A priori estimates and existence of solutions for nonlinear problems involving the p-laplacian

In this work we study the existence of positive solutions for a nonlinear Dirichlet problem involving the m-laplacian. The nonlinearity considered depends on the first derivatives; in such case, variational methods cannot be applied. So, we make use of topological methods to prove the existence of solutions. We combine a blow-up argument and a Lionville type theorem to obtain a priori estimates. Some Harnack type inequalities which are needed in our reasonings are also proved.