Logarithmic Harnack inequalities and gradient estimates for nonlinear $p$-Laplace equations on weighted Riemannian manifolds

Abstract

In this paper, we prove the logarithmic Harnack inequalities for $L^p$-Log-Sobolev function on $n$-dimensional weighted Riemannian manifolds with $m$-Bakry-Émery Ricci curvature bounded below by $-K$ $(m\ge n, K\ge0)$. Under the assumption of nonnegative $m$-Bakry-Émery Ricci curvature, we obtain a global Li-Yau type gradient estimate and a Hamilton type estimate for the positive solutions to the weighted parabolic $p$-Laplace equation with logarithmic nonlinearity. As applications, the corresponding Harnack inequalities are derived.