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

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 ${\mathbb{R}}^N$, $N\ge 2$, centered at the origin and $1<p<p_S$, with $p_S=+\mathrm{\infty }$ if $N=2$ and $p_S=\frac{N+2}{N-2}$ if $N\ge 3$. 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 $N\ge 2$.