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 ( P p ) { − Δ u = u p  in  Ω u = 0  on  ∂ Ω u > 0  in  Ω where Ω ⊂ R 2 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 ( P p ) , for p large. This proves, in dimension two and for p large, a longstanding conjecture.