### Morse decompositions in the absence of uniqueness

DOI: http://dx.doi.org/10.12775/TMNA.2001.032

#### Abstract

In this paper we define attractors and Morse

decompositions in an abstract framework of curves in a metric

space. We establish some basic properties of these concepts

including their stability under perturbations. This extends

results known for flows and semiflows on metric spaces to large

classes of ordinary or partial differential equations with

possibly nonunique solutions of the Cauchy problem. As an

application, we first prove a Morse equation in the context of a

Conley index theory

which was recently defined in

[M. Izydorek and K. P. Rybakowski, < i> On the Conley index in Hilbert spaces in the

absence of uniqueness< /i> , Fund. Math.] for problems without uniqueness,

and then apply this

equation to give an elementary proof of two multiplicity results for strongly

indefinite elliptic systems previously obtained in

[S. Angenent and R. van der Vorst, < i> A superquadratic indefinite elliptic system and

its Morse–Conley–Floer homology< /i> , Math. Z. < b> 231< /b> (1999), 203–248] using Morse-Floer

homology.

decompositions in an abstract framework of curves in a metric

space. We establish some basic properties of these concepts

including their stability under perturbations. This extends

results known for flows and semiflows on metric spaces to large

classes of ordinary or partial differential equations with

possibly nonunique solutions of the Cauchy problem. As an

application, we first prove a Morse equation in the context of a

Conley index theory

which was recently defined in

[M. Izydorek and K. P. Rybakowski, < i> On the Conley index in Hilbert spaces in the

absence of uniqueness< /i> , Fund. Math.] for problems without uniqueness,

and then apply this

equation to give an elementary proof of two multiplicity results for strongly

indefinite elliptic systems previously obtained in

[S. Angenent and R. van der Vorst, < i> A superquadratic indefinite elliptic system and

its Morse–Conley–Floer homology< /i> , Math. Z. < b> 231< /b> (1999), 203–248] using Morse-Floer

homology.

#### Keywords

Attractor-repeller pairs; Morse decompositions; Conley index; strongly indefinite elliptic systems

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