Asymptotic profile and Morse index of the radial solutions of the H\'enon equation

Abstract

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.