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 ,

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∈RN 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 1234567 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 in u (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]×RN, 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 ′ .

By the assumption (A) and the condition (AQ1′), 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 ) ,

for ∀ u ∈ R N and a.e. t∈[0,T].

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

(AQ3′) 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 1.1 Assume that (AQ1′), (AQ₂), (AQ3′) hold and W(t,u) is even in u. Then (1.1) possesses infinitely many solutions.

Remark The conditions (AQ₁) and (AQ₃) are stronger than (AQ1′) and (AQ3′). Then Theorem 1.1 above is different from Theorem 1.1 of 6.

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 ) .

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 〉

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 .

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) Φλ 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 of E;

(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 every k, 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 Φλ on our working space E by

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

and

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

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).

We firstly establish the following lemmas.

Lemma 3.1 Assume that (AQ1′) 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 } ,

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 (AQ3′), there exists a constant R 3 > R 1 ′ such that

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

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

| u ( t ) | ≥ R 3 , ∀ t ∈ Λ u

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∥≥R3/ϵ. This implies B(u)→∞ as ∥u∥→∞ on Y. □

Lemma 3.2 Assume that (AQ1′), (AQ₂) and (AQ3′) hold. Then there exist a positive integer k 1 and two sequences 0 < r k < ρ k → 0 as k→∞ such that

and

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

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), Φλ maps bounded sets to bounded sets uniformly for λ∈[1,2]. Obviously, Φλ(−u)=Φλ(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 k1 is given there. Therefore, by Theorem 2.1, for each k≥k1, there exist λn→1 and uλn∈Yn 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 → ∞ .

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 {un} 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 .

Now, we prove that {un} 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 λn→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∈RN 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 {un} has a convergent subsequence in E.

Since {un} 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 → ∞

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 .

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. □

The authors declare that they have no competing interests.

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