Morse homology for strongly indefinite functionals on Banach spaces

TL;DR

This paper develops Morse homology for strongly indefinite functionals on Banach spaces, providing a foundational framework applicable to geometric variational problems and establishing existence results for certain elliptic systems.

Contribution

It introduces a local theory for a class of strongly indefinite functionals on Banach manifolds and formulates general conditions for Morse homology to be well-defined in this setting.

Findings

Established local Morse theory for strongly indefinite functionals

Proved existence of solutions for certain quasilinear elliptic problems

Formulated conditions for well-defined Morse homology in Banach spaces

Abstract

In this paper we lay the foundations for the Morse theoretical study of strongly indefinite functionals on Banach manifolds by developing the local theory for a specific model class that captures several key analytical features also arising in the variational formulations of geometric problems such as Dirac-harmonic maps. As a corollary, we obtain existence results of solutions to certain systems of quasilinear elliptic problems involving the $p$-area functional. Abstracting from the concrete setting, we then formulate general conditions ensuring that Morse homology is well-defined for strongly indefinite functionals on a Banach space.