A Morse-Smale index theorem for indefinite elliptic systems and bifurcation

TL;DR

This paper extends the Morse index theorem to indefinite elliptic systems on star-shaped domains and applies it to analyze bifurcation phenomena as the domain shrinks, broadening previous scalar equation results.

Contribution

The authors generalize the Morse index theorem to indefinite elliptic systems and demonstrate its application to bifurcation analysis in semilinear systems with domain shrinking.

Findings

Established a Morse-Smale index theorem for indefinite elliptic systems.

Applied the theorem to identify bifurcation points as domains shrink.

Extended previous scalar equation bifurcation results to systems.

Abstract

We generalise the semi-Riemannian Morse index theorem to elliptic systems of partial differential equations on star-shaped domains. Moreover, we apply our theorem to bifurcation from a branch of trivial solutions of semilinear systems, where the bifurcation parameter is introduced by shrinking the domain to a point. This extends recent results of the authors for scalar equations.