Liouville type theorems of a nonlinear elliptic equation for the V-Laplacian

In this paper, we consider Liouville type theorems for positive solutions to the following nonlinear elliptic equation: ΔVu+aulogu=0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Delta _V u+au\log u=0, \end{aligned}$$\end{document}where a is a nonzero real constant. By using gradient estimates, we obtain upper bounds of |∇u|\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\nabla u|$$\end{document} with respect to supu\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sup u$$\end{document} and the lower bound of Bakry-Emery Ricci curvature. In particular, for complete noncompact manifolds with a<0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a<0$$\end{document}, we prove that any positive solution must be u≡1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u\equiv 1$$\end{document} under a suitable condition for a with respect to the lower bound of Bakry-Emery Ricci curvature. It generalizes a classical result of Yau.