Morse index and uniqueness of positive solutions of the Lane-Emden problem in planar domains

TL;DR

This paper calculates the Morse index of spike solutions to a semilinear elliptic problem in planar domains and establishes the uniqueness of positive solutions for large exponents in convex domains, confirming a conjecture by Gidas-Ni-Nirenberg.

Contribution

It provides the first computation of the Morse index for 1-spike solutions in 2D and proves the uniqueness of positive solutions for large p in convex domains.

Findings

Morse index of spike solutions computed for large p

Uniqueness of solutions established in convex domains for large p

Confirms Gidas-Ni-Nirenberg conjecture in 2D for large p

Abstract

We compute the Morse index of $1$-spike solutions of the semilinear elliptic problem \begin{equation}\label{abstr} \tag{$\mathcal P_p$} \begin{cases} -\Delta u= u^p & \text{in $\Omega$} \\ u=0 & \text{on $\partial\Omega$} \\ u>0 & \text{in $\Omega$.} \end{cases} \end{equation} where $\Omega\subset \mathbb{R}^2$ is a smooth bounded domain and $p>1$ is sufficiently large. When $\Omega$ is convex, our result, combined with the characterization in [22], a result in [41] and with recent uniform estimates in \cite{Sirakov}, gives the uniqueness of the solution to \eqref{abstr}, for $p$ large. This proves, in dimension two and for $p$ large, a conjecture by Gidas-Ni-Nirenberg [29].