A superquadratic indefinite elliptic system and its Morse-Conley-Floer homology (Q1298021)

The authors use Floer's homology construction [\textit{A. Floer}, Commun. Math. Phys. 120, No. 4, 575-611 (1989; Zbl 0755.58022)] to study the critical points of functionals of the type \(f_H(u,v)= \int_\Omega\{ \nabla u\cdot \nabla v-H(x,u,v)\}dx\), where \(H:\Omega \times \mathbb{R}^2\to \mathbb{R}\) is a \(C^2\)-Hamiltonian and \(\Omega \subset \mathbb{R}^n\) is a bounded domain with smooth boundary. Precise conditions on \(H\) for which this theory works are given. Compactness properties of the set of bounded trajectories for the gradient flow on a suitable Sobolev space \(E_\alpha\) are established. Using the generalized Morse index for the \(E_\alpha\) gradient flow of \(f_H\), the definition and continuation properties of the Floer homology groups for isolated neighborhoods are obtained. Based on the constructed theory, various existence results for critical points are proved.