Symmetry Results for Cooperative Elliptic Systems via Linearization

Abstract

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