The paper concerns universal a priori estimates for positive solutions to a large class of elliptic quasilinear equations and systems involving the p-Laplacian operator on arbitrary domains of RN and a convective term in the reaction. Our main theorems, new even for the Laplacian operator, extend previous estimates by Pol´aˇcik, Quitter and Souplet in [38] to very general nonlinearities admitting solely a lower bound, yielding a curious dichotomy. The main ingredients are a key doubling property, a rescaling argument, different from the classical blow-up technique of Gidas and Spruck, and Liouville-type theorems for inequalities. A discussion on the sharpness of the exponent in the power type term is also included.
