Existence of homoclinic solutions for a class of second-order Hamiltonian systems with subquadratic growth

By properly constructing a functional and by using the critical point theory, we establish the existence of homoclinic solutions for a class of subquadratic second-order Hamiltonian systems. Our result generalizes and improves some existing ones. An example is given to show that our theorem applies, while the existing results are not applicable.

Consider the following second-order Hamiltonian system:

where q = ( q 1 , q 2 , … , q n ) ∈ R n , L ∈ C ( R , R n × n ) is a symmetric matrix-valued function, and W ( t , q ) ∈ C 1 ( R × R n , R ) , W q ( t , q ) ∈ C 1 ( R × R n , R n ) is the gradient of W about q. As usual we say that a solution q ( t ) of (HS) is homoclinic (to 0) if q ∈ C 2 ( R , R n ) such that q ( t ) → 0 and q ˙ ( t ) → 0 as | t | → ∞ . If q ( t ) ≢ 0 , q ( t ) is called a nontrivial homoclinic solution.

By now, the existence and multiplicity of homoclinic solutions for second-order Hamiltonian systems have been extensively investigated in many papers (see, e.g., [1-17] and the references therein) via variational methods. More precisely, many authors studied the existence and multiplicity of homoclinic solutions for (HS); see [5-17]. Some of them treated the case where L ( t ) and W ( t , u ) are either independent of t or periodic in t (see, for instance, [5-7]), and a more general case is considered in the recent paper [7]. If L ( t ) is neither constant nor periodic in t, the problem of the existence of homoclinic solutions for (HS) is quite different from the one just described due to the lack of compactness of the Sobolev embedding. After the work of Rabinowitz and Tanaka [8], many results [9-17] were obtained for the case where L ( t ) is neither constant nor periodic in t.

Recently, Zhang and Yuan [15] obtained the existence of a nontrivial homoclinic solution for (HS) by using a standard minimizing argument. In this paper, ( ⋅ , ⋅ ) : R n × R n → R denotes the standard inner product in R n , and subsequently, | ⋅ | is the induced norm. If q = ( q 1 , q 2 , … , q n ) ∈ R n , then | q | = q 1 2 + q 2 2 + ⋯ + q n 2 .

Theorem 1.1 (See [[15], Theorem 1.1])

Assume thatLandWsatisfy the following conditions:

(H1) L ( t ) ∈ C ( R , R n × n ) is a symmetric matrix for all t ∈ R , and there is a continuous function α : R → R such that α ( t ) > 0 for all t ∈ R and ( L ( t ) q , q ) ≥ α ( t ) | q | 2 and α ( t ) → + ∞ as | t | → + ∞ .

(H2) W ( t , q ) = a ( t ) | q | γ where a ( t ) : R → R + is a positive continuous function such that a ( t ) ∈ L 2 ( R , R ) ∩ L 2 2 − γ ( R , R ) and 1 < γ < 2 is a constant.

Then (HS) possesses at least one nontrivial homoclinic solution.

In [15-17], the authors considered the case where W ( t , q ) is subquadratic as | q | → ∞ . However, there are many functions with subquadratic growth but they do not satisfy the condition (H2) in [15] and the corresponding conditions in [16,17]. For example,

where 1 < γ < 2 , a ( t ) , b ( t ) : R → R + are positive continuous functions such that a ( t ) , b ( t ) ∈ L 2 ( R , R ) ∩ L 2 2 − γ ( R , R ) .

In this paper, our aim is to revisit (HS) and study the subquadratic case which is not included in [15-17]. Now, we state our main result.

Theorem 1.2Let the above condition (H1) hold. Moreover, assume that the following conditions hold:

(H3) W ( t , q ) ≥ a ( t ) | q | γ , ∀ ( t , q ) ∈ ( R , R n ) , where a ( t ) : R → R + is a positive continuous function such that a ( t ) ∈ L 2 ( R , R ) ∩ L 2 2 − γ ( R , R ) and 1 < γ < 2 is a constant.

(H4) | W q ( t , q ) | ≤ f 1 ( t ) | q | γ − 1 + f 2 ( t ) , ∀ ( t , q ) ∈ ( R , R n ) where f 1 ( t ) , f 2 ( t ) : R → R + are positive continuous functions such that f 1 ( t ) , f 2 ( t ) ∈ L 2 ( R , R ) ∩ L 2 2 − γ ( R , R ) .

Then (HS) possesses at least one nontrivial homoclinic solution.

Remark 1.1 Obviously, the condition (H2) is a special case of (H3)-(H4). If (H2) holds, so do (H3)-(H4); however, the reverse is not true. W ( t , q ) defined in (1) can satisfy the conditions (H3) and (H4), but W ( t , q ) cannot satisfy the condition (H2). So, we generalize and significantly improve Theorem 1.1 in [15].

Remark 1.2 We still consider the function W ( t , q ) defined in (1),

Due to inf t ∈ R a ( t ) = 0 , there are no constants b , r 1 > 0 such that

so W ( t , q ) does not satisfy the conditions (W2) and (W3) in [16]. Moreover, for any given 1 < γ < 3 2 , W ( t , q ) does not satisfy the condition (W2) in [17]. Therefore, we also extend Theorem 1.2 in [16] and Theorem 1.1 in [17].

Example 1.1 Consider the following second-order Hamiltonian system with n = 3 :

where

Let α ( t ) = t 2 , γ = 5 4 and a 1 ( t ) = 1 1 + | t | 3 , a 2 ( t ) = 1 1 + | t | 2 , W q ( t , q ) = 5 4 a 1 ( t ) | q | − 3 4 q + 3 a 2 ( t ) e sin 3 | q | | q | − 1 q sin 2 | q | cos | q | ≤ 5 4 a 1 ( t ) | q | 1 4 + 3 a 2 ( t ) e , W q ( t , 0 ) = 0 . Clearly, (H1), (H3), and (H4) hold. Therefore, by applying Theorem 1.2, the Hamiltonian system (2) possesses at least one nontrivial homoclinic solution.

Remark 1.3 It is easy to see that (H2) in Theorem 1.1 is not satisfied, so we cannot obtain the existence of homoclinic solutions for the Hamiltonian system (2) by Theorem 1.1. On the other hand, W does not satisfy the conditions (W2) and (W5) of [17], then we cannot obtain the existence of homoclinic solutions for the Hamiltonian system (2) by Theorem 1.1 in [17].

The remainder of this paper is organized as follows. In Section 2, some preliminary results are presented. In Section 3, we give the proof of Theorem 1.2.

In order to establish our result via the critical point theory, we firstly describe some properties of the space on which the variational associated with (HS) is defined. Like in [15], let

Then the space E is a Hilbert space with the inner product

and the corresponding norm ∥ x ∥ 2 = 〈 x , x 〉 . Note that

for all p ∈ [ 2 , + ∞ ) with the embedding being continuous. Here L p ( R , R n ) ( 2 ≤ p < + ∞ ) and H 1 ( R , R n ) denote the Banach spaces of functions on ℝ with values in R n under the norms

and

respectively. In particular, for p = + ∞ , there exists a constant C > 0 such that

here ∥ q ∥ ∞ : = ess sup { | q ( t ) | : t ∈ R } .

Lemma 2.1There exists a constant β > 0 such that if q ∈ E , then

Proof From (H1), we can imply that there exists a constant β > 0 such that

for all t ∈ R and q ∈ R n . By the above inequality, one has

So, the lemma is proved. □

Lemma 2.2 ([[9], Lemma 1])

Suppose thatLsatisfies (H1). Then the embedding ofEin L 2 ( R , R n ) is compact.

Lemma 2.3Suppose that (H1) and (H4) are satisfied. If q k ⇀ q (weakly) inE, then W q ( t , q k ) → W q ( t , q ) in L 2 ( R , R n ) .

Proof Assume that q k ⇀ q in E. Then there exists a constant d 1 > 0 such that, by the Banach-Steinhaus theorem and (3),

Since 1 < γ < 2 , by (H4) there exists a constant d 2 > 0 such that

for all k ∈ N and t ∈ R . Hence,

On the other hand, by Lemma 2.2, q k → q in L 2 , passing to a subsequence if necessary, which implies q k ( t ) → q ( t ) for almost every t ∈ R . Then using Lebesgue’s convergence theorem, the lemma is proved. □

Now, we introduce more notation and some necessary definitions. Let E be a real Banach space, I ∈ C 1 ( E , R ) , which means that I is a continuously Fréchet-differentiable functional defined on E. Recall that I ∈ C 1 ( E , R ) is said to satisfy the (PS) condition if any sequence { u j } j ∈ N ⊂ E , for which { I ( u j ) } j ∈ N is bounded and I ′ ( u j ) → 0 as j → + ∞ , possesses a convergent subsequence in E.

Lemma 2.4 ([[18], Theorem 2.7])

LetEbe a real Banach space, and let us have I ∈ C 1 ( E , R ) satisfying the (PS) condition. IfIis bounded from below, then

is a critical value ofI.

Now, we are going to establish the corresponding variational framework to obtain homoclinic solutions of (HS). Define the functional I : E → R

Lemma 3.1Under the assumptions of Theorem 1.2, we have

which yields that

Moreover, Iis a continuously Fréchet-differentiable functional defined onE, i.e., I ∈ C 1 ( E , R ) and any critical point ofIonEis a classical solution of (HS) with q ( ± ∞ ) = 0 = q ˙ ( ± ∞ ) .

Proof We firstly show that I ∈ C 1 ( E , R ) . Let q ∈ E , by (3), (H4), and the Hölder inequality, we have

Combining (5) and (8), we show that I : E → R . Next, we prove that I ∈ C 1 ( E , R ) . Rewrite I as follows:

where

It is easy to check that I 1 ∈ C 1 ( E , R ) and

Thus, it is sufficient to show that this is the case for I 2 . In the process we will see that

which is defined for all q , v ∈ E . For any given q ∈ E , let us define J ( q ) : E → R as follows:

It is obvious that J ( q ) is linear. Now, we show that J ( q ) is bounded. Indeed, for any given q ∈ E , by (3) and (H4), there exists a constant d 3 > 0 such that

for all t ∈ R , which yields that by (4) and the Hölder inequality,

Moreover, for any q , v ∈ E , by the mean value theorem, we have

where h ( t ) ∈ ( 0 , 1 ) . Therefore, by Lemma 2.3 and the Hölder inequality, one has

as v → 0 in E. Combining (11) and (12), we see that (10) holds. It remains to prove that I 2 ′ is continuous. Suppose that q → q 0 in E and note that

By Lemma 2.3 and the Hölder inequality, we obtain that

as q → q 0 , which implies the continuity of I 2 ′ and I ∈ C 1 ( E , R ) .

Lastly, we check that critical points of I are classical solutions of (HS) satisfying q ( t ) → 0 and q ˙ ( t ) → 0 as | t | → + ∞ . We know that E ⊂ H 1 ( R , R n ) ⊂ C 0 ( R , R n ) , the space of continuous functions q on ℝ such that q ( t ) → 0 as | t | → + ∞ . Moreover, if q is one critical point of I, by (6) we have

which yields that q ∈ C 2 ( R , R n ) , i.e., q is a classical solution of (HS). Since q is one critical point of I, we have

It follows from q ( t ) → 0 as | t | → + ∞ and the above equality that

Hence, q satisfies q ˙ ( t ) → 0 as | t | → + ∞ . This proof is complete. □

Lemma 3.2Under the assumptions of Theorem 1.2, Isatisfies the (PS) condition.

Proof In fact, assume that { q j } j ∈ N ⊂ E is a sequence such that { I ( q j ) } j ∈ N is bounded and I ′ ( q j ) → 0 as j → ∞ . Then there exists a constant C 1 > 0 such that

for every j ∈ N .

We firstly prove that { q j } j ∈ N is bounded in E. By (5) and (8), we have

Combining (13) and (14), we obtain that

Since 1 < γ < 2 , the above inequality shows that { q j } j ∈ N is bounded in E. By Lemma 2.2, the sequence { q j } j ∈ N has a subsequence, again denoted by { q j } j ∈ N , and there exists q ∈ E such that

q j ⇀ q , weakly in E,

q j → q , strongly in L 2 ( R , R n ) .

Hence,

as j → + ∞ . Moreover, an easy computation shows that

So, ∥ q j − q ∥ → 0 as j → + ∞ , i.e., I satisfies the Palais-Smale condition. □

Now, we can give the proof of Theorem 1.2.

Proof of Theorem 1.2 By (5) and (8), for every r ∈ R ∖ { 0 } and q ∈ E ∖ { 0 } , we have

Since 1 < γ < 2 , (16) implies that I ( r q ) → + ∞ as | r | → + ∞ . Consequently, I is a functional bounded from below. By Lemmas 3.2 and 2.4, I possesses a critical value c = inf q ∈ E I ( q ) , i.e., there is a q ∈ E such that

On the other hand, take c 0 ∈ R n with | c 0 | ≠ 0 , and let φ ∈ E be given by

where − ∞ < t 1 < t 2 < + ∞ . Then we obtain that

which yields that I ( r φ ) < 0 as | r | small enough since 1 < γ < 2 , i.e., the critical point obtained above is nontrivial. □

The authors declare that they have no competing interests.

The authors declare that the work was realized in collaboration with same responsibility. All authors read and approved the final manuscript.

This work is supported by the Research Foundation of Education Bureau of Hunan Province, China (No.11C0594). The authors would like to thank the anonymous referees very much for helpful comments and suggestions which led to the improvement of presentation and quality of the work.

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