Unbounded sets of solutions of non-cooperative elliptic systems on symmetric spaces

TL;DR

This paper demonstrates that for certain non-cooperative elliptic systems on symmetric spaces, any continuum of solutions branching from trivial solutions is unbounded, using degree theory and symmetry considerations.

Contribution

It introduces an approach to prove unboundedness of solution continua in symmetric elliptic systems using equivariant degree theory.

Findings

Any bifurcating solution continuum is unbounded.

The method applies to elliptic systems on compact symmetric spaces.

The analysis leverages the torus-equivariant structure of eigenspaces.

Abstract

The aim of this paper is to show that, for a class of non-cooperative elliptic systems on compact symmetric spaces, any continuum of nontrivial solutions bifurcating from the set of trivial solutions is unbounded. The main tool is the degree for invariant strongly indefinite functionals. The analysis relies on the torus-equivariant structure of the Laplace--Beltrami eigenspaces. The result is obtained by ruling out return to the trivial branch in an equivariant version of the Rabinowitz alternative.