Optimal Liouville theorems for the Lane-Emden equation on Riemannian manifolds

TL;DR

The paper investigates Liouville-type nonexistence results for subcritical Lane-Emden–type equations on complete Riemannian manifolds with nonnegative Ricci curvature. It develops universal Cheng–Yau–type gradient estimates for positive solutions of $-\Delta_p u=f(u)$, enabling global bounds on $|\nabla\ln u|$ and leading to Liouville theorems when the nonlinearity is subcritical (i.e., $\alpha<p_s$). The authors extend classical Euclidean results of Gidas–Spruck and Serrin–Zou to the Riemannian setting, providing a unified framework of local and global gradient bounds, divergence-identity tools, and integral estimates that yield nonexistence results for both compact and noncompact manifolds. The methods combine Moser iteration, Saloff-Coste Sobolev inequalities, Bochner-type identities for the $p$-Laplacian, and Obata-type vector-field arguments to handle various regimes of the nonlinearity.

Abstract

We study degenerate quasilinear elliptic equations on Riemannian manifolds and obtain several Liouville theorems. Notably, we provide rigorous proof asserting the nonexistence of positive solutions to the subcritical Lane-Emden-Fowler equations over complete Riemannian manifolds with nonnegative Ricci curvature. These findings serve as a significant generalization of Gidas and Spruck's pivotal work (Comm. Pure Appl. Math. 34, 525-598, 1981) which focused on the semilinear case, as well as Serrin and Zou's contributions (Acta Math. 189, 79-142, 2002) within the context of Euclidean geometries.

Theorems & Definitions (42)

• Definition 2.1
• theorem 2.1
• theorem 2.2
• theorem 2.3
• theorem 2.4
• theorem 2.5
• Corollary 2.6
• theorem 2.7
• theorem 2.8
• theorem 2.9