  We consider the H\'enon equation \begin{equation}\label{alphab} -\Delta u =
|x|^{\alpha}|u|^{p-1}u \ \ \textrm{in} \ \ B^N, \quad
  u = 0 \ \ \textrm{on}\ \ \partial B^N,
  \tag{$P_{\alpha}$} \end{equation} where $B^N\subset \mathbb{R}^N$ is the open
unit ball centered at the origin, $N\geq 3$, $p>1$ and $\alpha> 0$ is a
parameter. We show that, after a suitable rescaling, the two-dimensional
Lane-Emden equation \[ -\Delta w = |w|^{p-1}w\quad \text{in}\ B^2,\quad
w=0\quad \text{on}\ \partial B^2, \] where $B^2 \subset \mathbb{R}^2$ is the
open unit ball, is the limit problem of \eqref{alphab}, as $\alpha \to \infty$,
in the framework of radial solutions. We exploit this fact to prove several
qualitative results on the radial solutions of \eqref{alphab} with any fixed
number of nodal sets: asymptotic estimates on the Morse indices along with
their monotonicity with respect to $\alpha$; 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.
