Mountain pass lemma and new periodic solutions of the singular second order Hamiltonian system
© Li and Li; licensee Springer. 2014
Received: 2 February 2014
Accepted: 18 February 2014
Published: 27 February 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.
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 , .
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 andwe havewhere
(V4) There exists such that ,
(V5) as uniformly for .
Then the system (1.1) has at least a non-constant T-periodic solution.
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.
- (i)Let and , then we have Wirtinger’s inequality:
- (ii)Let and , then we have Sobolev’s inequality:
- (iii)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 )
Definition 2.1 (Palais and Smale )
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 ()
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.
and has a weakly convergent subsequence in Λ, then we say that f satisfies the condition.
Lemma 2.5 (Ambrosetti-Rabinowitz , mountain pass lemma)
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
then the critical point of on Λ is a T-periodic solution of (1.1).
has a strongly convergent subsequence and the limit is in Λ.
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.
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.
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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