Asymptotic profile and Morse index of nodal radial solutions to the H\'enon problem

Abstract

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.