Liouville theorems and new gradient estimatesfor positive solutions to Δpu + auq= 0 on a complete manifold

We use the Saloff-Coste Sobolev inequality and the Nash-Moser iteration method to study the local and global behaviors of positive solutions to the nonlinear elliptic equation p u + au q = 0 defined on a complete Riemannian manifold (M, g) with Ricci lower bound, where p > 1 is a constant and p u = div(|u| p-2 u) is the usual p-Laplace operator.Under certain assumptions on a, p and q, we derive some gradient estimates and Liouville type theorems for positive solutions to the above equation.In pa