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(Pp){−Δu=up in Ωu=0 on ∂Ωu>0 in Ω where Ω⊂R2 is a smooth bounded domain and p>1 is sufficiently large.When Ω is convex, our result, combined with the characterization in [21], a result in [40] and with recent uniform estimates in [37], gives the uniqueness of the solution to (Pp), for p large. This proves, in dimension two and for p large, a longstanding conjecture.