Asymptotic profile and Morse index of nodal radial solutions to the Hénon problem

We compute the Morse index of nodal radial solutions to the H'enon problem [left{egin{array}{ll} -Delta u = |x|^alpha |u|^p-1 u qquad & ext{ in } B, ewline u= 0 & ext{ on } partial B, end{array} ight. ] where $B$ stands for the unit ball in $mathbb R^N$ in dimension $Nge 3$, $alpha>0$ and $p$ is near at the threshold exponent for existence of solutions $p_alpha=racN+2+2alphaN-2$, obtaining that egin{align*} m(u_p) & = m sumlimits_j=0^1+left[alpha/2 ight] N_j quad & mbox if $alpha$ is not an even integer, or ewline m(u_p)& = msumlimits_j=0^ alpha /2 N_j + (m-1) N_1+alpha/ 2 & mbox if $alpha$ is an even number. end{align*} Here $N_j$ denotes the multiplicity of the spherical harmonics of order $j$. The computation builds on a characterization of the Morse index by means of a one dimensional singular eigenvalue problem, and is carried out by a detailed picture of the asymptotic behavior of both the solution and the singular eigenvalues and eigenfunctions. In particular it is shown that nodal radial solutions have multiple blow-up at the origin. As side outcome we see that solutions are nondegenerate for $p$ near at $p_alpha$, and we give an existence result in perturbed balls.