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

TL;DR

The paper proves a new nonexistence region for the Lane-Emden system $- abla^2 u = v^p$, $- abla^2 v = u^q$ in $\mathbb{R}^n$ by leveraging an Obata-type integral inequality, Picone's identity, and the system's scaling invariance. By parameterizing the exponents with $d_1,d_2$ and carefully selecting auxiliary coefficients, the authors derive universal growth and positivity estimates that lead to a Liouville-type nonexistence result for $n\ge 5$ when $\frac{1}{p+1}+\frac{1}{q+1}\ge 1-\frac{2}{n}+\frac{4}{n^2}$. The argument blends analytic inequalities with targeted numerical verification (via Mathematica) to establish positivity of critical coefficients and a cross-term condition $\beta_1\beta_2>\gamma_1\gamma_2$, enabling a contradiction under the assumed existence of positive solutions. This expands the known region in exponents where the Lane-Emden conjecture holds, especially in higher dimensions, and demonstrates how combined analytic and computational techniques can resolve Liouville-type questions in nonlinear elliptic systems.

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 $n \geq 5$ and exponents satisfying $p\geq 1,q\geq 1$, and \[ \frac{1}{p+1} + \frac{1}{q+1} \geq 1 - \frac{2}{n} + \frac{4}{n^2}. \]

Theorems & Definitions (27)

• Conjecture
• Theorem 1.1
• Remark 1.2
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• proof
• Lemma 2.5
• proof