Gradient estimates and Harnack inequalities for nonlinear parabolic and nonlinear elliptic equations on Riemannian manifolds

Abstract

We derive the gradient estimates and Harnack inequalities for positive solutions of nonlinear parabolic and nonlinear elliptic equations (Δ − ∂ ∂t ) u(x, t) + h(x, t)u α(x, t) = 0 and Δu + b · ▽u + hu α = 0 on Riemannian manifolds. We also obtain a theorem of Liouville type for positive solutions of the nonlinear elliptic equation.