Symmetry results for cooperative elliptic systems in unbounded domains

TL;DR

This paper establishes symmetry properties for solutions of cooperative elliptic systems in unbounded domains, showing solutions are foliated Schwarz symmetric under Morse index bounds, and deriving nonexistence results.

Contribution

It provides new symmetry results for classical solutions of fully coupled cooperative elliptic systems in unbounded domains, under convexity and Morse index conditions.

Findings

Solutions are foliated Schwarz symmetric if Morse index is bounded.

Nonexistence theorems are derived as a consequence of symmetry.

Results apply to systems in R^N and exterior of a ball.

Abstract

In this paper we prove symmetry results for classical solutions of semilinear cooperative elliptic systems in R^N, or in the exterior of a ball. We consider the case of fully coupled systems and nonlinearities which are either convex or have a convex derivative. The solutions are shown to be foliated Schwarz symmetric if a bound on their Morse index holds. As a consequence of the symmetry results we also obtain some nonexistence theorems.