Existence results and a priori estimates for solutions of quasilinear problems with gradient terms

A priori estimates and an existence theorem of positive solutions for a Dirichlet problem on a bounded smooth domain in $mathbb R^N$ with a nonlinearity involving gradient terms are given. The existence result is proved with no use of a Liouville theorem for the limit problem obtained via the usual blow up method, in particular we refer to the modified version by Ruiz in J. Diff. Eqs. in 2004. In particular our existence theorem extends a result by Lorca and Ubilla in J. Math. Anal. Appl. (2010) in two directions, namely by considering a nonlinearity which includes in the gradient term a power of u and by removing the growth condition for the nonlinearity f at u=0.