Morse index computation for radial solutions of the Hénon problem in the disk

Abstract

We compute the Morse index m(up) of any radial solution up of the semilinear problem: (P)−Δu=|x|α|u|p−1uinBu=0on∂Bwhere B is the unit ball of R2 centered at the origin, α≥0 is fixed and p>1 is sufficiently large. In the case α=0, i.e. for the Lane–Emden problem, this leads to the following Morse index formula m(up)=4m2−m−2,for p large enough, where m is the number of nodal domains of u.