Gradient estimates and Liouville-type theorems for a weighted nonlinear elliptic equation

We consider gradient estimates for positive solutions to the following nonlinear elliptic equation on a smooth metric measure space (M,g,e−fdv)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(M, g,e^{-f}\,dv)$\end{document}: Δfu+aulogu+bu=0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta_{f} u+au\log u+bu=0, $$\end{document} where a, b are two real constants. When the ∞-Bakry–Émery Ricci curvature is bounded from below, we obtain a global gradient estimate which is not dependent on |∇f|\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$|\nabla f|$\end{document}. In particular, we find that any bounded positive solution of the above equation must be constant under some suitable assumptions.