Symmetry results for cooperative elliptic systems in unbounded domains Lucio DamascelliFrancesca GladialiF. Pacella 35B0635B5035J4735G60Cooperative elliptic systemsSymmetryMaximum PrincipleMorse index In this paper, we prove symmetry results for classical solutions of semilinear cooperative elliptic systems in $\mathbb{R}^N$, $N\geq2$, 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. Indiana University Mathematics Journal 2014 text pdf 10.1512/iumj.2014.63.5255 10.1512/iumj.2014.63.5255 en Indiana Univ. Math. J. 63 (2014) 615 - 649 state-of-the-art mathematics http://iumj.org/access/