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

Abstract

We consider the Hénon equation(Pα)−Δu=|x|α|u|p−1uinBN,u=0on∂BN, where BN⊂RN 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−Δw=|w|p−1winB2,w=0on∂B2, where B2⊂R2 is the open unit ball, is the limit problem of (Pα), as α→∞, in the framework of radial solutions. We exploit this fact to prove several qualitative results on the radial solutions of (Pα) 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 BN. All these results are proved for both positive and nodal solutions.