A New proof of Liouville type theorems for a class of semilinear elliptic equations

TL;DR

This paper presents a novel proof of Liouville theorems for specific semilinear elliptic equations using an integral identity and invariant tensor method, applicable in Euclidean space and on manifolds with nonnegative Ricci curvature.

Contribution

It introduces a new proof technique for Liouville theorems based on integral identities and invariant tensors, extending classical results to broader settings.

Findings

Reestablishment of classical Liouville theorems

Development of a new integral identity method

Application to equations on Euclidean space and manifolds with nonnegative Ricci curvature

Abstract

We study certain typical semilinear elliptic equations in Euclidean space $\bR^{n}$ or on a closed manifold $M$ with nonnegative Ricci curvature. Our proof is based on a crucial integral identity constructed by the invariant tensor method. Together with suitable integral estimates, some classical Liouville theorems will be reestablished.