Asymptotic profile and Morse index of the radial solutions of the Hénon equation

We consider the Henon equation (Pα) − Δ u = | x | α | u | p − 1 u in B N , u = 0 on ∂ B N , where B N ⊂ R N is the open unit ball centered at the origin, N ≥ 3 , p > 1 and α > 0 is a parameter. We show that, after a suitable rescaling, the two-dimensional Lane-Emden equation − Δ w = | w | p − 1 w in B 2 , w = 0 on ∂ B 2 , where B 2 ⊂ R 2 is the open unit ball, is the limit problem of ( P α ) , as α → ∞ , in the framework of radial solutions. We exploit this fact to prove several qualitative results on the radial solutions of ( P α ) with any fixed number of nodal sets: asymptotic estimates on the Morse indices along with their monotonicity with respect to α; asymptotic convergence of their zeros; blow up of the local extrema and on compact sets of B N . All these results are proved for both positive and nodal solutions.