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Homoclinic Solutions for a Class of Second Order Nonautonomous Singular Hamiltonian Systems
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.
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  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  of Gordon in 1975 and  of Rabinowitz in 1978. In , the notion of strong force is introduced to deal with singular problems, while in , 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.
Remark 2. The assumption () is the so-called strong force condition (see Gordon ), 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  under essentially the same assumptions as above. In , Tanaka used a minimax argument from Bahri and Rabinowitz  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 , who showed existence of two homoclinic orbits, and by Bessi , who used Lyusternick-Schnirelman category to prove the existence of distinct homoclinics for potentials satisfying a pinching condition. For , Janczewska and Maksymiuk in , 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  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  was substantially improved in  where, using the same idea, the authors showed that a nondegeneracy variational condition introduced in  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  constructed infinitely many multibump homoclinic solutions for of the form , with being almost periodic and satisfying (A) and ()–(). Recently, Izydorek and Janczewska in  investigated the existence of at least two connecting orbits in some more general sense. For the case that , recently, the authors in , 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 . 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 . 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.
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 , 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 ).
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.
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 , we can also obtain that
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.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
The project is supported by the National Natural Science Foundation of China (Grant no. 11101304).
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