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Morse homology for strongly indefinite functionals on Banach spaces

L. Asselle, S. Cingolani, M. Starostk

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.

Morse homology for strongly indefinite functionals on 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 -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.
Paper Structure (5 sections, 13 theorems, 108 equations)

This paper contains 5 sections, 13 theorems, 108 equations.

Table of Contents

  1. Introduction
  2. Preliminary computations, relative Morse index, and non-degeneracy
  3. The Palais-Smale condition, essentially vertical sets, and precompactness
  4. Fredholm theory and transversality
  5. Functoriality

Key Result

Theorem 1.2

Let $f$ be as in eq:functional, with $G$ satisfying eq:growthG. Then, for a generic choice of $G$, the Morse homology $HM_*(f;\mathds{Z}_2)$ is well-defined and isomorphic to $H_*(X,\mathds{Z}_2)$. As a corollary, the system of quasilinear elliptic problems eq:system has at least one solution for ev

Theorems & Definitions (31)

  • Definition 1.1
  • Theorem 1.2
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • Remark 2.4
  • Lemma 3.1
  • proof
  • ...and 21 more