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

This paper studies gradient estimates for positive solutions of the nonlinear elliptic equation Delta V(up)+lambda u=0,p >= 1,$$\begin{aligned} \Delta _V(u<^>p)+\lambda u=0,\quad p\ge 1, \end{aligned}$$\end{document}on a Riemannian manifold (M, g) with k-Bakry-emery 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.