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 $$\begin{cases}-\Delta u = f(|x|,u) \qquad & \text{ in } \Omega, \newline u= 0 & \text{ on } \partial \Omega, \end{cases}$$ where $\Omega$ 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.