# Monotonicity of the Morse index of radial solutions of the H\'enon equation in dimension two

## TL;DR

This paper studies the Morse index of radial solutions to a weighted nonlinear PDE in 2D, showing it increases with the weight parameter and diverges as the parameter grows large.

## Contribution

It proves the monotonicity of the Morse index with respect to the weight parameter and establishes a lower bound that diverges as the parameter tends to infinity.

## Key findings

- Morse index is non-decreasing with respect to 
- Morse index tends to infinity as  increases
- Provides lower bounds for Morse indices

## Abstract

We consider the equation \[ -\Delta u = |x|^{\alpha} |u|^{p-1}u, \ \ x \in B, \ \ u=0 \quad \text{on} \ \ \partial B, \] where $B \subset {\mathbb R}^2$ is the unit ball centered at the origin, $\alpha \geq0$, $p>1$, and we prove some results on the Morse index of radial solutions. The contribution of this paper is twofold. Firstly, fixed the number of nodal sets $n\geq1$ of the solution $u_{\alpha,n}$, we prove that the Morse index $m(u_{\alpha,n})$ is monotone non-decreasing with respect to $\alpha$. Secondly, we provide a lower bound for the Morse indices $m(u_{\alpha, n})$, which shows that $m(u_{\alpha, n}) \to +\infty$ as $\alpha \to + \infty$.