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arXiv · 2510.06613

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

Kui Li·Mingxiang Li·Juncheng Wei
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Abstract

We study the Lane-Emden conjecture, which asserts the non-existence of non-trivial, non-negative solutions to the Lane-Emden system \[ -\Delta u = v^p, \quad -\Delta v = u^q, \quad x \in \mathbb{R}^n\] 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 $n \geq 5$ and exponents satisfying $p\geq 1,q\geq 1$, and \[ \frac{1}{p+1} + \frac{1}{q+1} \geq 1 - \frac{2}{n} + \frac{4}{n^2}. \]

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Kui Li, Mingxiang Li, Juncheng Wei. 2025-10-08. On a new region for the Lane-Emden conjecture in higher dimensions. https://arxiv.org/abs/2510.06613

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