Abstract

By use of the Cerami-Palais-Smale condition, we generalize the classical Weierstrass minimizing theorem to the singular case by allowing functions which attain infinity at some values. As an application, we study certain singular second-order Hamiltonian systems with strong force potential at the origin and show the existence of new periodic solutions with fixed periods.

1. Introduction

We are mainly interested in the existence of periodic solutions , with a prescribed period, of the second-order differential equation with (, ) and where denotes the gradient of the function defined on . The study of the periodic solutions of such equations has a substantial literature with the works [1–21] of particular importance for our purpose.

In the 1975 paper of Gordon [10], variational methods were used to study periodic solutions of planar 2-body type problems under assumptions which have come to be known as Gordon’s Strong Force condition ():there exists a neighborhood of 0 and a function such that(i);(ii) for every and ;(iii).

For 2-body type problems in , one can see the works of Ambrosetti-Coti Zelati, Bahri-Rabinowitz, Greco, and other mathematicians from [1–5, 18–21]. In this paper we wish to highlight two main results among them, that is, Theorems 1 and 2. Firstly, we must specify three separate conditions may satisfy about its behavior at infinity. Suppose that is -periodic in ; then, (uniformly for ) and for every , ;there exist such that, for every and with ,(i);(ii);there exist , , , such that, for every , ,(i);(ii).

Set for every ; there hold the following results.

Theorem 1 (Greco [11]). If and and one of hold, then in (1) there is at least one nonconstant -periodic solution.

Theorem 2 (Bahri-Rabinowitz [3] and Greco [11]). Suppose , so that , and the following condition holds:
is compact (or empty).
If and one of hold, then in (1) there exist infinitely many nonconstant -periodic solutions.

In this paper, we prove the following new theorem.

Theorem 3. Suppose satisfies condition and conditionsfor the given , ;there exists such that, , as uniformly for .

Then the system (1) has a -periodic solution.

Corollary 4. Suppose , , , , , and then, , the system (1) has a -periodic solution. From the above example, we see that our potential does not satisfy any of conditions , , and .

2. A Few Lemmas

In order to prove Theorem 3, we will need to recall the following useful lemmas.

Lemma 5 (Sobolev-Rellich-Kondrachov [15]). It is well known that and the imbedding is compact.

Lemma 6 (Eberlein-Shmulyan [15]). A Banach space is reflexive if and only if every bounded sequence in has a weakly convergent subsequence.

Lemma 7 (Ekeland [8]). Let be a Banach space, and suppose defined on is Gateaux-differentiable, lower semicontinuous, and bounded from below. Then there is a sequence such that

Definition 8 (see [8]). Let be a Banach space and . We say satisfies the condition if whenever such that then has a strongly convergent subsequence.

Interestingly, Cerami [22] considers a weaker compact condition on a Banach space than the classical condition. Here we introduce a similar condition in an open subset of a Banach space.

Definition 9 (see [8]). Let be a Banach space; is an open subset; and suppose defined on is Gateaux-differentiable. We say that satisfies the condition if whenever is a sequence such that then has a strongly convergent subsequence in .

With this definition, we can deduce a minimizing result in an open subset of a Banach space, the proof of which is similar to the standard one.

Lemma 10 (see Mawhin-Willem [15]). Let be a Banach space, an open subset, and . Assume has a lower bound on the closure of , and let . If satisfies on and as , then is a critical value for .

Lemma 11 (see [10]). Let , and satisfies the Gordon’s Strong Force condition ; let Define Then as .

3. The Proof of Theorem 3

Let and .

Lemma 12 (see [2]). Suppose satisfies condition and define Then the critical point of is a -periodic solution of (1).

Lemma 13. If satisfies and in Theorem 1, then satisfies the Cerami-Palais-Smale condition for any ; that is, for any , if then has a strongly convergent subsequence and the limit is in .

Proof. By condition and Lemma 11, we must have as . Since , we know that, for any given , there exists such that when , there holds the inequality The limit implies and so Using condition together with the limits and inequalities (11), (12), and (14), we can choose such that when is large enough, there holds which implies is bounded.
In the following we prove that is bounded; otherwise, there is a subsequence, still denoted by , such that Then by Newton-Leibniz’s formula, we have Now, by and , we have which contradicts the limit (14).
Hence, is bounded; has a weakly convergent subsequence; we still denote it by , and let the limit be . We can show in a standard fashion that this subsequence is strongly convergent in . To complete the proof, we write it out.
Since the sequence is bounded in , so, by Sobolev’s embedding inequality, we know it is also bounded in maximum norm, and, by condition and Lemma 11, we know that when is large, By , when is large, is also uniformly bounded in maximum norm; we have Taking and in the above equation, we get Since , hence ; furthermore, since is bounded, so . Hence, by (21) and the uniformly bounded property for , we have By weakly, we have By Sobolev Embedding Theorem, has a subsequence, still denoted by subject to .
We notice That is, strongly in . Then by Lemma 10 the proof of Theorem 3 is complete.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The author sincerely thanks the referee for his/her valuable comments and suggestions.