Let m > 1 be a real number and let Ω ⊂ ℝn, n ≥ 2, be a connected smooth domain. Consider the system of quasi-linear elliptic differential equations div (|∇u|m-2∇u) + f (u,v) = in Omega;, div (|∇v|m-2∇v) + g (u,v) = in Omega;, where u ≥ 0, v ≥ 0, f and g are real functions. Relations between the Liouville non-existence and a priori estimates and existence on bounded domains are studied. Under appropriate conditions, a variety of results on a priori estimates, existence and non-existence of positive solutions have been established. © 2008 RAS(DoM) and LMS.