A sharp gradient estimate for the weighted p-Laplacian

Let M be an n-dimensional complete noncompact Riemannian manifold with sectional curvature bounded from below, d?? = e ^h (x) dV (x) the weighted measure and ??_??,p the weighted p-Laplacian. In this paper we consider the non-linear elliptic equation $$ \Delta _{\mu ,p} u = - \lambda _{\mu ,p} |u|^{p - 2} u $$ for p ?? (1, 2). We derive a sharp gradient estimate for positive smooth solutions of this equation. As applications, we get a Harnack inequality and a Liouville type theorem..