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Research Article | Open Access

Local and global properties of positive solutions for Δ u + f ( u ) = 0 on Riemannian manifolds

Fan Chen1, Jingxia Huang2, Qihua Ruan3( )
Fujian Key Laboratory of Financial Information Processing, Putian University, Putian 351100, China
Key Laboratory of Applied Mathematics of Fujian Province University, Putian University, Putian 351100, China
Key Laboratory of Financial Mathematics of Fujian Province University, Putian University, Putian 351100, China
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Abstract

We establish a local gradient estimate for positive solutions of the nonlinear elliptic equation Δ u + f ( u ) = 0 on complete Riemannian manifolds with Ricci curvature bounded below. The proof relies on a new P-function and the Moser iteration technique. As its applications, we prove new Liouville theorems that extend several known results.

Keywords

gradient estimateLiouville theoremMoser iteration
CLC number: 35B35, 35B45, 35B53

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AIMS Mathematics
Volume 11 Issue 4,
April 2026
Pages 11012-11030
DOI: 10.3934/math.2026452

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Cite this article:
Chen F, Huang J, Ruan Q. Local and global properties of positive solutions for Δ u + f ( u ) = 0 on Riemannian manifolds. AIMS Mathematics, 2026, 11(4): 11012-11030. https://doi.org/10.3934/math.2026452

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Received: 23 December 2025
Revised: 06 April 2026
Accepted: 14 April 2026
Published: 20 April 2026
©2026 the Author(s), licensee AIMS Press.

This is an open access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0)