Exact Morse index computation for nodal radial solutions of Lane-Emden problems

TL;DR

This paper precisely computes the Morse index of certain sign-changing solutions to the Lane-Emden problem in two dimensions, revealing it to be exactly 12 for large p, and also determines the first eigenvalue of a related limit problem.

Contribution

It provides an exact computation of the Morse index for least energy sign-changing radial solutions in 2D Lane-Emden problems, a novel explicit eigenvalue calculation for a limit weighted problem.

Findings

Morse index of the solution is exactly 12 for large p in 2D.

Explicit first eigenvalue of a limit weighted problem is obtained.

Results enhance understanding of solution stability in nonlinear PDEs.

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 $\mathbb R^N$, $N\geq2$, centered at the origin and $1<p<p_S$, with $p_S=+\infty$ if $N=2$ and $p_S=\frac{N+2}{N-2}$ if $N\geq3$. Our main result is to prove that in dimension $N=2$ the Morse index of the least energy sign-changing radial solution $u_p$ of \eqref{problemAbstract} is exactly $12$ if $p$ is sufficiently large. As an intermediate step we compute explicitly the first eigenvalue of a limit weighted problem in $\mathbb R^N$ in any dimension $N\geq2$.