About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2014 (2014), Article ID 829052, 8 pages
http://dx.doi.org/10.1155/2014/829052
Research Article

Homoclinic Solutions for a Class of Second Order Nonautonomous Singular Hamiltonian Systems

1Department of Mathematics, Tianjin Polytechnic University, Tianjin 300387, China
2Department of Mathematics, College of Science, Hohai University, Nanjing 210098, China
3School of ELectrical and Electronic Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798

Received 3 December 2013; Accepted 10 January 2014; Published 19 February 2014

Academic Editor: Jifeng Chu

Copyright © 2014 Ziheng Zhang et al. 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 are concerned with the existence of homoclinic solutions for the following second order nonautonomous singular Hamiltonian systems , (HS) where , , is a continuous bounded function, and the potential has a singularity at , and is the gradient of at . The novelty of this paper is that, for the case that and (HS) is nonautonomous (neither periodic nor almost periodic), we show that (HS) possesses at least one nontrivial homoclinic solution. Our main hypotheses are the strong force condition of Gordon and the uniqueness of a global maximum of . Different from the cases that (HS) is autonomous or (HS) is periodic or almost periodic, as far as we know, this is the first result concerning the case that (HS) is nonautonomous and . Besides the usual conditions on , we need the assumption that for all to guarantee the existence of homoclinic solution. Recent results in the literature are generalized and significantly improved.

1. Introduction

As is known to all, the search for periodic as well as homoclinic and heteroclinic solutions of Hamiltonian systems has a long and rich history. In present paper, we particularly focus our attention on the existence of homoclinic solutions of second order nonautonomous singular Hamiltonian systems. For the results on the literature of periodic solutions for such singular systems, we refer the reader to the book [1] of Ambreosetti and Zelati.

Second order Hamiltonian systems are systems of the following form: where and . Roughly speaking, they are the Euler-Lagrange equations of the functional where the integration is taken over a finite interval or all real and the Lagrangian has the form Clearly, when the potential is -periodic in , it is natural to look for -periodic solutions of as critical points of the functional over a suitable space of -periodic functions. Also, in such a case, one can look for homoclinic solutions at the origin as limits of -periodic solutions (subharmonic solutions) as , or alternatively, as critical points of the functional over a suitable space of functions on the whole space (typically, ).

For singular systems, one assumes that and for some . Although the study of singular systems is perhaps as old as Kepler’s classical problem in mechanics, (and, also, the -body problem), the interest in such problems was renewed by the pioneering papers [2] of Gordon in 1975 and [3] of Rabinowitz in 1978. In [2], the notion of strong force is introduced to deal with singular problems, while in [3], the use of variational methods is brought into the study of periodic solutions of Hamiltonian systems.

The present paper is concerned with the existence of homoclinic solutions for the following second order nonautonomous singular Hamiltonian systems: where , ( ), is a continuous bounded function, and the potential has a singularity at ; is the gradient of at . We recall that a homoclinic solution of is a solution such that for all and Throughout this paper, we assume that the following hypotheses are supposed: (A) is a continuous function such that where and are two positive constants and ( ) for some ;( ) for all and if and only if and ;( )there is a constant such that for all , where ;( ) ;( )there is a neighbourhood of in and a function such that as and

Now we are in the position to state our main result.

Theorem 1. Under the conditions of (A) and ( )–( ), has at least one nontrivial homoclinic solution.

Remark 2. The assumption ( ) is the so-called strong force condition (see Gordon [2]), which is used to verify the Palais-Smale condition for the functional corresponding to the approximating problem defined below. For example, ( ) is satisfied when ( ) in a neighbourhood of . The assumption ( ) is a kind of concavity condition for near . In particular, ( ) holds for small when is negative definite.

In the case of autonomous singular Hamiltonian systems, the first result on existence of a homoclinic orbit using variational methods was obtained by Tanaka [4] under essentially the same assumptions as above. In [4], Tanaka used a minimax argument from Bahri and Rabinowitz [5] in order to get approximating solutions of the following boundary value problem: as critical points of the corresponding functional and obtained uniform estimates to show that those solutions converge weakly to a nontrivial homoclinic solution of . Regarding multiplicity of homoclinics, still in the autonomous singular case, early results were obtained by Caldiroli [6], who showed existence of two homoclinic orbits, and by Bessi [7], who used Lyusternick-Schnirelman category to prove the existence of distinct homoclinics for potentials satisfying a pinching condition. For , Janczewska and Maksymiuk in [8], via use of variational methods and geometrical arguments, investigated the existence of homoclinic orbits. In addition, different kinds of multiplicity results were obtained in [9, 10] (still for conservative systems) by exploiting the topology of , the domain of the potential, when the set is such that the fundamental group of is nontrivial.

In the case of planar autonomous systems, more extensive existence and multiplicity results were obtained. Indeed, under essentially the same conditions as above with , Rabinowitz showed in [11] that has at least a pair of homoclinic solutions by exploiting the topology of the plane and minimizing the energy functional on classes of sets with a fixed winding number around the singularity . The result in [11] was substantially improved in [12] where, using the same idea, the authors showed that a nondegeneracy variational condition introduced in [12] is in fact necessary and sufficient for the minimum problem to have a solution in the class of sets with winding number greater than and, therefore, proved a result on existence of infinitely many homoclinic solutions. For the recent results, we refer the reader to [13, 14].

On the other hand, in the case of -periodic time dependent Hamiltonians in , existence of infinitely many homoclinic orbits was obtained for smooth Hamiltonians by using a variational procedure due to Séré in [15, 16] for the first systems and in [17, 18] for second order systems. In case , using theses ideas, Rabinowitz [19] constructed infinitely many multibump homoclinic solutions for of the form , with being almost periodic and satisfying (A) and ( )–( ). Recently, Izydorek and Janczewska in [20] investigated the existence of at least two connecting orbits in some more general sense. For the case that , recently, the authors in [21], using the category theory, for the first time considered the existence of infinitely many geometrically distinct homoclinic solutions, under the assumptions that is periodic and satisfies ( )–( ) and the following condition on at infinity:( )there exists such that and for large.

Here we must point out that all the results mentioned above are obtained for the case that is autonomous or periodic or almost periodic. Motivated mainly by the works of [4, 21], in present paper, we focus our attention on the case that and is non-autonomous (neither periodic nor almost periodic). To the best of our knowledge, this is the first result on the existence of homoclinic solutions for second order nonautonomous singular Hamiltonian systems in ( ). The proof of Theorem 1 will be demonstrated in the following sections. To this end, we employ the technique used in [4]. Explicitly, considering the approximating problem: solutions of are obtained as critical points of the functional defined in Section 2. We show the existence of critical points of via a minimax argument, which is essentially due to Bahri and Rabinowitz [5]. Furthermore, we also get some estimates, which are uniform with respect to , for minimax values and corresponding critical points . These uniform estimates permit us to let ; for a suitable sequence and a subsequence , we see that converges weakly to a homoclinic solution of as .

The remaining part of this paper is organized as follows. In Section 2, via a minimax argument, for any , we show that has at least one nontrivial solution . In Section 3, some uniform estimates for solutions are obtained to investigate homoclinic solution of . In Section 4, we are devoted to accomplishing the proof of Theorem 1.

2. Approximating Problem

In this section, we investigate the approximating problem via a minimax argument. Denote by the usual Sobolev space on with values in under the norm Let Clearly, is an open subset of . Define the functional as follows: Under the assumptions of Theorem 1, as usual, one can show that and Then there is an one-to-one correspondence between critical points of and classical solutions of .

In order to obtain a critical point of , we use a minimax argument. To do so, must satisfy the Palais-Smale condition ((PS) condition) on ; that is, for any sequence such that is bounded and as , possesses a subsequence converging to some .

Proposition 3. If and satisfy (A), ( ), ( ), ( ), and ( ), then satisfies the (PS) condition.

Proof. Let be a sequence such that is bounded and as . Then, by ( ) and the definition of , is bounded in . Hence, we can extract a subsequence of , still denoted by , such that converges to (the closure of in ) weakly in . On the other hand, by Lemma  2.1 in [22], if , then That is, . Hence, . Subsequently, in view of ( ) and the Sobolev compact embedding theorem, it follows that strongly in .

Since satisfies the (PS) condition, we have the following deformation theorem (see [23]).

Lemma 4. Suppose that is not a critical value of . Then, for all , there are an and such that (1) if ;(2) for all ;(3) ,where denotes the level set defined by

Now, one introduces a minimax procedure for . Let For , one observes that for all Since , one can consider for each a map defined by One denotes by the Brouwer degree of a map . Let It is obvious that . Define a minimax value of by Then one has the following.

Proposition 5. is a critical value of .

Proof. will be seen later in Proposition 6. Here we assume it and prove that is a critical value of . On the contrary, we suppose that is not a critical value. Taking and in Lemma 4, we have a deformation flow with the properties (1)–(3). Moreover, we can verify In fact, since (see (1)), we have for . On the other hand, due to (2), we have Hence, for all and . Thus we have Therefore, for ; that is, (22) holds.
Choose such that and consider . Then, in virtue of (3), we obtain which contradicts the definition of . Therefore, is a critical value of .

Proposition 6. There is a constant independent of such that

Proof. For any given , we define ( ) by Then, we can easily see the following: (1) , that is, for all ;(2) for all and .Therefore, we get In what follows, we prove the existence of a constant   such that for all . For any given , we have where is defined in ( ). Otherwise, we can easily observe that . Hence, there is such that Since , there is an such that By the Schwarz inequality, we have for where Thus, we have which yields that Combining with (28) and (35), we obtain the desired conclusion.

In view of Propositions 5 and 6, we deduce the following proposition concerned with the uniform boundness of critical value of .

Proposition 7. For , possesses a solution such that where and are independent of .

3. Estimates for Solutions of

In this section, we give some estimates on the solutions of to allow go to in Section 4. Firstly, from the definition of and Proposition 7, we have the following.

Lemma 8. There is a constant which is independent of such that

In what follows, we denote by , , , various constants which are independent of .

Proposition 9. Consider for all .

Proof. Suppose that for some . On account of , we can find an interval such that and for all , which, combining with Lemma 8, implies that On the other hand, we have where is defined in (33). Combining with (38) and (39), we deduce that which yields that As a result, we conclude that

Lemma 10. Define the function as follows: Then, is nondecreasing on . Furthermore, one has

Proof. In view of (A) and ( ), it deduces that which implies that is nondecreasing on . Thus, we have for all . Integrating (43) over and by Lemma 8, we have for all . We observe the fact that . Otherwise, and then we have by the uniquemess of the solution of the initial value problem: which contradicts the fact that . Consequently, combining with and in view of (46), we have
It only remains to show that as . On the contrary, due to the facts that is increasing on , , and it occurs that However, this contradicts the fact that for all .

The following proposition gives us an -bound from below on for all . Here, it must be pointed that the condition ( ) is used only in this proposition.

Proposition 11. Consider for all .

Proof. Using (43), we get Due to the fact that (the reason has been explained in Lemma 10) and the condition ( ), we obtain that Suppose that takes its maximum at . From the above inequality, we deduce that . Thus, we have

By Proposition 11, we can find two numbers such that , , and for all . For and , we need the following property.

Lemma 12. Consider , as .

Proof. Let be a solution of the following initial value problem: By the continuous dependence of on the initial data, for any , there is an such that By (44), for any , we can find such that That is, for . Therefore, we have as . Similarly, we can obtain as .

4. Proof of Theorem 1

In this section, we construct a homoclinic solution of as a limit of the solutions of when goes to and complete the proof of Theorem 1. In order to accomplish such a process, for each , we define by Then it directly follows from Lemma 8 and Proposition 9 that (i) is a solution of in ;(ii) for all ;(iii) , , and are uniformly bounded for .By (iii), we can extract a subsequence such that converges to some with in the following sense: Moreover, we have Due to the strong force condition ( ), similar to Lemma  2.1 in [22], we can also obtain that

In what follows, we focus our attentions to show that is exactly right the homiclinic solution of that we need.

Proposition 13. is a nontrivial solution of on .

Proof. In view of (61), it is sufficient to verify that for any By Lemma 12, we can choose such that for all . By property (i) of , we have for all On account of (58) and (59), we get (62) as . On the other hand, as a direct consequence of (58) and the property (ii) of , we have ; that is, is nontrivial.

As the last step of the proof of Theorem 1, we show that and satisfy the following property.

Proposition 14. Consider , as .

Proof. Here, we just check the case that , since it is the same as . First, we prove as . On the contrary, we assume that as . Then, for some sequence and for some , we have On the other hand, in view of ( ) and (60), it follows that Hence, there is a sequence such that . Thus, must intersect and infinitely as . However, this contradicts and (60). In fact, suppose that is an interval such that Then, we obtain where . On account of Schwarz inequality, it follows that If intersects and infinitely as , we can find infinitely many disjoint intervals with the property (66). Thus, we have which contradicts and (60). Therefore, we obtain as .
Since satisfies , is bounded on each compact interval by (iii) and (61). Thus, converges to in . Hence, Since is increasing, as and , it is obvious that as .

Up to now, we are in the position to give the proof of our main result.

Proof. In view of Propositions 13 and 14, it is obvious that is one nontrivial homoclinic solution of .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The project is supported by the National Natural Science Foundation of China (Grant no. 11101304).

References

  1. A. Ambrosetti and V. C. Zelati, Periodic Solutions of Singular Lagrangian Systems, vol. 10 of Progress in Nonlinear Differential Equations and Their Applications, Birkhäuser, Boston, Mass, USA, 1993. View at MathSciNet
  2. W. B. Gordon, “Conservative dynamical systems involving strong force,” Transactions of the American Mathematical Society, vol. 204, pp. 113–115, 1975. View at Publisher · View at Google Scholar · View at MathSciNet
  3. P. H. Rabinowitz, “Periodic solutions of Hamiltonian systems,” Communications on Pure and Applied Mathematics, vol. 31, no. 2, pp. 157–184, 1978. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. K. Tanaka, “Homoclinic orbits for a singular second order Hamiltonian system,” Annales de l'Institut Henri Poincaré C, vol. 7, pp. 427–438, 1990. View at Zentralblatt MATH · View at MathSciNet
  5. A. Bahri and P. H. Rabinowitz, “A minimax method for a class of Hamiltonian systems with singular potentials,” Journal of Functional Analysis, vol. 82, no. 2, pp. 412–428, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  6. P. Caldiroli, “Existence and multiplicity of homoclinic orbits for potentials on unbounded domains,” Proceedings of the Royal Society of Edinburgh A, vol. 124, no. 2, pp. 317–339, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. U. Bessi, “Multiple homoclinic orbits for autonomous singular potentials,” Proceedings of the Royal Society of Edinburgh A, vol. 124, no. 4, pp. 785–802, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. J. Janczewska and J. Maksymiuk, “Homoclinic orbits for a class of singular second order Hamiltonian systems in 3,” Central European Journal of Mathematics, vol. 10, no. 6, pp. 1920–1927, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  9. M. L. Bertotti and L. Jeanjean, “Multiplicity of homoclinic solutions for singular second-order conservative systems,” Proceedings of the Royal Society of Edinburgh A, vol. 126, no. 6, pp. 1169–1180, 1996. View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  10. P. Caldiroli and C. de Coster, “Multiple homoclinics for a class of singular Hamiltonian systems,” Journal of Mathematical Analysis and Applications, vol. 211, no. 2, pp. 556–573, 1997. View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  11. P. H. Rabinowitz, “Homoclinics for a singular Hamiltonian system,” in Geometric Analysis and the Calculus of Variations, pp. 267–296, International Press, Cambridge, Mass, USA, 1996. View at Zentralblatt MATH · View at MathSciNet
  12. P. Caldiroli and L. Jeanjean, “Homoclinics and heteroclinics for a class of conservative singular hamiltonian systems,” Journal of Differential Equations, vol. 136, no. 1, pp. 76–114, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  13. M. J. Borges, “Heteroclinic and homoclinic solutions for a singular Hamiltonian system,” European Journal of Applied Mathematics, vol. 17, no. 1, pp. 1–32, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  14. M. Izydorek and J. Janczewska, “The shadowing chain lemma for singular Hamiltonian systems involving strong forces,” Central European Journal of Mathematics, vol. 10, no. 6, pp. 1928–1939, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. E. Séré, “Existence of infinitely many homoclinic orbits in Hamiltonian systems,” Mathematische Zeitschrift, vol. 209, no. 1, pp. 27–42, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  16. E. Sere, “Looking for the Bernoulli shift,” Annales de l'Institut Henri Poincaré C, vol. 10, pp. 561–590, 1993. View at Zentralblatt MATH · View at MathSciNet
  17. P. Caldiroli and P. Montecchiari, “Homoclinic orbits for second order Hamiltonian systems with potential changing sign,” Communications on Applied Nonlinear Analysis, vol. 1, pp. 97–129, 1994. View at Zentralblatt MATH · View at MathSciNet
  18. V. C. 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. View at Zentralblatt MATH · View at MathSciNet
  19. P. H. Rabinowitz, “Multibump solutions for an almost periodically forced singular Hamiltonian system,” Electronic Journal of Differential Equations, vol. 1995, no. 12, pp. 1–21, 1995. View at Zentralblatt MATH · View at MathSciNet
  20. M. Izydorek and J. Janczewska, “Connecting orbits for a periodically forced singular planar Newtonian system,” Journal of Fixed Point Theory and Applications, vol. 12, no. 1-2, pp. 59–67, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. D. G. Costa and H. Tehrani, “On a class of singular second-order Hamiltonian systems with infinitely many homoclinic solutions,” Journal of Mathematical Analysis and Applications, vol. 412, no. 1, pp. 200–211, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  22. C. Greco, “Periodic solutions of a class of singular Hamiltonian systems,” Nonlinear Analysis, vol. 12, no. 3, pp. 259–269, 1988. View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  23. P. H. Rabinowitz, “Periodic and heteroclinic orbits for a periodic Hamiltonian system,” Annales de l'Institut Henri Poincaré C, vol. 6, pp. 331–346, 1989. View at Zentralblatt MATH · View at MathSciNet