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 , is a symmetric matrix-valued function, and , is the gradient of W about q. As usual we say that a solution of (HS) is homoclinic (to 0) if such that and as . If , 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 and 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 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 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, denotes the standard inner product in , and subsequently, is the induced norm. If , then .

Theorem 1.1 (See [[15], Theorem 1.1])

Assume thatLandWsatisfy the following conditions:

(H1) is a symmetric matrix for all, and there is a continuous functionsuch thatfor allandandas.

(H2) whereis a positive continuous function such thatandis a constant.

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

In [15-17], the authors considered the case where is subquadratic as . 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 , are positive continuous functions such that .

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) , , whereis a positive continuous function such thatandis a constant.

(H4) , whereare positive continuous functions such that.

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. defined in (1) can satisfy the conditions (H3) and (H4), but cannot satisfy the condition (H2). So, we generalize and significantly improve Theorem 1.1 in [15].

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

Due to , there are no constants such that

so does not satisfy the conditions (W2) and (W3) in [16]. Moreover, for any given , 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 :

where

Let , and , , , . 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 . Note that

for all with the embedding being continuous. Here () and denote the Banach spaces of functions on ℝ with values in under the norms

and

respectively. In particular, for , there exists a constant such that

here .

Lemma 2.1There exists a constantsuch that if, then

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

for all and . By the above inequality, one has

So, the lemma is proved. □

Lemma 2.2 ([[9], Lemma 1])

Suppose thatLsatisfies (H1). Then the embedding ofEinis compact.

Lemma 2.3Suppose that (H1) and (H4) are satisfied. If (weakly) inE, thenin.

Proof Assume that in E. Then there exists a constant such that, by the Banach-Steinhaus theorem and (3),

Since , by (H4) there exists a constant such that

for all and . Hence,

On the other hand, by Lemma 2.2, in , passing to a subsequence if necessary, which implies for almost every . 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, , which means that I is a continuously Fréchet-differentiable functional defined on E. Recall that is said to satisfy the (PS) condition if any sequence , for which is bounded and as , possesses a convergent subsequence in E.

Lemma 2.4 ([[18], Theorem 2.7])

LetEbe a real Banach space, and let us havesatisfying 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

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., and any critical point ofIonEis a classical solution of (HS) with.

Proof We firstly show that . Let , by (3), (H4), and the Hölder inequality, we have

Combining (5) and (8), we show that . Next, we prove that . Rewrite I as follows:

where

It is easy to check that and

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

which is defined for all . For any given , let us define as follows:

It is obvious that is linear. Now, we show that is bounded. Indeed, for any given , by (3) and (H4), there exists a constant such that

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

Moreover, for any , by the mean value theorem, we have

where . Therefore, by Lemma 2.3 and the Hölder inequality, one has

as in E. Combining (11) and (12), we see that (10) holds. It remains to prove that is continuous. Suppose that in E and note that

By Lemma 2.3 and the Hölder inequality, we obtain that

as , which implies the continuity of and .

Lastly, we check that critical points of I are classical solutions of (HS) satisfying and as . We know that , the space of continuous functions q on ℝ such that as . Moreover, if q is one critical point of I, by (6) we have

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

It follows from as and the above equality that

Hence, q satisfies as . This proof is complete. □

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

Proof In fact, assume that is a sequence such that is bounded and as . Then there exists a constant such that

for every .

We firstly prove that is bounded in E. By (5) and (8), we have

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

Since , the above inequality shows that is bounded in E. By Lemma 2.2, the sequence has a subsequence, again denoted by , and there exists such that

, weakly in E,

, strongly in .

Hence,

as . Moreover, an easy computation shows that

So, as , 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 and , we have

Since , (16) implies that as . Consequently, I is a functional bounded from below. By Lemmas 3.2 and 2.4, I possesses a critical value , i.e., there is a such that

On the other hand, take with , and let be given by

where . Then we obtain that

which yields that as small enough since , 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.

1. Tang, C: Periodic solutions for nonautonomous second order systems with sublinear nonlinearity. Proc. Am. Math. Soc.. 126, 3263–3270 (1998). Publisher Full Text

2. Ambrosetti, A, Coti Zelati, V: Multiple homoclinic orbits for a class of conservative systems. Rend. Semin. Mat. Univ. Padova. 89, 177–194 (1993)

3. Zhang, X, Zhou, Y: Periodic solutions of non-autonomous second order Hamiltonian systems. J. Math. Anal. Appl.. 345, 929–933 (2008). Publisher Full Text

4. Cordaro, G, Rao, G: Three periodic solutions for perturbed second order Hamiltonian systems. J. Math. Anal. Appl.. 359, 780–785 (2009). Publisher Full Text

5. Bonanno, G, Livrea, R: Multiple periodic solutions for Hamiltonian systems with not coercive potential. J. Math. Anal. Appl.. 363, 627–638 (2010). Publisher Full Text

6. Paturel, E: Multiple homoclinic orbits for a class of Hamiltonian systems. Calc. Var. Partial Differ. Equ.. 12, 117–143 (2001). Publisher Full Text

7. Izydorek, M, Janczewska, J: Homoclinic solutions for a class of second order Hamiltonian systems. J. Differ. Equ.. 219, 375–389 (2005). Publisher Full Text

8. Rabinowitz, PH, Tanaka, K: Some results on connecting orbits for a class of Hamiltonian systems. Math. Z.. 206, 473–499 (1991). Publisher Full Text

9. Omana, W, Willem, M: Homoclinic orbits for a class of Hamiltonian systems. Differ. Integral Equ.. 5, 1115–1120 (1992)

10. Ou, Z, Tang, C: Existence of homoclinic solutions for the second order Hamiltonian systems. J. Math. Anal. Appl.. 291, 203–213 (2004). Publisher Full Text

11. Tang, X, Xiao, L: Homoclinic solutions for non-autonomous second-order Hamiltonian systems with a coercive potential. J. Math. Anal. Appl.. 351, 586–594 (2009). Publisher Full Text

12. Tang, X, Lin, X: Homoclinic solutions for a class of second order Hamiltonian systems. J. Math. Anal. Appl.. 354, 539–549 (2009). Publisher Full Text

13. Yang, J, Zhang, F: Infinitely many homoclinic orbits for the second-order Hamiltonian systems with super-quadratic potentials. Nonlinear Anal., Real World Appl.. 10, 1417–1423 (2009). Publisher Full Text

14. Zhang, Z, Yuan, R: Homoclinic solutions for some second-order non-autonomous systems. Nonlinear Anal.. 71, 5790–5798 (2009). Publisher Full Text

15. Zhang, Z, Yuan, R: Homoclinic solutions for a class of non-autonomous subquadratic second-order Hamiltonian systems. Nonlinear Anal.. 71, 4125–4130 (2009). Publisher Full Text

16. Ding, Y: Existence and multiplicity results for homoclinic solutions to a class of Hamiltonian systems. Nonlinear Anal.. 25, 1095–1113 (1995). Publisher Full Text

17. Zhang, Q, Liu, C: Infinitely many homoclinic solutions for second order Hamiltonian systems. Nonlinear Anal.. 72, 894–903 (2010). Publisher Full Text

18. Rabinowitz, PH: Minimax Methods in Critical Point Theory with Applications to Differential Equations, Am. Math. Soc., Providence (1986)