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

This paper studies gradient estimates for positive solutions of the nonlinear elliptic equation $$begin{aligned} Delta _V(u^p)+lambda u=0,quad pge 1, end{aligned}$$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.