# Rabinowitz alternative for non-cooperative elliptic systems on geodesic balls

## TL;DR

This paper investigates the properties of solution sets for non-cooperative elliptic systems on geodesic balls in spheres, revealing unbounded continua and symmetry-breaking bifurcations using equivariant bifurcation theory.

## Contribution

It demonstrates the unboundedness of solution continua and the occurrence of symmetry-breaking bifurcations for elliptic systems on geodesic balls, applying equivariant bifurcation techniques.

## Key findings

- Solution continua are unbounded on hemispheres.
- Global symmetry-breaking bifurcation occurs.
- Uses degree theory for SO(n)-invariant functionals.

## Abstract

The purpose of this paper is to study properties of continua (closed connected sets) of nontrivial solutions of non-cooperative elliptic systems considered on geodesic balls in $S^n$. In particular, we have shown that if the geodesic ball is a hemisphere, then these continua are unbounded. It is also shown that the phenomenon of global symmetry-breaking bifurcation of such solutions occurs. Since the problem is variational and SO(n)-symmetric, we apply the techniques of equivariant bifurcation theory to prove the main results of this article. As the topological tool we use the degree theory for SO(n)-invariant strongly indefinite functionals defined in A. Go{\l}\c{e}biewska, S. Rybicki, \emph{Global bifurcations of critical orbits of $G$-invariant strongly indefinite functionals}, Nonl. Anal. {74} (2011), 1823-1834..