A priori estimates, existence and Liouville theorems for semilinear elliptic systems with power nonlinearities

We prove a priori estimates and existence of positive solutions of semilinear elliptic systems with power nonlinearities and homogeneous Dirichlet boundary conditions. In general, our systems are nonvariational and noncooperative, and our estimates are optimal in the class of very weak solutions. We also provide new Liouville-type theorems.