On a new region for the Lane-Emden conjecture in higher dimensions

Abstract

We study the Lane-Emden conjecture, which asserts the non-existence of non-trivial, non-negative solutions to the Lane-Emden system \[ -Δu = v^p, \quad -Δv = u^q, \quad x \in \mathbb{R}^n\] in the subcritical regime. By employing an Obata-type integral inequality, Picone's identity, and exploiting the scaling invariance of the system, we prove that the conjecture holds for any dimension $n\ge 5$ and exponents satisfying $p\ge 1,q\ge 1$, and \[ \frac{1}{p+1} + \frac{1}{q+1} \geq 1 - \frac{2}{n} + \frac{4}{n^2}. \]

27 pages. Comments are welcome!