  We compute the Morse index of nodal radial solutions to the H\'enon problem
  \[\left\{\begin{array}{ll}
  -\Delta u = |x|^{\alpha}|u|^{p-1} u \qquad & \text{ in } B, \newline
  u= 0 & \text{ on } \partial B,
  \end{array} \right.
  \]
  where $B$ stands for the unit ball in ${\mathbb R}^N$ in dimension $N\ge 3$,
$\alpha>0$ and $p$ is near at the threshold exponent for existence of solutions
$p_{\alpha}=\frac{N+2+2\alpha}{N-2}$, obtaining that
  \begin{align*}
  m(u_p) & = m \sum\limits_{j=0}^{1+\left[{\alpha}/{2}\right]} N_j \quad &
\mbox{ if $\alpha$ is not an even integer, or} \newline
  m(u_p)& = m\sum\limits_{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, where each node converges (up to
a suitable rescaling) to the bubble shaped solution of a limit problem.
  As side outcome we see that solutions are nondegenerate for $p$ near at
$p_{\alpha}$, and we give an existence result in perturbed balls.

Analysis of PDEs