@article{Chen2026, 
author = {Fan Chen and Jingxia Huang and Qihua Ruan},
title = {Local and global properties of positive solutions for    Δ  u  +  f  (  u  )  =  0 on Riemannian manifolds},
year = {2026},
journal = {AIMS Mathematics},
volume = {11},
number = {4},
pages = {11012-11030},
keywords = {gradient estimate, Liouville theorem, Moser iteration},
url = {https://www.sciopen.com/article/10.3934/math.2026452},
doi = {10.3934/math.2026452},
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.}
}