In this paper we prove nonexistence of positive C-2 solutions for systems o
f semilinear elliptic inequalities, for polyharmonic semilinear inequalitie
s in cones and, under better conditions on the nonlinearity, for bounded po
sitive solutions of elliptic semilinear equations in half spaces. Using a b
low-up argument. these results allow us to prove a-priori bounds for a clas
s of semilinear elliptic systems of equations in bounded domains.