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

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} -Δu= u^p & \text{in $Ω$} \\ u=0 & \text{on $\partialΩ$} \\ u>0 & \text{in $Ω$.} \end{cases} \end{equation} where $Ω\subset \mathbb{R}^2$ is a smooth bounded domain and $p>1$ is sufficiently large. When $Ω$ 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].