A symmetry result for semilinear cooperative elliptic systems

TL;DR

This paper establishes symmetry properties for solutions of nonlinear cooperative elliptic systems in balls or annuli, showing that solutions with low Morse index are foliated Schwarz symmetric under certain conditions.

Contribution

It proves that solutions with Morse index less than the dimension are foliated Schwarz symmetric when the nonlinearity has convex derivative and the system is fully coupled.

Findings

Solutions with Morse index less than N are foliated Schwarz symmetric.

Symmetry holds under convexity of the nonlinearity's derivative.

Results apply to classical solutions in balls or annuli.

Abstract

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.