Symmetry results for cooperative elliptic systems via linearization

In this paper we prove symmetry results for classical solutions of nonlinear cooperative elliptic systems in a ball or in annulus in $\text{RN}$, $N\ge 2$. More precisely we prove that solutions having Morse index $j\le N$ are foliated Schwarz symmetric if the nonlinearity is convex and a full coupling condition is satisfied along the solution.