Gradient estimates and Harnack inequalities for nonlinear parabolic and nonlinear elliptic equations on Riemannian manifolds (Q1178601)

The author derives gradient estimates and Harnack inequalities for positive solutions of non-linear parabolic and non-linear elliptic equations \((\Delta-\partial/\partial t)u(x,t)+h(x,t)u^ \alpha(x,t)=0\) and \(\Delta+b\cdot\nabla u +hu^{\alpha}=0\) on complete Riemannian manifolds under conditions on the Ricci tensor and the function \(h\) according to \(\alpha > 1\), \(\alpha = 1\) and \(0 < \alpha < 1\) or \(0 < \alpha < n/(n-1)\) (\(n\) is the dimension of the manifold). Some results in \textit{P. Li} and \textit{S. T. Yau} [Acta Math. 156, 154-201 (1986; Zbl 0611.58045)]are treated in the author's framework. Finally a theorem of Liouville type for positive solutions of the non-linear elliptic equation is proved.