Homoclinic Orbits for a Class of Subquadratic Second Order Hamiltonian Systems

Abstract

The existence and multiplicity of homoclinic orbits are considered for a class of subquadratic second order Hamiltonian systems . Recent results from the literature are generalized and significantly improved. Examples are also given in this paper to illustrate our main results.

1. Introduction and Main Results

In this paper, we will study the existence and multiplicity of homoclinic orbits for the second order Hamiltonian systems of the type: where is a symmetric matrix valued function and . As usual, we say that is a nontrivial homoclinic orbit (to 0) if and as (see [1]). In the following, denotes the standard inner product in and is the induced norm.

Hamiltonian system theory, a classical as well as a modern study area, widely consists in mathematical sciences, life sciences, and various aspects of social science. Lots of mechanical and field theory models even exist in the form of Hamiltonian system. Solutions of Hamiltonian system can be divided into periodic solution, subharmonic solution, homoclinic orbit, heteroclinic orbit, and so on. Homoclinic orbit was discovered among the models of nonlinear dynamical system and affects the nature of the whole system significantly. Starting from the Poincaré era, the study of homoclinic orbits in the nonlinear dynamical system exploits Perturbation method mainly. It is not until the recent decade that variational principle has been widely used. The existence and multiplicity of homoclinic orbits for the second order Hamiltonian systems have been extensively investigated in many recent papers (see [1–16]). The main feature of the problem is the lack of global compactness due to unboundedness of domain. To overcome the difficulty, many authors have considered the periodic case, autonomous case, or asymptotically periodic case (see [1–4]). Some papers treat the coercive case (see [5–8]). Recently, the symmetric case has been dealt with (see [9–11]). Compared with the superquadratic case, the case that is subquadratic as has been considered only by a few authors. As far as the author is aware, the author in [5] first discussed the subquadratic case. Later, the authors in [12] dealt with this case by use of a standard minimizing argument, which is the following theorem.

Theorem 1 (see [12]). Assume that and satisfy the following conditions: (A₁) is a symmetric and positive definite matrix for all and there is a continuous function such that and as ;(A₂), where is a continuous function such that and is a constant.
Then, (1) has at least one nontrivial homoclinic orbit.

Subsequently, the condition () was generalized, respectively, in [13, 14] by the following conditions:(A₃), where is a continuous function such that and is a constant.(A₄), , where are continuous functions such that , , and are constants, , for all .

The authors in [13, 14] got infinitely many homoclinic orbits by using the variant fountain theorem in [17].

Motivated by these papers, we also consider the subquadratic case. Main results are the following theorems.

Theorem 2. Assume that () is satisfied and the following conditions hold: (W₁)there exists a constant such that for all and , where and ;(W₂) for all and , where ;(W₃) uniformly for ;(W₄)there exists a constant such that for all and , where is a positive constant.
Then, (1) has at least one nontrivial homoclinic orbit.

Remark 3. Theorem 2 generalizes Theorem 1 (see [12]). Obviously, the condition () is stronger than the conditions ()–(). On the other hand, there are functions satisfying our Theorem 2 and not satisfying the corresponding result in [12]. For example, let where and . Moreover, our result is different from the corresponding results in [5, 15], since does not satisfy conditions () and () in [5] and our condition () on is different from the condition () in [15].

Theorem 4. Assume that (), ()–() are satisfied and the following condition holds: (W₅) and for all and .
Then, (1) has infinitely many homoclinic orbits.

Remark 5. Theorem 4 generalizes Theorem 1.2 in [13]. It is easy to see that the condition () is stronger than our conditions ()–(). On the other hand, there are functions satisfying Theorem 4 and not satisfying the corresponding result in [13]. For example, let where and .

Theorem 6. Assume that (), ()–() are satisfied and the following conditions hold: (W₆)there exist constants and such that for all and , where and ;(W₇) for all and , where .
Then, (1) has infinitely many homoclinic orbits.

Remark 7. Theorem 6 generalizes Theorem 1.2 in [14]. It is obvious that the condition () is stronger than our conditions ()–(). Besides, there are functions satisfying our Theorem 6 and not satisfying the corresponding result in [14]. For example, let where , and .

Remark 8. We should point out that there exist errors in the proofs of [13, 14]. On one hand, in Lemma 2.2, in [13], can not imply that since the convergence of the general term is only the necessary but not the sufficient condition for the convergence of the progression. On the other hand, the proof of Lemma 2.2 in [14] is not like the proof of Lemma 2.2 in [12] as the authors said. It is easy to see that plays an important role in the proof of Lemma 2.2 in [12], however, , in condition () do not belong to . In the proofs of our results, we take some methods to avoid such mistakes.

2. Proof of Main Results

Let

Then, is a Hilbert space with the norm given by

Obviously, is continuously embedded in for . Thus, we have where . For , let

In the following, we always denote by any suitable positive constant.

Lemma 9 (see [5]). Assume that satisfies (). Then, is compactly embedded in for .

Lemma 10. Under the conditions of ()–(), one has that and that

Proof. Let for all . For any , it follows from () and () that
Then, by the Hölder inequality and (7), we have for all . Therefore, is well defined. Moreover, and
In fact, for any given , by (), the Hölder inequality, and (7), one has for all . For any , by the mean value theorem, we have where . Besides, for any given , by (), the Hölder inequality, and (7), there exists a positive constant such that for all . The combination of (13)–(15) shows as in , which gives (12) and (9) immediately. In addition, is weakly continuous. In fact, let in . By Lemma 9, in and . Since as , there exists a positive constant such that
It follows from () and the Hölder inequality such that
Therefore, for any , since in , one can take such that
On the other hand, one has
Hence, there exists such that
proving the weak continuity of . Moreover, is compact for is weakly continuous (see [5]).

Lemma 11. Under the conditions of (), (), and (), one has that and that

Proof. Since the proof is exactly similar to the proof of Lemma 10, we omit it here.

By Lemmas 10 and 11, it is routine to verify that any critical point of on is a classical solution of (1) with . Now, we state the critical point theorem used in [12].

Lemma 12 (see [18]). Let be a real Banach space and let us have satisfying the (PS) condition. If is bounded from below, then is a critical value of .

Proof of Theorem 2. We divide our proof into three steps.
Step  1. is bounded from below. In fact, using (11), we have for all . Since and is an arbitrary positive constant, as and then is bounded from below.
Step 2.   satisfies the (PS) condition. Assume that is a sequence such that
By (11), one gets for all . Since and is an arbitrary positive constant, it is obvious that is bounded. Noting that is compact, a routine verification shows that has a convergent subsequence, proving the (PS) condition. Now, by Lemma 12, there is a such that Step  3. is nontrivial. In fact, let where . By () one obtains for . Since , we can choose small enough such that . Hence,  . The proof is complete.

In order to prove Theorems 4 and 6, we need the Dual fountain theorem. For the reader’s convenience, we recall it here.

Lemma 13 (see [19]). Let be a Banach space with the norm and with . For any , define
Assume that (B₁)the compact group acts isometrically on . The spaces are invariant and there exists a finite dimensional space such that, for every , and the action of on is admissible.
Besides, let be an invariant functional. If for every , there exist such that (B₂);(B₃);(B₄) as ;(B₅) satisfies the condition for every .
Then, has a sequence of negative critical values converging to 0.

Proof of Theorem 4. Let be the completely orthogonal basis of and . Then, and . For any , define
In the following, we will check that all conditions in Lemma 13 are satisfied and we divide our proof into several steps.
Step 1. Conditions and hold. Let
We claim that as . In fact, it is obvious that and then for some as . For every , there is such that and . It follows from the definition of that in . Then, by Lemma 9, we have in as and thus . Take in (11), then for all , one gets
Since , there exists such that
Similar to the third step in the proof of Theorem 2, we can show that . Since as , we get as .
Step 2. Condition holds. We claim that there exists such that
If not, there exists a sequence with such that for all . Since we have , it follows from the compactness of the unit sphere of that, going to a subsequence if necessary,   converges to some in and . By the equivalence of the norms on the finite-dimensional space , one has as . It is easy to check that there exist and such that
To be specific, by negation, we have for all . Then by (7), one has as , which contradicts that . Therefore, (37) holds. Thus, define and . Combining (35) and (37), one has
for all . Selecting , we have
Then, one gets which implies that for , which is in contradiction to (36). Hence, (22) holds. Define
It follows from (), (7), (), and (22) that for all with . Since , there exists such that
Step 3. Condition holds. Since it is similar to the second step in the proof of Theorem 2, we omit it here.
At last, it is standard to verify that condition () holds and is invariant. So, Theorem 4 is proved by Lemma 13 immediately.

Proof of Theorem 6. Since the proof of Theorem 6 is exactly similar to the proof of Theorem 4, we omit it here.

Conflict of Interests

The author declares that there is no conflict of interests regarding this work.

Acknowledgments

The paper is supported by General Project of Educational Department in Sichuan (no. 13ZB0182) and Doctor Research Foundation of Southwest University of Science and Technology (no. 11zx7130).