Symmetry results for cooperative elliptic systems in unbounded domains

In this paper we prove symmetry results for classical solutions of semilinear cooperative elliptic systems in $\R^N$, $N\geq 2$ 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.