In this paper, we consider the Hénon 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 2π/n. By considerations based on the Morse index we see that, depending on the values of α and n, such least energy solutions can be radial, or nonradial and different one from another.

中文翻译：

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渐近线性 Hénon 问题

在本文中，我们考虑具有狄利克雷边界条件的球中的 Hénon 问题。我们研究了径向解的渐近分布，然后推导出它们的摩尔斯指数的精确计算，当指数p接近1. 接下来，我们关注平面情况并描述一些解决方案的渐近分布，这些解决方案最小化函数之间的能量，这些函数对于给定角度的反射和旋转是不变的2π/n. 通过基于莫尔斯指数的考虑，我们看到，取决于α和n，这种最小能量的解决方案可以是径向的，也可以是非径向的，并且彼此不同。