Morse index computation for radial solutions of the {H\'e}non problem in the disk

TL;DR

This paper calculates the Morse index of radial solutions to a semilinear PDE in a disk, revealing a specific formula for the Lane-Emden case when the parameter p is large.

Contribution

It provides an explicit Morse index formula for radial solutions of the Hénon problem in the disk, especially for large p in the Lane-Emden case, extending understanding of solution stability.

Findings

Morse index formula for Lane-Emden problem: 4m^2 - m - 2

Morse index computed for large p

Explicit relation between Morse index and nodal domains

Abstract

We compute the Morse index $\textsf{m}(u_{p})$ of any radial solution $u_{p}$ of the semilinear problem: \begin{equation} \label{problemaAbstract}\tag{P} \left\{ \begin{array}{lr} -\Delta u=|x|^{\alpha}|u|^{p-1}u & \mbox{in } B\\ u=0 & \mbox{ on }\partial B \end{array} \right. \end{equation} where $B$ is the unit ball of $\mathbb R^{2}$ centered at the origin, $\alpha\geq 0$ is fixed and $p>1$ is sufficiently large. In the case $\alpha=0$, i.e. for the \emph{Lane-Emden problem}, this leads to the following Morse index formula \[\textsf{m}(u_{p}) = 4m^{2}-m-2, \] for $p$ large enough, where $m$ is the number of nodal domains of $u$.