Mountain pass lemma and new periodic solutions of the singular second order Hamiltonian system
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Mountain pass lemma and new periodic solutions of the singular second order Hamiltonian system Bingyu Li1 and Fengying Li2* * Correspondence: [email protected] 2 School of Economic and Mathematics, Southwestern University of Finance and Economics, Chengdu, 611130, China Full list of author information is available at the end of the article
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 Keywords: Ambrosetti-Rabinowitz’s mountain pass lemma; singular second-order Hamiltonian systems; periodic solutions; Cerami-Palais-Smale condition
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),
(.)
where = RN – {} (N ∈ N, N ≥ ) and V ∈ C (R × , R); V (t, ·) denotes the gradient of the function V (t, ·) defined on . In , Gordon [] firstly used variational methods to study periodic solutions of planar -body type problems, he assumed the condition nowadays called Gordon’s strong force condition. Condition (V ): There exists a neighborhood N of and a function U ∈ C (, R) such that: (i) limx→ U(x) = –∞; (ii) –V (t, x) ≥ |U (x)| for every x ∈ N – {} and t ∈ [, T]. Moreover, (iii) limx→ V (t, x) = –∞. In the s and s, Ambsosetti-Coti Zelati, Bahri-Rabinowitz, Greco etc. [–, – ] further studied -body type problems in RN (N ≥ ). 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. Condition (V ): lim|x|→∞ V (t, x) = , lim|x|→∞ V (t, x) = (uniformly for t) and V (t, x) < for every t ∈ [, T], x ∈ . ©2014 Li and Li; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Li and Li Boundary Value Problems 2014, 2014:49 http://www.boundaryvalueproblems.com/content/2014/1/49
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Condition (V ): There exist c , M , R , ν > such that, for every t ∈ [, T] and x ∈ RN with |x| ≥ R : (i) |V (t, x)| ≤ M ; (ii) V (t, x) ≥ c |x|ν . Condition (V ): There exist c , R > , θ > , ν > such that, for every t ∈ [, T], |x| ≥ R : (i) θ V (t, x)x ≤ V (t, x); (ii) V (t, x) ≥ c |x|ν . Setting K = {x ∈ |V (t, x) = for every t ∈ [, T]}, they got the following results. Theorem . (Greco []) If (V ) and one of (V )-(V ) hold, and moreover K = ∅, then there is at least one non-constant T-periodic C solution. Theorem . (Bahri-Rabinowitz [], Greco []) Suppo
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