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

We study the Lane-Emden conjecture, which asserts the non-existence of non-trivial, non-negative solutions to the Lane-Emden system $$-\mathrm{\Delta }u=v^p,-\mathrm{\Delta }v=u^q,x\in {\mathbb{R}}^n$$ 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 $$\frac{1}{p+1}+\frac{1}{q+1}\ge 1-\frac{2}{n}+\frac{4}{n^2}\mathrm{.}$$