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

TL;DR

This paper establishes Liouville theorems for the Lane-Emden equation on Riemannian manifolds, proving nonexistence of positive solutions under certain curvature and subcritical conditions, extending classical Euclidean results.

Contribution

It generalizes classical Liouville theorems to Riemannian manifolds for the Lane-Emden equation, including nonnegative Ricci curvature cases, and broadens the scope beyond Euclidean spaces.

Findings

Proves nonexistence of positive solutions on manifolds with nonnegative Ricci curvature.

Extends classical Euclidean Liouville theorems to Riemannian geometry.

Provides rigorous proofs for subcritical Lane-Emden equations on manifolds.

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.