  In this paper we consider the H\'enon problem in the ball with Dirichlet
boundary conditions. We study the asymptotic profile of radial solutions and
then deduce the exact computation of their Morse index when the exponent $p$ is
close to $1$. Next we focus on the planar case and describe the asymptotic
profile of some solutions which minimize the energy among functions which are
invariant for reflection and rotations of a given angle $2\pi/n$. By
considerations based on the Morse index we see that, depending on the values of
$\alpha$ and $n$, such least energy solutions can be radial, or nonradial and
different one from another.
