Symmetry results for cooperative elliptic systems via linearization

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 convexity and coupling conditions.

Contribution

It provides new symmetry results for classical solutions of cooperative elliptic systems based on Morse index, convexity, and coupling assumptions.

Findings

Solutions with Morse index ≤ N are foliated Schwarz symmetric.

Symmetry holds under convexity of the nonlinearity.

Full coupling condition is essential for the symmetry results.

Abstract

In this paper we prove symmetry results for classical solutions of nonlinear cooperative elliptic systems in a ball or in annulus in $\RN$, $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.