Index estimates for strongly indefinite functionals, periodic orbits and homoclinic solutions of first order Hamiltonian systems

Abstract

We improve Benci and Rabinowitz's Linking theorem for strongly indefinite functionals, giving estimates for a suitably defined relative Morse index of critical points. Such abstract result is applied to the existence problem of periodic orbits and homoclinic solutions of first order Hamiltonian systems in cases where the Palais-Smale condition does not hold.

Morse homology on Hilbert spaces

“…If V is a closed subspace of E, P V will denote the orthogonal projection on V , and V ⊥ the orthogonal complement of V in E. The following definition was given in [1] (see [4] for an updated presentation): DEFINITION 1.1 Two closed subspaces V and W of E are said to be commensurable if the operator P V − P W is compact.…”

Nonnegative solutions for nonlinear elliptic systems

“…Applying linking type theorems in [1] and [2], we find a critical point (u j , v j ) of I j with the relative Morse index M(u j , v j ) 1. This allows us to establish our main results:…”

“…E − is the subspace of E on which the quadratic form Q(z) is negative, see [2] and [8] for further development.…”

Spike-layered solutions for an elliptic system with Neumann boundary conditions

“…in E − ). If (u, v) is a solution of (P) j , we denote by m(u, v) its relative Morse index, as defined in [1,2] (the definition of m(u, v) is recalled in the proof of Lemma 1.2 below). Now we can state the main result of this section.…”

“…[19]. Since, anyway, it is our aim to prove a priori bounds for solutions of (P), we find it more convenient to consider a modified problem, by combining ideas from [2], [3], [18].…”