A superquadratic indefinite elliptic system and its Morse–Conley–Floer homology

Abstract.

We study critical points of the indefinite functional $f_H(u, v)=\int\{\nabla u\cdot\nabla v- H(x, u, v) \}\mathrm{d} x$ by applying Floer's homology construction to the ordinary gradient flow of the functional f on a suitable Sobolev space. One of our main observations is that even though this flow is well posed in both time directions and lacks any kind of smoothing property one can still obtain compactness of connecting orbit spaces and thus define the Floer homology for $f_H$.