Abstract
We generalize the classical Ambrosetti-Rabinowitz mountain pass lemma with the Palais-Smale condition for C 1 functional to some singular case with the Cerami-Palais-Smale condition and then we study the existence of new periodic solutions with a fixed period for the singular second-order Hamiltonian systems with a strong force potential.MSC:34C15, 34C25, 58F.
Highlights
1 Introduction Many authors [ – ] studied the existence of periodic solutions t → x(t) ∈ , with a prescribed period, of the following second-order differential equations: x = –V (t, x),
We prove the following new theorem
By Lemmas . and . , f has a critical value C >, and the corresponding critical point is a T-periodic solution of the system ( . )
Summary
1 Introduction Many authors [ – ] studied the existence of periodic solutions t → x(t) ∈ , with a prescribed period, of the following second-order differential equations: x = –V (t, x), Suppose that V (t, x) is T-periodic in t; as regards the behavior of V (t, x) at infinity, they suppose that one of the following conditions holds. (Greco [ ]) If (V ) and one of (V )-(V ) hold, and K = ∅, there is at least one non-constant T-periodic C solution.
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