On the asymptotically linear H\'enon problem

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.