Abstract and Applied Analysis

Volume 2013 (2013), Article ID 328630, 12 pages

http://dx.doi.org/10.1155/2013/328630

Research Article

Existence and Multiplicity of Homoclinic Orbits for Second-Order Hamiltonian Systems with Superquadratic Potential

Ying Lv and Chun-Lei Tang

School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

Received 25 September 2012; Accepted 11 January 2013

Academic Editor: Wenming Zou

Copyright © 2013 Ying Lv and Chun-Lei Tang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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

when and are both large enough.

By Lemma 10, possesses a critical value given by

where

Hence there is a such that

Therefore is a nontrivial homoclinic orbit of problem (1). Theorem 3 is proved now.

Acknowledgements

The authors would like to thank the referee for valuable suggestions. This paper is supported by National Natural Science Foundation of China (no. 11071198) and Fundamental Funds for the Central Universities (XDJK2009C075).

References

H. Poincaré, Les Méthods Nouvelles de la Mécanique Céleste, Gauthier-Villars, Paris, France, 1897–1899.

V. Coti Zelati, I. Ekeland, and E. Séré, “A variational approach to homoclinic orbits in Hamiltonian systems,” Mathematische Annalen, vol. 288, no. 1, pp. 133–160, 1990.

P. L. Felmer and E. A. de Barros e Silva, “Homoclinic and periodic orbits for Hamiltonian systems,” Annali della Scuola Normale Superiore di Pisa. Classe di Scienze, vol. 26, no. 2, pp. 285–301, 1998.

V. Coti Zelati and P. H. Rabinowitz, “Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potentials,” Journal of the American Mathematical Society, vol. 4, no. 4, pp. 693–727, 1991.

E. Serra, M. Tarallo, and S. Terracini, “Subharmonic solutions to second-order differential equations with periodic nonlinearities,” Nonlinear Analysis: Theory, Methods & Applications, vol. 41, no. 5-6, pp. 649–667, 2000.

Y. Ding and C. Lee, “Homoclinics for asymptotically quadratic and superquadratic Hamiltonian systems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 5-6, pp. 1395–1413, 2009.

R. Yuan and Z. Zhang, “Homoclinic solutions for a class of second order Hamiltonian systems,” Results in Mathematics, vol. 61, no. 1-2, pp. 195–208, 2012.

E. Paturel, “Multiple homoclinic orbits for a class of Hamiltonian systems,” Calculus of Variations and Partial Differential Equations, vol. 12, no. 2, pp. 117–143, 2001.

P. H. Rabinowitz, “Homoclinic orbits for a class of Hamiltonian systems,” Proceedings of the Royal Society of Edinburgh. Section A, vol. 114, no. 1-2, pp. 33–38, 1990.

M. Izydorek and J. Janczewska, “Homoclinic solutions for a class of the second order Hamiltonian systems,” Journal of Differential Equations, vol. 219, no. 2, pp. 375–389, 2005.

X. H. Tang and L. Xiao, “Homoclinic solutions for a class of second-order Hamiltonian systems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 3-4, pp. 1140–1152, 2009.

C. O. Alves, P. C. Carrião, and O. H. Miyagaki, “Existence of homoclinic orbits for asymptotically periodic systems involving Duffing-like equation,” Applied Mathematics Letters, vol. 16, no. 5, pp. 639–642, 2003.

P. C. Carrião and O. H. Miyagaki, “Existence of homoclinic solutions for a class of time-dependent Hamiltonian systems,” Journal of Mathematical Analysis and Applications, vol. 230, no. 1, pp. 157–172, 1999.

P. H. Rabinowitz and K. Tanaka, “Some results on connecting orbits for a class of Hamiltonian systems,” Mathematische Zeitschrift, vol. 206, no. 3, pp. 473–499, 1991.

W. Omana and M. Willem, “Homoclinic orbits for a class of Hamiltonian systems,” Differential and Integral Equations, vol. 5, no. 5, pp. 1115–1120, 1992.

X. Lv, S. Lu, and P. Yan, “Existence of homoclinic solutions for a class of second-order Hamiltonian systems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no. 1, pp. 390–398, 2010.

Z. Zhang and R. Yuan, “Homoclinic solutions for a class of non-autonomous subquadratic second-order Hamiltonian systems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 9, pp. 4125–4130, 2009.

Y. H. Ding, “Existence and multiplicity results for homoclinic solutions to a class of Hamiltonian systems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 25, no. 11, pp. 1095–1113, 1995.

M.-H. Yang and Z.-Q. Han, “Infinitely many homoclinic solutions for second-order Hamiltonian systems with odd nonlinearities,” Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 7, pp. 2635–2646, 2011.

Z.-Q. Ou and C.-L. Tang, “Existence of homoclinic solution for the second order Hamiltonian systems,” Journal of Mathematical Analysis and Applications, vol. 291, no. 1, pp. 203–213, 2004.

J. Yang and F. Zhang, “Infinitely many homoclinic orbits for the second-order Hamiltonian systems with super-quadratic potentials,” Nonlinear Analysis: Real World Applications, vol. 10, no. 3, pp. 1417–1423, 2009.

Q. Zhang and C. Liu, “Infinitely many homoclinic solutions for second order Hamiltonian systems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no. 2, pp. 894–903, 2010.

P. Korman and A. C. Lazer, “Homoclinic orbits for a class of symmetric Hamiltonian systems,” Electronic Journal of Differential Equations, vol. 1994, pp. 1–10, 1994.

Y. Lv and C.-L. Tang, “Existence of even homoclinic orbits for second-order Hamiltonian systems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 67, no. 7, pp. 2189–2198, 2007.

J. Sun, H. Chen, and J. J. Nieto, “Homoclinic solutions for a class of subquadratic second-order Hamiltonian systems,” Journal of Mathematical Analysis and Applications, vol. 373, no. 1, pp. 20–29, 2011.

W. Zou, “Variant fountain theorems and their applications,” Manuscripta Mathematica, vol. 104, no. 3, pp. 343–358, 2001.

M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and Their Applications, 24, Birkhäuser, Boston, Mass, USA, 1996.

P.-L. Lions, “The concentration-compactness principle in the calculus of variations. The locally compact case. II,” Annales de l'Institut Henri Poincaré. Analyse Non Linéaire, vol. 1, no. 4, pp. 223–283, 1984.

G. Arioli and A. Szulkin, “Homoclinic solutions for a class of systems of second order differential equations,” Topological Methods in Nonlinear Analysis, vol. 6, no. 1, pp. 189–197, 1995.

P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, vol. 65 of CBMS Regional Conference Series in Mathematics, published for the Conference Board of the Mathematical Sciences, Washington, DC, American Mathematical Society, Providence, RI, USA, 1986.

P. Bartolo, V. Benci, and D. Fortunato, “Abstract critical point theorems and applications to some nonlinear problems with “strong” resonance at infinity,” Nonlinear Analysis: Theory, Methods & Applications, vol. 7, no. 9, pp. 981–1012, 1983.