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

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.}.. (read more)

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