Gradient Estimates and Liouville Theorems for a Class of Quasilinear Elliptic Equations on Complete Riemannian Manifolds

In this paper, we consider the gradient estimates of positive solutions to the following quasilinear elliptic equation<br /> Delta(p)u + bu(q )F(ln(u + 1)) = 0<br /> defined on a complete Riemannian manifold (M, g), where p > 1 and F is a smooth function. We employ the Nash-Moser iteration technique to obtain some refined gradient estimates of the solutions to the above equation, if (M, g) satisfies Ric >= -(n-1)kappa, where n is the dimension of M and kappa is a nonnegative constant. By the obtained gradient estimates, we also derive a Liouville type theorem for the above equation under some suitable geometric and analysis conditions.