On the Lane-Emden conjecture

TL;DR

This paper proves the Lane-Emden conjecture in a new parameter region and provides an alternative proof for the case when the space dimension is four or less, using Sobolev embeddings and scale invariance.

Contribution

It introduces a novel approach to confirm the Lane-Emden conjecture in a broader setting and offers a different proof for low-dimensional cases.

Findings

Conjecture holds in a new parameter region

Alternative proof for space dimension N ≤ 4

Method based on Sobolev embeddings and scale invariance

Abstract

We consider the Lane-Emden conjecture which states that there is no non-trivial non-negative solution for the Lane-Emden system whenever the pair of exponents is subcritical. By Sobolev embeddings on $S^{N-1}$ and scale invariance of the solutions, we show this conjecture holds in a new region. Our methods can also be used to prove the Lane-Emden conjecture in space dimension $N\leq 4$, that is to give a different proof of the main result of Souplet in Adv. Math. 2009.