\mid }x\mathrm{\mid },u)& \text{ in }\mathrm{\Omega },\\ u=0& \text{ on }\mathrm{\partial }\mathrm{\Omega },\end{array}$$ where $\mathrm{\Omega }$ is a bounded radially symmetric domain of ${\mathbb{R}}^N$ ($N\ge 2$)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.