A symmetry result for semilinear cooperative elliptic systems
  In this paper we prove symmetry results for classical solutions of nonlinear cooperative elliptic systems in a ball or in annulus in R^N, with N greather or equal than 2. More precisely we prove that solutions having Morse index j smaller than N are foliated Schwarz symmetric if the nonlinearity has a convex derivative and a full coupling condition is satisfied along the solution. 
L.Damascelli, F.Gladiali, F.Pacella
math.AP