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

TL;DR

This paper links the Morse index of radial solutions to semilinear PDEs with a singular eigenvalue problem, providing a detailed characterization that aids in understanding solution stability and symmetry properties.

Contribution

It introduces a novel characterization of the Morse index and degeneracy for solutions to semilinear PDEs via a singular eigenvalue problem, including symmetry analysis.

Findings

Characterization of Morse index through a singular eigenvalue problem

Detailed analysis of eigenfunction symmetries

Provides a lower bound for Morse index in future work

Abstract

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.