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

We compute the Morse index of 1-spike solutions of the semilinear elliptic problem {-Delta u = u(p )in Omega u = 0 on partial derivative Omega u > 0 in Omega (P-p) where Omega subset of R-2 is a smooth bounded domain and p > 1 is sufficiently large. When Omega 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. (C) 2019 Elsevier Masson SAS. All rights reserved.