Abstract
We use the variational minimizing method to study the existence of new nontrivial periodic solutions with a prescribed energy for second order Hamiltonian systems with singular potential , which may have an unbounded potential well. MSC:34C15, 34C25, 58F.
Highlights
Introduction and main results For singularHamiltonian systems with a fixed energy h ∈ R, q + V (q) =, ( . )|q| + V (q) = h.Ambrosetti-Coti Zelati [, ] used Ljusternik-Schnirelmann theory on an C manifold to get the following theorem.Theorem . (Ambrosetti-Coti Zelati [ ]) Suppose V ∈ C (Rn\{ }, R) satisfies (A )V (u) → –∞, u →, (A )V (u) · u + V (u)u, u =,V (u) · u >, u =,(A ) ∃α >, s.t.V (u) · u ≤ –αV (u)
If u ∈ such that f (u ) = and f (u ) >, we find that q(t) = u (t/T) is a non-constant T-periodic solution for ( . )-( . )
Proof By Lemma . and Lemmas . - . , we know that the functional f (u) attains the infimum in ; we claim that inf f (u) >
Summary
(Ambrosetti-Coti Zelati [ ]) Suppose V ∈ C (Rn\{ }, R) satisfies (A ) By symmetry condition (V ), similar to Ambrosetti-Coti Zelati [ ], let (Gordon [ ]) Let V satisfy the so-called Gordon Strong Force condition: There exist a neighborhood N of and a function U ∈ C ( , R) such that: (i) lims→ U(x) = –∞; (ii) –V (x) ≥ |U (x)| for every x ∈ N – { }. Assume φ(u) ≡ +∞ and is weakly lower semi-continuous on ∩ E, and that it is coercive on ∩ E: φ(u) → +∞, u → +∞
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
