Research

# Infinitely many periodic solutions for subquadratic second-order Hamiltonian systems

Hua Gu* and Tianqing An

Author Affiliations

College of Science, Hohai University, Nanjing, 210098, China

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Boundary Value Problems 2013, 2013:16  doi:10.1186/1687-2770-2013-16

 Received: 8 November 2012 Accepted: 10 January 2013 Published: 6 February 2013

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

In this paper, we investigate the existence of infinitely many periodic solutions for a class of subquadratic nonautonomous second-order Hamiltonian systems by using the variant fountain theorem.

### 1 Introduction

Consider the second-order Hamiltonian systems

{ u ¨ ( t ) + u W ( t , u ) = 0 , t R , u ( 0 ) = u ( T ) , u ˙ ( 0 ) = u ˙ ( T ) , T > 0 , (1.1)

where W ( t , u ) is also T-periodic and satisfies the following assumption (A):

(A) W ( t , u ) is measurable in t for all u R N , continuously differentiable in u for a.e. t [ 0 , T ] and there exist a C ( R + , R + ) and b L 1 ( [ 0 , T ] , R + ) such that

| W ( t , u ) | a ( | u | ) b ( t ) , | u W ( t , u ) | a ( | u | ) b ( t )

for all u R N and a.e. t [ 0 , T ] .

Here and in the sequel, , and | | always denote the standard inner product and the norm in R N respectively.

There have been many investigations on the existence and multiplicity of periodic solutions for Hamiltonian systems via the variational methods (see [1-7] and the references therein). In [6], Zhang and Liu studied the asymptotically quadratic case of W ( t , u ) = 1 2 U ( t ) u , u + W 1 ( t , u ) under the following assumptions:

(AQ1) W 1 ( t , u ) 0 for all ( t , u ) [ 0 , T ] × R N , and there exist constants μ ( 0 , 2 ) and R 1 > 0 such that

u W 1 ( t , u ) , u μ W 1 ( t , u ) , t [ 0 , T ]  and  | u | R 1 ;

(AQ2) lim | u | 0 W 1 ( t , u ) | u | 2 = uniformly for t [ 0 , T ] , and there exist constants c 2 , R 2 > 0 such that

W 1 ( t , u ) c 2 | u | , t [ 0 , T ]  and  | u | R 2 ;

(AQ3) lim inf | u | W 1 ( t , u ) | u | d > 0 uniformly for t [ 0 , T ] .

They obtained the existence of infinitely many periodic solutions of (1.1) provided W 1 ( t , u ) is even inu (see Theorem 1.1 of [6]).

The subquadratic condition (AQ1) is widely used in the investigation of nonlinear differential equations. This condition was weakened by some researchers; see, for example, [4] of Jiang and Tang. This paper considers the case of U ( t ) 0 , then W ( t , u ) = W 1 ( t , u ) . Motivated by [4] and [6], we replace (AQ1) with the following condition:

( AQ 1 ) W ( t , u ) 0 for all ( t , u ) [ 0 , T ] × R N , and

The condition ( AQ 1 ) implies that for some constant R 1 > 0 ,

u W ( t , u ) , u 2 W ( t , u ) , t [ 0 , T ]  and  | u | R 1 . (1.2)

By the assumption (A) and the condition ( AQ 1 ), for any ϵ > 0 , there exists a δ > 0 such that

W ( t , u ) ϵ | u | 2 + max s [ 0 , δ ] a ( s ) b ( t ) , (1.3)

for u R N and a.e. t [ 0 , T ] .

Meanwhile, we weaken the condition (AQ3) to ( AQ 3 ) as follows:

( AQ 3 ) There exists a constant ϱ ( 0 , 1 ] such that

lim inf | u | W ( t , u ) | u | ϱ d > 0 uniformly for  t [ 0 , T ] .

Then our main result is the following theorem.

Theorem 1.1Assume that ( AQ 1 ), (AQ2), ( AQ 3 ) hold and W ( t , u ) is even inu. Then (1.1) possesses infinitely many solutions.

Remark The conditions (AQ1) and (AQ3) are stronger than ( AQ 1 ) and ( AQ 3 ). Then Theorem 1.1 above is different from Theorem 1.1 of [6].

### 2 Preliminaries

In this section, we establish the variational setting for our problem and give the variant fountain theorem. Let E = H T 1 be the usual Sobolev space with the inner product

u , v E = 0 T u ( t ) , v ( t ) d t + 0 T u ˙ ( t ) , v ˙ ( t ) d t .

We define the functional on E by

Φ ( u ) = 1 2 0 T | u ˙ | 2 d t Ψ ( u ) ,

where Ψ ( u ) = 0 T W ( t , u ( t ) ) d t . Then Φ and Ψ are continuously differentiable and

Φ ( u ) , v = 0 T u ˙ , v ˙ d t 0 T u W ( t , u ) , v d t .

Define a self-adjoint linear operator B : L 2 ( [ 0 , T ] ; R N ) L 2 ( [ 0 , T ] ; R N ) by

0 T B u , v d t = 0 T u ˙ ( t ) , v ˙ ( t ) d t

with the domain D ( B ) = E . Then ℬ has a sequence of eigenvalues σ k = 4 k 2 π 2 T 2 ( k = 0 , 1 , 2 , ). Let { e j } j = 0 + be the system of eigenfunctions corresponding to { σ j } j = 0 + , it forms an orthogonal basis in L 2 . Denote by E + = { u E | 0 T u ( t ) d t = 0 } , E 0 = R N , it is well known that

and E possesses orthogonal decomposition E = E 0 E + . For u E , we have

u = u 0 + u + E 0 E + .

We can define on E a new inner product and the associated norm by

u , v 0 = B u + , v + L 2 + u 0 , v 0 L 2 ,

and

u = u , u 0 1 2 .

Therefore, Φ can be written as

Φ ( u ) = 1 2 u + 2 Ψ ( u ) . (2.1)

Direct computation shows that

Ψ ( u ) , v = 0 T u W ( t , u ) , v d t , Φ ( u ) , v = u + , v + 0 Ψ ( u ) , v (2.2)

for all u , v E with u = u 0 + u + and v = v 0 + v + respectively. It is known that Ψ : E E is compact.

Denote by | | p the usual norm of L P , then there exists a τ p > 0 such that

| u | p τ p u , u E . (2.3)

We state an abstract critical point theorem founded in [8]. Let E be a Banach space with the norm and E = j N X j ¯ with dim X j < for any j N . Set Y k = j = 1 k X j and Z k = j = k X j ¯ . Consider the following C 1 -functional Φ λ : E R defined by

Φ λ ( u ) : = A ( u ) λ B ( u ) , λ [ 1 , 2 ] .

Theorem 2.1 [[8], Theorem 2.2]

Assume that the functional Φ λ defined above satisfies the following:

(T1) Φ λ maps bounded sets to bounded sets uniformly for λ [ 1 , 2 ] , and Φ λ ( u ) = Φ λ ( u ) for all ( λ , u ) [ 1 , 2 ] × E ;

(T2) B ( u ) 0 for all u E , and B ( u ) as u on any finite-dimensional subspace ofE;

(T3) There exist ρ k > r k > 0 such that

α k ( λ ) : = inf u Z k , u = ρ k Φ λ ( u ) 0 > β k ( λ ) : = max u Y k , u = r k Φ λ ( u ) , λ [ 1 , 2 ]

and

ξ k ( λ ) : = inf u Z k , u ρ k Φ λ ( u ) 0 as   k   uniformly for   λ [ 1 , 2 ] .

Then there exist λ n 1 , u λ n Y n such that

Φ λ n Y n ( u λ n ) = 0 , Φ λ n ( u λ n ) η k [ ξ k ( 2 ) , β k ( 1 ) ] as   n .

Particularly, if { u λ n } has a convergent subsequence for everyk, then Φ 1 has infinitely many nontrivial critical points { u k } E { 0 } satisfying Φ 1 ( u k ) 0 as k .

In order to apply this theorem to prove our main result, we define the functionals A, B and Φ λ on our working space E by

A ( u ) = 1 2 u + 2 , B ( u ) = 0 T W ( t , u ) d t (2.4)

and

Φ λ ( u ) = A ( u ) λ B ( u ) = 1 2 u + 2 λ 0 T W ( t , u ) d t (2.5)

for all u = u 0 + u + E = E 0 + E + and λ [ 1 , 2 ] . Then Φ λ C 1 ( E , R ) for all λ [ 1 , 2 ] . Let X j = span { e j } , j = 0 , 1 , 2 ,  . Note that Φ 1 = Φ , where Φ is the functional defined in (2.1).

### 3 Proof of Theorem 1.1

We firstly establish the following lemmas.

Lemma 3.1Assume that ( AQ 1 ) and ( AQ 3 ) hold. Then B ( u ) 0 for all u E and B ( u ) as u on any finite-dimensional subspace of E.

Proof Since W ( t , u ) 0 , by (2.4), it is obvious that B ( u ) 0 for all u E .

By the proof of Lemma 2.6 of [6], for any finite-dimensional subspace Y E , there exists a constant ϵ > 0 such that

m ( { t [ 0 , T ] : | u | ϵ u } ) ϵ , u Y { 0 } , (3.1)

where m ( ) is the Lebesgue measure.

For the ϵ given in (3.1), let

Λ u = { t [ 0 , T ] : | u | ϵ u } , u Y { 0 } .

Then m ( Λ u ) ϵ . By ( AQ 3 ), there exists a constant R 3 > R 1 such that

W ( t , u ) d | u | ϱ / 2 , t [ 0 , T ]  and  | u | R 3 , (3.2)

where R 1 is the constant given in (1.2). Note that

| u ( t ) | R 3 , t Λ u (3.3)

for any u Y with u R 3 / ϵ . Thus,

B ( u ) = 0 T W ( t , u ) d t Λ u W ( t , u ) d t Λ u d | u | ϱ / 2 d t d ϵ ϱ u ϱ m ( Λ u ) / 2 d ϵ ϱ + 1 u ϱ / 2

for any u Y with u R 3 / ϵ . This implies B ( u ) as u on Y. □

Lemma 3.2Assume that ( AQ 1 ), (AQ2) and ( AQ 3 ) hold. Then there exist a positive integer k 1 and two sequences 0 < r k < ρ k 0 as k such that

(3.4)

(3.5)

and

β k ( λ ) : = max u Y k , u = r k Φ λ ( u ) < 0 , k N , (3.6)

where Y k = j = 0 k X j = span { e 0 , e 1 , , e k } and Z k = j = k X j ¯ = span { e k , e k + 1 , } ¯ for all k N .

Proof Comparing this lemma with Lemma 2.7 of [6], we find that these two lemmas have the same condition (AQ2) which is the key in the proof of Lemma 2.7 of [6]. We can prove our lemma by using the same method of [6], so the details are omitted. □

Now it is the time to prove our main result Theorem 1.1.

Proof of Theorem 1.1 By virtue of (1.3), (2.3) and (2.5), Φ λ maps bounded sets to bounded sets uniformly for λ [ 1 , 2 ] . Obviously, Φ λ ( u ) = Φ λ ( u ) for all ( λ , u ) [ 1 , 2 ] × E since W ( t , u ) is even in u. Consequently, the condition (T1) of Theorem 2.1 holds. Lemma 3.1 shows that the condition (T2) holds, whereas Lemma 3.2 implies that the condition (T3) holds for all k k 1 , where k 1 is given there. Therefore, by Theorem 2.1, for each k k 1 , there exist λ n 1 and u λ n Y n such that

Φ λ n Y n ( u λ n ) = 0 , Φ λ n ( u λ n ) η k [ ξ k ( 2 ) , β k ( 1 ) ] as  n . (3.7)

For the sake of notational simplicity, in the following we always set u n = u λ n for all n N .

Step 1. We firstly prove that { u n } is bounded in E.

Since { u n } satisfies (3.7), one has

lim n ( Φ λ n Y n ( u n ) , u n 2 Φ λ n ( u n ) ) = 2 η k .

More precisely,

lim n 0 T ( u W ( t , u n ) , u n 2 W ( t , u n ) ) d t = 2 η k . (3.8)

Now, we prove that { u n } is bounded. Otherwise, without loss of generality, we may assume that

u n as  n .

Put z n = u n u n , we have z n = 1 . Going to a subsequence if necessary, we may assume that

z n z in  E , z n z in  L 2 and z n ( t ) z ( t ) for a.e.  t [ 0 , T ] .

By (1.3), we have

Φ λ n ( u n ) = 1 2 u n + 2 λ n 0 T W ( t , u n ) d t 1 2 u n 2 1 2 u n 0 2 λ n ( ϵ 0 T | u n | 2 d t + max s [ 0 , δ ] a ( s ) 0 T b ( t ) d t ) 1 2 u n 2 ( 1 2 + λ n ϵ ) 0 T | u n | 2 d t λ n c 1 ,

where c 1 = max s [ 0 , δ ] a ( s ) 0 T b ( t ) d t . Therefore, one obtains

Φ λ n ( u n ) u n 2 1 2 ( 1 2 + λ n ϵ ) 0 T ( | u n | u n ) 2 d t λ n c 1 u n 2 = 1 2 ( 1 2 + λ n ϵ ) z n 2 2 λ n c 1 u n 2 .

Passing to the limit in the inequality, by using Φ λ n ( u n ) η k and λ n 1 as n , we obtain

1 2 ( 1 2 + ϵ ) z 2 2 0 .

Thus, z 0 on a subset Ω of [ 0 , T ] with positive measure.

By (1.2), we have

u W ( t , u ) , u 2 W ( t , u ) 0 , t [ 0 , T ]  and  | u | R 1 ,

and by the assumption (A), we obtain

u W ( t , u ) , u 2 W ( t , u ) c 3 b ( t ) , for all  | u | R 1  and a.e.  t [ 0 , T ] ,

where c 3 = ( 2 + R 1 ) max [ 0 , R 1 ] a ( s ) . So, we get

u W ( t , u ) , u 2 W ( t , u ) c 3 b ( t )

for all u R N and a.e. t [ 0 , T ] . Hence,

An application of Fatou’s lemma yields

Ω ( u W ( t , u n ) , u n 2 W ( t , u n ) ) d t as  n ,

which is a contradiction to (3.8).

Step 2. We prove that { u n } has a convergent subsequence in E.

Since { u n } is bounded in E, E is reflexible and dim E 0 < , without loss of generality, we assume

u n 0 u 0 0 , u n + u 0 + and u n u 0 as  n (3.9)

for some u 0 = u 0 0 + u 0 + E = E 0 E + .

Note that

0 = Φ λ n Y n ( u n ) = u n + λ n P n Ψ ( u n ) , n N ,

where P n : E Y n is the orthogonal projection for all n N , that is,

u n + = λ n P n Ψ ( u n ) , n N . (3.10)

In view of the compactness of Ψ and (3.9), the right-hand side of (3.10) converges strongly in E and hence u n + u 0 + in E. Together with (3.9), we have u n u 0 in E.

Now, from the last assertion of Theorem 2.1, we know that Φ = Φ 1 has infinitely many nontrivial critical points. The proof is completed. □

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

HG wrote the first draft and TA corrected and improved the final version. All authors read and approved the final draft.

### Acknowledgements

The authors thank the referee for his/her careful reading of the manuscript. The work is supported by the Fundamental Research Funds for the Central Universities and the National Natural Science Foundation of China (No. 61001139).

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