A refined approach for non-negative entire solutions of $\Delta u + u^p = 0$ with subcritical Sobolev growth

TL;DR

This paper refines existing methods to prove that non-negative solutions of the Lane-Emden equation are trivial across all dimensions, extending classical results with a new approach.

Contribution

It introduces an improved method for proving Liouville type theorems for the Lane-Emden equation in all dimensions, expanding previous lower-dimensional results.

Findings

Proves non-negative solutions are trivial in all dimensions

Extends classical Liouville theorems to higher dimensions

Provides an alternative proof method for the Lane-Emden equation

Abstract

Let $N\geq 2$ and $1 < p < (N+2)/(N-2)_{+}$. Consider the Lane-Emden equation $\Delta u + u^p = 0$ in $\mathbb{R}^N$ and recall the classical Liouville type theorem: if $u$ is a non-negative classical solution of the Lane-Emden equation, then $u \equiv 0$. The method of moving planes combined with the Kelvin transform provides an elegant proof of this theorem. A classical approach of J. Serrin and H. Zou, originally used for the Lane-Emden system, yields another proof but only in lower dimensions. Motivated by this, we further refine this approach to find an alternative proof of the Liouville type theorem in all dimensions.