#### Abstract

We investigate the existence and multiplicity of homoclinic orbits for second-order Hamiltonian systems with local superquadratic potential by using the Mountain Pass Theorem and the Fountain Theorem, respectively.

#### 1. Introduction and Main Result

Consider the second-order nonautonomous Hamiltonian systems where is a symmetric matrix valued function, , and . We say that a nonzero solution of problem (1) is homoclinic (to 0) if and as .

The existence of homoclinic orbits for Hamiltonian systems and their importance in the study of the behavior of dynamical systems have been already recognized by Poincaré [1]. Only during the last two decades such problem has been studied by using critical point theory.

If and are independent of or periodic in , many authors have studied the existence of homoclinic orbits for Hamiltonian systems, see, for instance, [2–9], and a more general case is considered in recent papers [10, 11]. In this case, the existence of homoclinic orbits is obtained by going to the limit of periodic solutions of approximating problems. In recent years, concentration compactness principle has also been widely used to deal with the perturbations of periodic or autonomous problems, for example, [12, 13].

If and are neither autonomous nor periodic, the problem is quite different from the ones just described, because of the lack of compactness of the Sobolev embedding. Rabinowitz and Tanaka [14] study without any periodicity assumption and obtain the existence of homoclinic orbits of problem (1) by using a variant of the Mountain Pass Theorem without the Palais-Smale condition under the following condition.

() is a symmetric and positively definite matrix for all , and there exists a continuous function such that for all and Assuming coercivity assumption (), Omana and Willem [15] obtain an improvement on the latter result by employing a new compact embedding theorem; in fact, they show that the (PS) condition is satisfied and obtain the existence and multiplicity of homoclinic orbits of problem (1) by using the usual Mountain Pass Theorem. After [14] and [15], many results [16–22] are obtained for the case where is neither constant nor periodic in .

Korman and Lazer [23] remove the technical coercivity in case that and are even in and is positively definite for all , by approximating homoclinic orbits from solutions of boundary value problems, which is complemented by [24].

Most of the papers mentioned previously tackle the superquadratic case (see [2–10, 14–16, 18–21, 23, 24]) and the subquadratic case (see [17–19, 22, 25]). The following Ambrosetti-Rabinowitz condition is widely used in almost all papers tackling the superquadratic case.

(AR) There exists a constant such that, for every and , Many recent papers have complemented the (AR) condition, for example, [6, 16, 20–22, 24].

There are also many papers that tackle the multiplicity of homoclinic orbits, for example [20, 21, 23–25]. In particular, based on the variant Fountain Theorem of [26], Yang and Han [19] consider the multiplicity of homoclinic orbits for problem (1).

Theorem A (see [19, Theorem 1.2]). *Suppose that satisfies () and . For some and , one of the following is true: *(i)* and , for all , or*(ii)* and , for all ,**where and and satisfies the following.*(W1)* for all .*(W2)*, for all .*(W3)* There exist such that
for all .*(W4)* uniformly for .*(W5)* There exist and such that
uniformly for .*(W6)* is an increasing function of , for all .*(W7)*, for all .**Then system (1) has infinitely many homoclinic solutions satisfying
**
as .*

In the present paper, based on the Fountain Theorem, we can prove the same result under more generic conditions, which generalizes Theorem A. Our first result can be stated as follows.

Theorem 1. *Assume that satisfies () and () and satisfies (W1), (W4), (W7), (W8), and (W9). *(W8)* For any ,
uniformly in .*(W9)* there exists , such that
for all and , where .**Then problem (1) has infinitely many homoclinic orbits satisfying
**as .*

*Remark 2. *Theorem 1 generalizes and improves Theorem A. Firstly, in Theorem 1 we remove the positiveness condition (W2) and the growth condition (W3), which are indispensable in Theorem A. Now we compare conditions (W5) and (W8), (W6), and (W9). Our condition (W8) is a local superquadratic condition and is really weaker than condition (W5). Under condition (W6), for all we have
for all , which means that (W9) holds in the case that . We consider the multiplicity of homoclinic orbits for problem (1) by using the Fountain Theorem in [27] which is simpler than the variant Fountain Theorem [26].

Moreover, under all conditions of Theorem 1 except (W7) we obtain an existence result.

Theorem 3. *Assume that satisfies () and () and satisfies (W1), (W4), (W8) and (W9). Then problem (1) possesses a nontrivial homoclinic orbit.*

*Remark 4. *In Theorem 3, we consider the existence of homoclinic orbits for problem (1) under a class of local superquadratic conditions without the (AR) condition and any periodicity assumptions on both and . There are functions and which satisfy Theorem 3, but do not satisfy the corresponding results in [2–10, 14–16, 18–21, 23, 24]. For example,
where is the unit matrix of order , .

#### 2. Preliminary Results

In order to establish our results via critical point theory, we firstly describe some properties of the space on which the variational functional associated with problem (1) is defined. Let Then the space is a Hilbert space with the inner product and the corresponding norm Note that for all with the embedding being continuous. In particular, for and , there exist constants and such that Here and denote the Banach spaces of functions on with values in under the norms respectively. is the Banach space of essentially bounded functions from into equipped with the norm

Lemma 5 (see [18]). *Suppose that assumption () holds. Then the embedding of into is compact for all .*

Denote by the self-adjoint extension of the operator with the domain .

Lemma 6 (see [18]). *If L satisfies and , then is continuously embedded in , and, consequently, one has
**
as , for all .*

Lemma 7. *Suppose that assumptions (W1), (W4) and (W9) hold. Then for all and .*

*Proof. *Given and , let
for ; then
By (W1) and (W9), we have
for all and . Hence,
for all , which shows that is nondecreasing in . It is clear that
On the other hand, by (W4) one has
Therefore
Now we get for all , which implies that
for all and .

Lemma 8. *Assume that assumptions () and (W4) hold and (weakly) in . Then in .*

*Proof. *Assume that in . Then there exists a constant such that
By (W4), for every , there exists such that
for all and with . Now we claim that given , for any , there exists such that
for all and all with . If not, there exists , for all , and there exists with and such that
On the other hand, by Lemma 5
as in . In view of (32) and (33), we have
when is large enough, which is a contradiction to the fact that
Hence, (31) holds. It follows from (29), (30), and (31) that
for all and . By Lemma 5, in , and for almost every and passing to a subsequence if necessary:
which implies
for and
for . Since is a Cauchy sequence in , so by (39) we know that is also a Cauchy sequence in , which together with (38) and the completeness of shows that
is well defined and
In consequence,
for all and . Consequently,
for all . Then using Lebesgue’s convergence theorem, the lemma is proved.

In our paper we will also use the following lemma which is a special case of Lemma 1.1 in [28], due to Arioli and Szulkin [29].

Lemma 9 (see [28, 29]). *Let be a bounded sequence in , such that is bounded in , . If, in addition, there exists such that
**
as , then
**
in for all .*

Now we introduce some notations and some necessary definitions which will be used later. Let be a real Banach space, , which means that is continuously Frechet-differentiable functional defined on B. Recall that is said to satisfy the (PS) condition if any sequence , for which is bounded and as possesses a convergent subsequence in .

Moreover, let be the open ball in with the radius and centered at , and denotes its boundary; we obtain the existence of homoclinic orbits of problem (1) by the use of the following well-known Mountain Pass Theorem [30].

Lemma 10 (see [30]). *Let be a real Banach space and let satisfying the (PS) condition. Suppose that and that*(A1)* there are constants such that ,*(A2)* there is an such that .**Then possesses a critical value . Moreover can be characterized as
**
where
**
As shown in [31], a deformation lemma can be proved with the condition replacing the usual (PS) condition, and it turns out that Lemma 10 holds true under the condition.*

In order to prove the multiplicity of homoclinic orbits, we will use the Fountain Theorem. Since is a Hilbert space, then there exists a basis such that , where . Letting , now we show the following Fountain Theorem.

Lemma 11 (see [27]). *If satisfies the condition, , and for every , there exists such that*(i)*, as ;*(ii)*.**Then has a sequence of critical points such that as .*

In the proof of Theorem 1, the following lemma will also be used. A similar result with respect to elliptic problem has been proved in [27].

Lemma 12. *Suppose that ; then one has
**
as .*

*Proof. *It is clear that , so there exists such that
as for every . By the definition of , there exists with such that
for every and . Since is bounded, then there exists such that
as . Now since is a basis of , it follows that for all
as , which shows that . By Lemma 5 we have
in for all , which together with (49) and (50) implies that .

#### 3. Proof of Theorems

Define the functional by

Lemma 13. *Under the conditions (), (), and (W4), , and for all one has
**
Moreover, any critical point of on is a solution of problem (1) with and .*

*Proof. *We firstly show that is well defined. It follows from (30) that for any , there exists such that
for all and with . Letting , then , the space of continuous function on , such that as . Therefore there exists such that
for all . Hence, one has
so is well defined.

Next we prove that . Rewrite as follows
where
It is easy to check that and
It remains to show that . By the mean value theorem, for any and we have
where . For any , there exists such that
for all , so that
for all , which together with (15) and (30) implies

Then by Lebesgue’s convergence theorem, we have
Now we show that is continuous. Supposing that in , by an easy computation, one has
Hence by Lemma 8, we obtain
as uniformly with respect to , which implies the continuity of . Now we have proved

Finally, we show that any critical point of is a solution of problem (1) satisfying and . If is a critical point of , a standard argument shows that satisfies (1). By Lemma 6, we only need to show that is an element of . It follows from (30) and (57) that
for all . Hence, one has
so , which together with (1) implies that . This means; , and the proof is completed.

Lemma 14. *Under conditions (), (W4), and (W9), satisfies the condition.*

*Proof. * satisfies the condition; that is, for every , has a convergent subsequence if is bounded and as . Assume that is a sequence such that
is bounded and
as . Hence, we have

Firstly, we show that is bounded; if not, up to a subsequence we have
as . Letting , then is bounded in . By Lemma 5, we have
as . We claim the following.*Claim *1. consider
Otherwise, for some , up to a subsequence we have
We can choose such that
In view of in and (79), we have
when is large enough. By (80), there exists , such that the set has a positive Lebesgue measure. Moreover similar to (57), there exists such that for all , which implies that . For all , one has as , which together with (W8) shows
as uniformly for all . Hence by Lemma 6 and the fact that for all and , we have
as , which is a contradiction. Therefore we have proved Claim 1. Since is bounded, by Lemma 9, we have
in for all . Next, we will derive a contradiction. For any given , . Similar to (31), for defined in (56), there exists such that
for all and all , which together with (56) shows that
for all and . In view of (83), in , which implies that
as . We can derive from (W4) that
is bounded for all . Combining (86) and (87), we have
when is large enough. It follows from (85) and (88) that
when is large enough, which implies that
for any given . Choose a sequence , such that
Given , since is large enough, we have ; using (90) with , we obtain
for large enough, which together with the arbitrary of implies that
as . In view of (91) and the fact that , we have
By (W9), we get
for all and . It follows from (93) and (94) that
as , which contradicts (74). Therefore we have proved that is bounded.

By Lemma 5 and the fact that is bounded in , there exist , , and a subsequence of again denoted by such that
as . Arguing as in Lemma 8, we can also define
and . It is obvious that
as . By (), (97), and Lemma 8 one has
as , which implies that
as . Summing up (100) and (102), we have
as . On the other hand, by Lemma 7 and (97) we get
as . An easy computation shows that
Consequently, as .

*Proof of Theorem 1. *By Lemma 13 and Lemma 14, satisfies the condition and ; hence to prove Theorem 1 we should just show that has the geometric properties (i) and (ii) of Lemma 11.

(i) By Lemma 12
as for . We choose ; then as , and for every with , we have
Similar to (31), there exists such that
for all and all with , where is defined in (56). Consequently, by (56), for any
for all and all with . Hence, we have for all with
when is small enough. Therefore, one has
as .

(ii) Firstly, we claim that there exists a constant such that
for all . Otherwise, for every , there exists such that
Without loss of generality, we suppose that ; then there is
In view of the compactness of the unit sphere of , there exists a subsequence which is still denoted by such that converges to some as . It is clear that . Since all the norms in are equivalent, we have in as ; that is,
as . Thus there exist constants and such that
If not, we have
for all , which implies that
as . Hence , which contradicts . Therefore (116) holds. Let , , and . It follows from (114) and (116) that
We can choose large enough, such that
Then we have
for all . In consequence,
for all being large enough, which is a contradiction to (115). Hence (112) holds.

Similar to (31), for any there exists such that
for all and all with . Consequently, for all ,
for all , which implies that for all . By (W8), there exists such that
for all with and . Hence we have
for all and with , which together with Lemma 7 implies that
for all and . So we can choose ; then

Hence by Lemma 11, we obtain that problem (1) has infinitely many homoclinic solutions satisfying
as .

*Proof of Theorem 3. *We divide the proof of Theorem 3 into the following three steps.*Step *1. It is clear that , and we have proved that satisfies the condition in Lemma 13 and Lemma 14.*Step *2. Letting and , we have , where is defined in (16) and is defined in (56). Hence, by (56), for any
for all and with . In consequence, combining this with (54), we obtain
for all and with . Setting , the inequality (131) implies that
*Step *3. It remains to prove that there exists an such that and , where is defined in Step 2. By (W8), for any , there exists such that
for all and . Letting , where , hence for all . It is clear that when ,
for all , which together with (133) shows that when
for all . Combining (135), Lemma 7, and the fact that , we have