Existence of multiple critical points for an asymptotically quadratic functional with applications
Let \(X\) be a Hilbert space and let \(f\in{\mathcal C}^1(X)\) be an asymptotically quadratic functional with resonance at infinity, so \(f'(x)=Ax+o(|x|)\) as \(|x|\to\infty\) and the selfadjoint operator \(A\in{\mathcal{L}}(X)\) has a kernel. The abstract results of the paper yield the existence of one or two critical points provided a certain angle condition at infinity holds (plus some standard conditions). The proof depends on computations of the critical groups at infinity and at isolated, possibly degenerate critical points. The results are applied to the existence of periodic solutions of asymptotically linear second order Hamiltonian systems with resonance at infinity. The main new contributions are conditions on the Hamiltonian which imply the angle condition for the associated functional.
periodic solutions of Hamiltonian systems
scientific article; zbMATH DE number 1370660