Existence of positive solutions for the p-Laplacian with p-gradient term

In this paper, we study the existence of positive solutions for the p-Laplacian involving a p-gradient term. Due to the non-variational structure and the fact that the nonlinearity may be critical or supercritical, the variational method is no longer valid. Taking advantage of global C1,α estimates and the Liouville type theorems, we employ the blow-up argument to obtain the a priori estimates on solutions, and finally obtain the existence result based on the Krasnoselskii fixed point theorem.