Existence of Four Solutions of Some Nonlinear Hamiltonian System

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.

1. Introduction and Statements of Main Results

Let be a function defined on which is -periodic with respect to the first variable .In this paper, we investigate the number of -periodic nontrivial solutions of the following nonlinear Hamiltonian system

(1.1)

where , ,

(1.2)

is the identity matrix on , , and is the gradient of . Let , Then (1.1) can be rewritten as

(1.3)

We assume that satisfies the following conditions.

(H1)There exist constants such that

(1.4)

(H2)Let and be integers and , be any numbers (without loss of generality, we may assume ) such that . Suppose that there exist and such that and

(1.5)

(H 3), , and such that

(1.6)

(H4) is -periodic with respect to .

We are looking for the weak solutions of (1.1). Let . The -periodic weak solution of (1.3) satisfies

(1.7)

and coincides with the critical points of the induced functional

(1.8)

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

Let The scalar product in naturally extends as the duality pairing between and . It is known that if is -periodic, then it has a Fourier expansion with and : is the closure of such functions with respect to the norm

(2.1)

Let us set the functional

(2.2)

,

so that

(2.3)

Let denote the usual bases in and set

(2.4)

Then and are the subspaces of on which is null, positive definite and negative definite, and these spaces are orthogonal with respect to the bilinear form

(2.5)

associated with . Here, and If and , then the bilinear form is zero and . We also note that , and are mutually orthogonal in . Let be the projection from onto and the one from onto . Then the norm in is given by

(2.6)

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 each , E is compactly embedded in . In particular, there is an such that

(2.7)

for all .

Proposition 2.2.

Assume that . Then is , that is, is continuous and Fréchet differentiable in with Fréchet derivative

(2.8)

where and Moreover, the functional is

Proof.

For ,

(2.9)

We have

(2.10)

Thus, we have

(2.11)

Next, we prove that is continuous. For ,

(2.12)

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.

Let be a closed subset of with . Let be the relative category of in . We define the limit relative category of in , with respect to , by

(2.13)

We set

(2.14)

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.

Then there exists a lower critical point such that . If

(2.15)

then

(2.16)

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.

Let be a Hilbert space and let , where , , are three closed subspaces of with , of finite dimension. For a given subspace of , let be the orthogonal projection from onto . Set

(2.17)

and let be a function defined on a neighborhood of . Let , . One defines

(2.18)

Assume that

(2.19)

and that the condition holds for on , with respect to the sequrnce , for all , where

(2.20)

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 assume that . Let denote the usual bases in and set

(3.1)

Then is the topological direct sum of subspaces , , and , where and are finite dimensional subspaces. We also set

(3.2)

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

(H 2)^' suppose that there exist and such that and

(3.3)

Then there exist , , , and such that , and for any and with and

(3.4)

Proof.

Let . By we have

(3.5)

for some . Since , there exists such that if , then . Thus, . Moreover, if , then , so we have . Next, we will show that there exist , and such that if , then . Let with , , , where is a small number. Let for some and . Then and . By , there exists such that

(3.6)

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.

We notice that from Lemma 3.1, for fixed , the functional is weakly convex in , while, for fixed , the functional is strictly concave in . Moreover, is the critical point in with . So if is another critical point for , then we have

(3.7)

So we have .

Let be the orthogonal projection from onto and

(3.8)

Then is the smooth manifold with boundary. Let . Let us define a functional by

(3.9)

We have

(3.10)

Let us define the functional by

(3.11)

Then . We note that if is the critical point of and lies in the interior of , then is the critical point of . We also note that

(3.12)

Let us set

(3.13)

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

Lemma 3.3.

satisfies the condition with respect to for every real number such that

(3.14)

Proof.

Let be a sequence such that , be a sequence in such that , for all , and . Set (and hence ) and . We first consider the case in which , for all . Since for large , we have

(3.15)

By (3.9) and (3.10),

(3.16)

In the first case, the claim follows from the limit Palais-Smale condition for . In the second case, . We claim that is bounded. By contradiction, we suppose that and set . Up to a subsequence weakly for some . By the asymptotically linearity of , we have

(3.17)

We have

(3.18)

where . Passing to the limit we, get

(3.19)

Since and are bounded and in , . On the other hand, we have

(3.20)

Moreover, we have

(3.21)

Since converges to 0 weakly and is bounded, . Since , converges to 0 strongly, which is a contradiction. Hence, is bounded. Up to a subsequence, we can suppose that converges to for some . We claim that converges to strongly. We have

(3.22)

By and the boundedness of ,

(3.23)

That is, converges. Since , converges, so converges to strongly. Therefore, we have

(3.24)

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.

Assume that satisfies the conditions , , , . Then there exists a number such that for any and with and , there exist at least two nontrivial critical points , , in for the functional such that

(3.25)

where , , and are introduced in Lemma 3.1.

Proof.

First, we will find two nontrivial critical points for . By Lemma 3.1, satisfies the torus-sphere variational linking inequality, that is, there exist , , , and such that , and for any and with and

(3.26)

By Lemma 3.3, satisfies the condition with respect to for every real number such that

(3.27)

Thus by Theorem 2.5, there exist two critical points , for the functional such that

(3.28)

Setting , , we have

(3.29)

We claim that , that is , which implies that are the critical points for in , so are the critical points for in . For this we assume by contradiction that . From (3.12), , namely, , , are the critical points for . By Lemma 3.2, , which is a contradiction for the fact that

(3.30)

Lemma 3.2 implies that there is no critical point such that

(3.31)

Hence, , . This proves Proposition 3.4.

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

Assume that satisfies the conditions , , , and the condition

(H 2)^'' suppose that there exist and such that and

(3.32)

Then there exist , , , and such that , and for any and with and ,

(3.33)

Proof.

Let . By we have

(3.34)

for some . Since , there exists such that if , then . Thus we have . Moreover, if , then , so we have . Next, let with , where is a small number. We also let and . Then and . By , there exists such that

(3.35)

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.

By contradiction, we can suppose that there exist , a sequence , such that , with , and a sequence in such that and . We claim that is bounded. If we do not suppose that , let us set . We have up to a subsequence, that weakly for some . Furthermore,

(3.36)

so we have

(3.37)

Moreover,

(3.38)

so we have

(3.39)

Adding (3.37) and (3.39), we have

(3.40)

From (3.40) we have

(3.41)

We also have

(3.42)

Dividing by and going to the limit, we have

(3.43)

Thus

(3.44)

which is a contradiction since . So is bounded and we can suppose that for . From (3.42), we have

(3.45)

From (3.40),

(3.46)

Thus, converges to strongly. We claim that . Assume that . By (H 1) , for some and . If with for and ,

(3.47)

If , , and

(3.48)

Thus, we have

(3.49)

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

Let be the orthogonal projection from onto and

(3.50)

Then is the smooth manifold with boundary. Let . Let us define a functional by

(3.51)

We have

(3.52)

Let us define the functional by

(3.53)

Then . We note that if is the critical point of and lies in the interior of , then is the critical point of . We also note that

(3.54)

Let us set

(3.55)

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.

satisfies the condition with respect to for every real number such that

(3.56)

where , , and are introduced in Lemma 3.5.

Proposition 3.8.

Assume that satisfies the conditions , , , and . Then there exists a small number such that for any and with and , there exist at least two nontrivial critical points , , in for the functional such that

(3.57)

where , , and are introduced in Lemma 3.5.

Proof.

It suffices to find the critical points for . By Lemma 3.5, satisfies the torus-sphere variational linking inequality, that is, there exist , , , and such that , and for any and with

(3.58)

By Lemma 3.7, satisfies the condition with respect to for every real number such that

(3.59)

Then by Theorem 2.5, there exist two critical points , for the functional such that

(3.60)

Setting , , we have

(3.61)

We claim that , that is , which implies that are the critical points for , so are the critical points for . For this we assume by contradiction that . From (3.54), , namely, , , are the critical points for . By Lemma 3.6, , which is a contradiction for the fact that

(3.62)

It follows from Lemma 3.6 that there is no critical point such that

(3.63)

Hence, , . This proves Proposition 3.8.

Proof of Theorem 1.1.

Assume that satisfies conditions –. By Proposition 3.4, there exist , , , and such that for any and with ,(1.1) has at least two nontrivial solutions , , in for the functional such that

(3.64)

By Proposition 3.8, there exist , , , and such that for any and with and , (1.1) has at least two nontrivial solutions , , in for the functional such that

(3.65)

Let

(3.66)

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

Assume that satisfies conditions – with . Let us set

(4.1)

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

Proof of Theorem 1.2.

By the same arguments as that of the proof of Theorem 1.1, there exist , , , , , and such that for any and with , (1.1) has at least four nontrivial solutions, two of which are nontrivial solutions , , in with

(4.2)

and two of which are nontrivial solutions , ,in with

(4.3)