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

## TL;DR

This paper establishes a lower bound for the Morse index of radial solutions to Hénon type PDEs using a singular eigenvalue problem, showing the index tends to infinity as a parameter increases, and explores properties of solutions including nondegeneracy and nodal structure.

## Contribution

It introduces a new lower bound for the Morse index of radial solutions to Hénon problems and analyzes their degeneracy and nodal properties, extending previous characterizations.

## Key findings

- Morse index of solutions tends to infinity as α increases.
- Radial Morse index equals the number of nodal zones.
- Least energy nodal solutions are non-radial.

## Abstract

By using a characterization of the Morse index and the degeneracy in terms of a singular one dimensional eigenvalue problem given in a previous paper, we give a lower bound for the Morse index of radial solutions to H\'enon type problems \[ \left\{\begin{array}{ll} -\Delta u = |x|^{\alpha}f(u) \qquad & \text{ in } \Omega, u= 0 & \text{ on } \partial \Omega, \end{array} \right. \] where $\Omega$ is a bounded radially symmetric domain of $\mathbb R^N$ ($N\ge 2$), $\alpha>0$ and $f$ is a real function. From this estimate we get that the Morse index of nodal radial solutions to this problem goes to $\infty$ as $\alpha\to \infty$. Concerning the real H\'enon problem, $f(u)= |u|^{p-1}u$, we prove radial nondegeneracy, we show that the radial Morse index is equal to the number of nodal zones and we get that a least energy nodal solution is not radial.