A note on the radial solutions for the supercritical Henon equation

TL;DR

This paper proves the existence of positive radial solutions for the supercritical Hénon equation using a shooting method, revealing an increasing solution and suggesting many oscillating solutions through numerical experiments.

Contribution

It introduces a new existence proof for positive radial solutions of the supercritical Hénon equation via a shooting method, highlighting their increasing nature.

Findings

Existence of positive radial solutions for arbitrary growth.

Solutions are increasing functions of the radial variable.

Numerical experiments indicate many oscillating solutions.

Abstract

We prove the existence of a positive radial solution for the H\'enon equation with arbitrary growth. The solution is found by means of a shooting method and turns out to be an increasing function of the radial variable. Some numerical experiments suggest the existence of many positive oscillating solutions.