Gradient Estimates For Positive Solutions Of V-Laplacian Tpye

V-Laplace is the generalization of the classical Laplace and f-Laplace,If V is the zero vector field,the V-Laplace operator is reduced to the classical Laplacian.If V is the gradient of a function,the V-Laplace operator is reduced to the f-Laplace operator.We discuss the gradient estimates for positive solutions of nonlinear equations with V-Laplace,and obtain Liouville theorems and Hamack inequalities.We get results of more types of equations,and the consequences also contain the classical case.We choose the same auxiliary function as the classcial case,then we use the Max-imal principle,the Laplacian comparison theorem and the Cauchy inequality to get a One-Variable Quadratic inequality.Appling the theory about the existence of solutions for a One-Variable Quadratic equation,we hold the estimate of the auxiliary function,and then we can get the gradient estimate for the positive solution.Let(M,g)be a complete Riemann manifold of dimension n.We consider the gradient estimates for positive solutions of the following equations:a nonlinear elliptic equation ?Vu(x)+h(x)ua(x)=0 and a nonlinear parabolic equation(?V-(?))u(x,t)+h(x,t)ua(x,t)=0.When N-Bakry-Emery Ricci curvature and oo-Bakry-Emery Ricci curvature hold lower bounds respectively,we can get the gradient estimates and Hamack inequalities for positive solutions.In this paper,we obtain new results on the gradient estimates for the positive so-lution of the nonlinear parabolic equation under the conditon that the oo-Bakry-Emery Ricci curvature holds a lower bound.And the conclusions can be applied to the classical Laplacian and the f-Laplacian directly under the corresponding conditions.