LOGARITHMIC HARNACK INEQUALITIES AND GRADIENT ESTIMATES FOR NONLINEAR p-LAPLACE EQUATIONS ON WEIGHTED RIEMANNIAN MANIFOLDS

In this paper, we prove the logarithmic Harnack inequalities for L-p-Log-Sobolev function on n-dimensional weighted Riemannian manifolds with m-Bakry-Emery Ricci curvature bounded below by -K (m = n,K = 0). Under the assumption of non -negative m-Bakry-Emery Ricci curvature, we obtain a global Li-Yau type gradient estimate and a Hamilton type estimate for the positive solutions to the weighted par-abolic p-Laplace equation with logarithmic nonlinearity. As applications, the corre-sponding Harnack inequalities are derived.