# On the asymptotically linear H\'enon problem

## TL;DR

This paper investigates the asymptotic behavior and Morse index of radial solutions to the Hénon problem in a ball, revealing how symmetry and parameters influence solution types as the exponent approaches one.

## Contribution

It provides the first detailed analysis of the asymptotic profile and Morse index of solutions near the critical exponent for the Hénon problem, including symmetry-breaking phenomena.

## Key findings

- Exact Morse index computation for solutions near p=1
- Identification of radial and nonradial least energy solutions
- Dependence of solution symmetry on parameters α and n

## Abstract

In this paper we consider the H\'enon problem in the ball with Dirichlet boundary conditions. We study the asymptotic profile of radial solutions and then deduce the exact computation of their Morse index when the exponent $p$ is close to $1$. Next we focus on the planar case and describe the asymptotic profile of some solutions which minimize the energy among functions which are invariant for reflection and rotations of a given angle $2\pi/n$. By considerations based on the Morse index we see that, depending on the values of $\alpha$ and $n$, such least energy solutions can be radial, or nonradial and different one from another.