Morse index and sign changing bubble towers for 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 }\Omega\ u=0\qquad\qquad\qquad\mbox{ on }\partial \Omega \end{array} \right.\tag{${\mathcal{E}}_p$} \end{equation} where $p>1$ and $\mathrm{\Omega }$ is a smooth bounded symmetric domain of ${\mathbb{R}}^2$. We show that for families $(u_p)$ of sign-changing symmetric solutions of \eqref{problemAbstract} an upper bound on their Morse index implies concentration of the positive and negative part, $u_p^{\pm }$, at the same point, as $p\to +\mathrm{\infty }$. Then an asymptotic analysis of $u_p^{+}$ and $u_p^{-}$ shows that the asymptotic profile of $(u_p)$, as $p\to +\mathrm{\infty }$, is that of a tower of two different bubbles.