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

We compute the Morse index m ( u p ) of any radial solution u p of the semilinear problem: (P) − Δ u = | x | α | u | p − 1 u in B u = 0 on ∂ B where B is the unit ball of R 2 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 ( u p ) = 4 m 2 − m − 2 , for p large enough, where m is the number of nodal domains of u .