Symmetry results for cooperative elliptic systems in unbounded domains

Abstract. In this paper we prove symmetry results for classical solutions of semilinear cooperative elliptic systems in R^N, N > 1 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.