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

TL;DR

This paper calculates the Morse index of nodal radial solutions to the Hénon problem near the critical exponent, revealing their asymptotic blow-up behavior and nondegeneracy, with implications for solution existence in perturbed domains.

Contribution

It provides a detailed computation of the Morse index for nodal solutions to the Hénon problem, including asymptotic analysis and conditions for nondegeneracy.

Findings

Morse index expressed in terms of spherical harmonics multiplicities.

Nodal solutions exhibit multiple blow-up at the origin.

Solutions are nondegenerate near the critical exponent.

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.