Gradient estimates and Liouville type theorems for (p-1)p-1Δpu+aup-1logup-1=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(p-1)^{p-1}\Delta _pu+au^{p-1}\log u^{p-1}=0$$\end{document} on Riemannian manifolds

In this paper, we study gradient estimates of positive smooth solutions to the p-Laplace equation (p-1)p-1Δpu+aup-1logup-1=0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} (p-1)^{p-1}\Delta _pu+au^{p-1}\log u^{p-1}=0, \end{aligned}$$\end{document}which is related to the Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^p$$\end{document}-log-Sobolev constant on Riemannian manifolds, where a is a nonzero constant. As applications, some Liouville type results are provided.