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Mountain pass lemma and new periodic solutions of the singular second order Hamiltonian system
Boundary Value Problems volume 2014, Article number: 49 (2014)
We generalize the classical Ambrosetti-Rabinowitz mountain pass lemma with the Palais-Smale condition for 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.
where (, ) and ; denotes the gradient of the function defined on Ω.
In 1975, Gordon  firstly used variational methods to study periodic solutions of planar 2-body type problems, he assumed the condition nowadays called Gordon’s strong force condition.
Condition (V1): There exists a neighborhood of 0 and a function such that:
for every and .
Suppose that is T-periodic in t; as regards the behavior of at infinity, they suppose that one of the following conditions holds.
Condition (V2): , (uniformly for t) and for every , .
Condition (V3): There exist such that, for every and with :
Condition (V4): There exist , , such that, for every , :
Setting , they got the following results.
Theorem 1.1 (Greco )
If (V1) and one of (V2)-(V4) hold, and moreover , then there is at least one non-constant T-periodic solution.
Suppose that , so ; moreover suppose we have the following condition.
Condition (V5): K is compact (or empty).
Then, if (V1) and one of (V2)-(V4) hold, there exist infinitely many non-constant T-periodic solutions.
In this paper, we prove the following new theorem.
Theorem 1.3 Suppose satisfies the conditions:
(V1) For the given , .
(V2) , .
(V3) There is , such that for any given and
(V4) There exists such that ,
(V5) as uniformly for .
Then the system (1.1) has at least a non-constant T-periodic solution.
Corollary 1.1 Suppose , , , , and
then , (1.1) has at least a T-periodic solution.
2 A few lemmas
Lemma 2.1 (Sobolev-Rellich-Kondrachov )
and the embedding is compact.
Lemma 2.2 (Eberlein-Shmulyan )
A Banach space X is reflexive if and only if any bounded sequence in X has a weakly convergent subsequence.
Lemma 2.3 ()
Let and , then we have Wirtinger’s inequality:
Let and , then we have Sobolev’s inequality:
Let ϕ be a convex function on the real line; is a non-negative real-valued function which is Lebesgue-integrable, then
Lemma 2.4 (Ekeland )
Let X be a Banach space; suppose that Φ defined on X is Gateaux-differentiable and lower semi-continuous and bounded from below. Then there is a sequence such that
Definition 2.1 (Palais and Smale )
Let X be a Banach space; , if s.t.
and has a strongly convergent subsequence; then we say that f satisfies the condition.
Cerami  presented a weaker compact condition than the above classical condition.
Definition 2.2 ()
Let X be a Banach space, , and suppose that Φ is defined on Λ is Gateaux-differentiable, if the sequence is such that
then has a strongly convergent subsequence in Λ.
Then we say that f satisfies the condition.
We can give a weaker condition than the condition.
Definition 2.3 Let X be a Banach space, , and suppose that Φ defined on Λ is Gateaux-differentiable; if the sequence is such that
and has a weakly convergent subsequence in Λ, then we say that f satisfies the condition.
Lemma 2.5 (Ambrosetti-Rabinowitz , mountain pass lemma)
Let X be a Banach space, , . We have
If there are two points , such that
then , where . If f satisfies the condition on , furthermore, if as , then C is a critical value for f.
3 The proof of Theorem 1.3
If satisfies the conditions (V1)-(V2), let
then the critical point of on Λ is a T-periodic solution of (1.1).
Lemma 3.2 If V satisfies (V3), (V4) in Theorem 1.1, then f satisfies the Cerami-Palais-Smale condition for any , that is, for any :
has a strongly convergent subsequence and the limit is in Λ.
Proof By the condition (V3), it is well known  that as . Since , we know that for any given , there exists N such that when , we have
By , we have
So by (V4) and (3.2) and (3.4), we have such that when n large enough, we have
So is bounded. Then has a weakly convergence subsequence, and it is standard to further prove that this subsequence is strongly convergent in Λ.
Now we can prove our theorem.
In order to apply for Ambrosetti-Rabinowitz’s mountain pass lemma, we notice that
so by (V3) and Wirtinger’s inequality we have
It is easy to see that if , ϕ attains its infimum which is a positive number. , we can take , take , . By Sobolev’s inequality, we know that , so if , then the above proofs hold.
Let us choose constant value vector in , . Then by (V1) and (V5), we have
So if is large enough, we have
By Lemmas 2.5 and 3.2, f has a critical value , and the corresponding critical point is a T-periodic solution of the system (1.1). Furthermore, we claim that the critical point is non-constant; in fact, if otherwise, by the anti- periodic property, we know that the critical point must be constant zero, which is impossible since . □
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The authors declare that they have no competing interests.
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
About this article
- Ambrosetti-Rabinowitz’s mountain pass lemma
- singular second-order Hamiltonian systems
- periodic solutions
- Cerami-Palais-Smale condition