Multiple solutions of Hamiltonian systems via limit relative category
Let \(E\) be a Hilbert space, \(V\) a manifold of class \(C^ 2\) and \(f \in C^ 1(E \times V, \mathbb{R})\) a strongly indefinite functional. In [\textit{G. Fourier, D. Lupo, M. Ramos} and \textit{M. Willem}, [Dynamics reported. Expositions in dynamical systems. New series Vol. 3, 1-24 (1994; Zbl 0794.58007)] it has been shown that if \(f\) satisfies a compactness condition of Palais-Smale type and a certain linking condition, then \(f\) possesses at least \(1 + \text{cuplength}(V)\) critical points. The main result of this paper asserts that the number of critical points is at least \(2+2 \text{cuplength}(V)\) provided a stronger linking condition is satisfied (roughly speaking, there are two subsets of \(E\), say \(A\) and \(B\), such that \(A\) is linked to \(B\), \(B\) is linked to \(A\) and the corresponding minimax levels are distinct). The paper ends in an application of the above result to asymptotically linear Hamiltonian system with periodic nonlinearity and to a superquadratic Hamiltonian system.
scientific article; zbMATH DE number 711774