Gradient estimates of a nonlinear elliptic equation for the V-Laplacian

This paper studies gradient estimates for positive solutions of the nonlinear elliptic equation ΔV(up)+λu=0,p≥1,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Delta _V(u^p)+\lambda u=0,\quad p\ge 1, \end{aligned}$$\end{document}on a Riemannian manifold (M, g) with k-Bakry–Émery Ricci curvature bounded from below. We consider both the case where M is a compact manifold with or without boundary and the case where M is a complete manifold.