  In this paper we prove symmetry results for classical solutions of semilinear
cooperative elliptic systems in R^N, or in the exterior of a ball. We consider
the case of fully coupled systems and nonlinearities which are either convex or
have a convex derivative. The solutions are shown to be foliated Schwarz
symmetric if a bound on their Morse index holds. As a consequence of the
symmetry results we also obtain some nonexistence theorems.
