A Morse index formula for radial solutions of Lane-Emden problems

TL;DR

This paper derives a formula for the Morse index of radial solutions to the Lane-Emden problem in a unit ball, showing how the index depends on the number of nodal domains and the dimension, especially near the critical exponent.

Contribution

The paper provides a new explicit Morse index formula for radial solutions of the Lane-Emden problem close to the critical exponent, linking the index to nodal domains and dimension.

Findings

Morse index formula: m + N(m - 1) for solutions near p_S

Morse index depends on nodal domains and dimension

Valid for p close to the critical exponent p_S

Abstract

We consider the semilinear Lane-Emden problem: \begin{equation}\label{problemAbstract}\left\{\begin{array}{lr}-\Delta u= |u|^{p-1}u\qquad \mbox{ in }B u=0\qquad\qquad\qquad\mbox{ on }\partial B \end{array}\right.\tag{$\mathcal E_p$} \end{equation} where $B$ is the unit ball of $\mathbb R^N$, $N\geq3$, centered at the origin and $1<p<p_S$, $p_S=\frac{N+2}{N-2}$. We prove that for any radial solution $u_p$ of \eqref{problemAbstract} with $m$ nodal domains its Morse index $\mathsf{m}(u_p)$ is given by the formula \[\mathsf{m}(u_p)=m+N(m-1)\] if $p$ is sufficiently close to $p_S$.