- Research Article
- Open Access

# Existence of Four Solutions of Some Nonlinear Hamiltonian System

- Tacksun Jung
^{1}and - Q-Heung Choi
^{2}Email author

**Received:**25 August 2007**Accepted:**3 December 2007**Published:**14 January 2008

## Abstract

We show the existence of four -periodic solutions of the nonlinear Hamiltonian system with some conditions. We prove this problem by investigating the geometry of the sublevels of the functional and two pairs of sphere-torus variational linking inequalities of the functional and applying the critical point theory induced from the limit relative category.

## Keywords

- Weak Solution
- Bilinear Form
- Orthogonal Projection
- Nontrivial Solution
- Smooth Manifold

## 1. Introduction and Statements of Main Results

We assume that satisfies the following conditions.

(H4) is -periodic with respect to .

where .

Our main results are the following.

Theorem 1.1.

Assume that satisfies conditions – . Then there exists a number such that for any and with , , system (1.1) has at least four nontrivial -periodic solutions.

Theorem 1.2.

Assume that satisfies conditions – . Then there exists a number such that for any and , and , , system (1.1) has at least four nontrivial -periodic solutions.

Chang proved in [1] that, under conditions – ,system (1.1) has at least two nontrivial -periodic solutions. He proved this result by using the finite dimensional variational reduction method. He first investigate the critical points of the functional on the finite dimensional subspace and the condition of the reduced functional and find one critical point of the mountain pass type. He also found another critical point by the shape of graph of the reduced functional.

For the proofs of Theorems 1.1 and ,1.2 we first separate the whole space into the four mutually disjoint four subspaces , , , which are introduced in Section 3 and then we investigate two pairs of sphere-torus variational linking inequalities of the reduced functional and of on the submanifold with boundary and , respectively, and translate these two pairs of sphere-torus variational links of and into the two pairs of torus-sphere variational links of and , where and are the restricted functionals of to the manifold with boundary and , respectively. Since and are strongly indefinite functinals, we use the notion of the condition and the limit relative category instead of the notion of condition and the relative category, which are the useful tools for the proofs of the main theorems. We also investigate the limit relative category of torus in (torus, boundary of torus) on and , respectively. By the critical point theory induced from the limit relative category theory we obtain two nontrivial -periodic solutions in each subspace and , so we obtain at least four nontrivial -periodic solutions of (1.1).

In Section 2, we introduce some notations and some notions of condition and the limit relative category and recall the critical point theory on the manifold with boundary. We also prove some propositions. In Section 3, we prove Theorem 1.1 and in Section 4, we prove Theorem 1.2.

## 2. Recall of the Critical Point Theory Induced from the Limit Relative Category

which is equivalent to the usual one. The space with this norm is a Hilbert space.

We need the following facts which are proved in [2].

Proposition 2.1.

for all .

Proposition 2.2.

where and Moreover, the functional is

Proof.

Similarly, it is easily checked that is .

Now, we consider the critical point theory on the manifold with boundary induced from the limit relative category. Let be a Hilbert space and be the closure of an open subset of such that can be endowed with the structure of manifold with boundary. Let be a functional, where is an open set containing . The condition and the limit relative category (see [3]) are useful tools for the proof of the main theorem.

Let be a sequence of a closed finite dimensional subspace of with the following assumptions: where , for all ( and are subspaces of ), , , are dense in . Let , for any , be the closure of an open subset of and has the structure of a manifold with boundary in . We assume that for any there exists a retraction . For a given , we will write . Let be a closed subspace of .

Definition 2.3.

We have the following multiplicity theorem (for the proof, see [4]).

Theorem 2.4.

Let and assume that

(1) ,

(2) ,

(3)the condition with respect to holds.

Now, we state the following multiplicity result (for the proof, see [4, Theorem 4.6]) which will be used in the proofs of our main theorems.

Theorem 2.5.

Moreover, one assumes and has no critical points in with . Then there exist two lower critical points , for on such that , .

## 3. Proof of Theorem 1.1

We have the following two pairs of the sphere-torus variational linking inequalities.

Lemma 3.1. (First Sphere-Torus Variational Linking).

Assume that satisfies the conditions , , , and the condition

Proof.

Since , , and , there exist a small number and with and such that if and , then . Thus, we have . Moreover, if and , then we have . Thus, . Thus, we prove the lemma.

Lemma 3.2.

Let be the number introduced in Lemma 3.1. Then for any and with and , if is a critical point for , then .

Proof.

So we have .

We note that and have the same topological structure as , , and , respectively.

Lemma 3.3.

Proof.

So we proved the first case.

We consider the case , that is, . Then , for all . In this case, and . Thus, by the same argument as the first case, we obtain the conclusion. So we prove the lemma.

Proposition 3.4.

where , , and are introduced in Lemma 3.1.

Proof.

Hence, , . This proves Proposition 3.4.

Lemma 3.5. (Second Sphere-Torus Variational Linking).

Assume that satisfies the conditions , , , and the condition

Proof.

Since and , there exist a small number and with and such that if and , then . Thus we have .

Moreover, if , then . Thus we have . Thus we prove the lemma.

Lemma 3.6.

For any there exists a constant such that for any and with and , if is a critical point for with , then .

Proof.

*H*1) , for some and . If with for and ,

which is absurd because of and . Thus . We proved the lemma.

We note that and have the same topological structure as , , , and , respectively.

We have the following lemma whose proof has the same arguments as that of Lemma 3.5 except the space , , instead of the space , , .

Lemma 3.7.

where , , and are introduced in Lemma 3.5.

Proposition 3.8.

where , , and are introduced in Lemma 3.5.

Proof.

Hence, , . This proves Proposition 3.8.

Proof of Theorem 1.1.

Then for any and with and , (1.1) has at least four nontrivial solutions, two of which are in and two of which are in .

## 4. Proof of Theorem 1.2

Then the space is the topological direct sum of the subspaces , , , and , where and are finite dimensional subspaces.

Proof of Theorem 1.2.

## Declarations

### Acknowledgment

This research is supported in part by Inha University research grant.

## Authors’ Affiliations

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## Copyright

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