Liouville theorems and new gradient estimates for 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 particular, under certain assumptions on a, p and q we show whether or not the exact Cheng-Yau log-gradient estimates for the positive solutions to p u+ au q = 0 on (M, g) with Ricci lower bound hold true is equivalent to whether or not the positive solutions to this equation fulfill Harnack inequality, and hence some new Cheng-Yau log-gradient estimates are established.417 3.4.Moser iteration for positive solutions of (1-1) 420 4. Proof of the main theorem and its consequences 422