Existence and Multiplicity of Homoclinic Orbits for Second-Order Hamiltonian Systems with Superquadratic Potential
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 . 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  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  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  and , many results [16–22] are obtained for the case where is neither constant nor periodic in .
Korman and Lazer  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 .
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 , Yang and Han  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
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  which is simpler than the variant Fountain Theorem .
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 ). 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 ). 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 , due to Arioli and Szulkin .
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 .
Lemma 10 (see ). 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 , 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 ). 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 .
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