Morse index and sign changing bubble towers for Lane-Emden problems

TL;DR

This paper studies sign-changing solutions to the Lane-Emden problem in symmetric domains, showing that bounded Morse index solutions concentrate at a point and resemble a tower of two bubbles as the exponent grows large.

Contribution

It establishes a link between Morse index bounds and solution concentration, revealing a bubble tower structure in the asymptotic profile for large exponents.

Findings

Solutions concentrate at a single point as p increases.

The positive and negative parts form a bubble tower.

Morse index bounds imply concentration behavior.

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 }\Omega\\ u=0\qquad\qquad\qquad\mbox{ on }\partial \Omega \end{array} \right.\tag{$\mathcal E_p$} \end{equation} where $p>1$ and $\Omega$ 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 $p\to+\infty$. Then an asymptotic analysis of $u_p^+$ and $u_p^-$ shows that the asymptotic profile of $(u_p)$, as $p\to+\infty$, is that of a tower of two different bubbles.