Periodic Orbits of Radially Symmetric Keplerian-Like Systems with a Singularity

Li, Shengjun; Zhu, Yuming

doi:https://doi.org/10.1155/2016/7134135

Journal of Function Spaces

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Abstract Introduction Preliminaries Acknowledgments References Copyright Related Articles

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Existence and Asymptotic Behaviour of Solutions of Differential and Integral Equations in Some Function Spaces 2016

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Volume 2016 | Article ID 7134135 | https://doi.org/10.1155/2016/7134135

Periodic Orbits of Radially Symmetric Keplerian-Like Systems with a Singularity

Shengjun Li^1,2and Yuming Zhu³

Academic Editor: Leszek Olszowy

Received22 Dec 2015

Accepted15 Mar 2016

Published18 Apr 2016

Abstract

We study planar radially symmetric Keplerian-like systems with repulsive singularities near the origin and with some semilinear growth near infinity. By the use of topological degree theory, we prove the existence of two distinct families of periodic orbits; one rotates around the origin with small angular momentum, and the other one rotates around the origin with both large angular momentum and large amplitude.

1. Introduction

In recent few years, Fonda and his coworkers have studied the periodic, subharmonic, and quasiperiodic orbits for the radially symmetric Keplerian-like system in a systematic way, where may be singular at the origin. See [1–6]. As mentioned in [5], many phenomena of the nature obey laws of (1), such as the Newtonian equation for the motion of a particle subjected to the gravitational attraction of a sun which lies at the origin.

Setting , in [1], Fonda and Toader proved the case of solutions with large angular moment.

Theorem 1. Assume the hypotheses, uniformly for almost every . Then, there exists such that, for every integer , (1) has a periodic solution with minimal period , which makes exactly one revolution around the origin in the period time . Moreover, if denotes the angular momentum associated with , then

And in [2], they proved the case of solutions with small angular moment.

Theorem 2. Let the following two assumptions hold. There are an integer and two constants for which uniformly for every There are a positive constant and a continuous function such that

Then, there exists such that, for every integer , (1) has a periodic solution with a minimal period , which makes exactly one revolution around the origin in the period time Moreover, there is a constant such that, for every , and if denotes the angular momentum associated with then

Following the notion in [7], we say that (1) has a repulsive singularity at the origin if whereas (1) has an attractive singularity at the origin if

Concerned with singular differential equations or singular dynamical systems, the question of the existence of periodic solutions is one of the central topics and therefore has attracted much attention [2, 3, 8–16]. More general systems, of the type were studied by many authors, mainly by use of variational methods; the singularities are of attractive type (see [17–19]), where the potential is -periodic in and has singularities in . When the singularities are of repulsive type, for the scalar singular equation we recall the following results. Let , where and are -periodic satisfying the following strong force condition at : where is superlinear at : Fonda et al. [20] used the Poincaré-Birkhoff theorem to obtain the existence of positive periodic solutions, including all subharmonics. Similarly, when is superlinear at and satisfies the strong force condition at that states that there are positive constants such that andfor every and every sufficiently small, del Pino and Manásevich proved in [21] the existence of infinitely many periodic solutions to (10).

When is semilinear at , del Pino et al. [22] proved the existence of at least one positive -periodic solution of (10) if satisfies (13) near and the following nonresonance conditions at : there exists and a small constant such that for all and all . The result was later improved by Yan and Zhang [23], conditions (14) are removed, and the existence of at least one positive solution under suitable nonresonance conditions is obtained by using the topological degree theory. We note that conditions (14) are the uniform nonresonance conditions with respect to the Dirichlet boundary condition, not with respect to the periodic boundary condition.

It seems that the periodic boundary value problem for singular differential equations is closely related to the Dirichlet boundary value problem. A relationship between periodic and Dirichlet boundary value problems for second-order differential equations with singularities is established in [24]. Our main motivation is to obtain by [1, 2] that we will use such a relationship between the periodic boundary value problem and the Dirichlet boundary value problem to obtain the existence of two distinct families of periodic orbits to singular systems (1). Compared with Theorems 1 and 2, the main novelty in the paper is represented by the conditions at infinity, which remind us of a situation between the first and the second eigenvalue but are more general since the comparison involves the mean and the “weighted” eigenvalue associated with the functions controlling the ratio .

The main results in this paper are formulated in Theorem 6 and Theorem 17 (see them also for the precise statements). We summarize these two results informally.

Theorem 3. Let the following assumptions hold.(H₁)There exist and positive constants , such that for all and all (H₂)There exist -periodic continuous functions such that uniformly in . Moreover, here is the mean value, and is the -periodic eigenvalues of

Then system (1) has two distinct families of periodic orbits with the following distinct behavior: one rotates around the origin with large angular momentum and large amplitude, and the other one rotates around the origin with small angular momentum.

The rest of this paper is organized as follows. In Section 2, some preliminary results will be given. In Section 3, by the use of topological degree theory, we establish the existence of periodic orbits with large momentum and large amplitude. In Section 4, the periodic orbits with small momentum are established.

2. Preliminaries

In this section, we present some results which will be applied in Sections 3 and 4. Let us first introduce some known results on eigenvalues. Let be a -periodic potential such that . Consider the scalar eigenvalue problems of with the periodic boundary condition, or with the antiperiodic boundary condition We use to denote all eigenvalues of (19) with the Dirichlet boundary condition:

These eigenvalues, as a whole, are called the characteristic values of (19); the following are the standard results. See, for example, [25, 26].(E₁)With respect to the periodic and antiperiodic eigenvalues, there exist two sequences and such that where as .(E₂) is an eigenvalue of (19)-(20) if and only if or with being even; and is an eigenvalue of (19)–(21) if and only if or with being odd.(E₃)The comparison results hold for all of these eigenvalues. If then for any (E₄)The eigenvalues and can be recovered from the Dirichlet eigenvalues in the following way. For any , where denotes the translation of .(E₅), and are continuous in in the -topology of .(E₆) a.e.

In order to prove our results, we need two preliminary results. The first one is the following global continuation principle of Leray-Schauder.

Lemma 4 (see [27, Theorem ]). Let the operator be compact, where is a bounded open set in the Banach space . Then equation has a continuum of solutions in which connects the set with the set , if the following conditions are satisfied: (I);(II)(26) has no solution on .

To state the second preliminary result, we recall some notation and terminology from [28]. Define , a Fredholm mapping of index zero, with , where the Banach spaces are given as with their usual norms. Let be the Nemitzky operator from to induced by the map ; that is, Consider the equation

Lemma 5 (see [28, Theorem ]). Let be a bounded open subset and assume that there is no such that Then where denote the Schauder degree and the Brouwer degree, respectively.

We refer the reader to [29] for more details about degree theory.

3. Periodic Solutions with a Large Angular Momentum

We look for solutions which never attain the singularity, in the sense that Using the same idea in [1], we may write the solutions of (1) in polar coordinates Then we have the collisionless orbits if for every . Moreover, (1) is equivalent to the following system: where is the angular momentum of . Recall that is constant in time along with any solution. In the following, when considering a solution of (32), we will always implicitly assume that and

If is -radially periodic, then must be -periodic. We will look for solutions for which is -periodic. We thus consider the boundary value problem

Now we present our main result.

Theorem 6. Assume that and are satisfied. Then, (33) has a -periodic solution, and there exists such that, for every integer , system (1) has a periodic solution with minimal period , which makes exactly one revolution around the origin in the period time Moreover,

Now we begin by showing Theorem 6 and use topological degree theory. To this end, we deform the first equation of (33) to a simpler singular autonomous equation: where for some positive constant satisfy . Consider the following homotopy equation: where We need to find a priori estimates for the possible positive -periodic solutions of (36).

Note that satisfies the conditions with the same and minor changes of . Accordingly satisfies uniformly with respect to Moreover, for each , satisfies (16) with and . We will prove that and satisfy (17) uniformly in . The usual -norm is denoted by , and the supremum norm of is denoted by . This follows from the convexity of the first eigenvalues with respect to potentials.

Lemma 7. Given , then, for all ,

Proof. Put . ThenThis proves (37).
For (38), applying (37) to , where , we have for all . Thus Hence (38) holds.

Note thatApplying Lemma 7 to and , we have Thus and defined above satisfy (17) uniformly in .

Lemma 8. For every , there exists a constant such that if , and is a -periodic solution of (36), then .

Proof. Assume by contradiction that there exist and sequence , , and such that , , and is a -periodic solution of (36) for and with
From , there exist and such that and from , uniformly in ; one knows that there are some constants such thatMultiplying (36) by and integrating from to , we obtain Therefore, which is a contradiction with the fact that

Lemma 9. Fix as in Lemma 8. Given with , there exists such that if and is a positive -periodic solution of (36), then for some .

Proof. Let be a positive -periodic solution of (36). From , we know that there exist positive constants such that for all and Thus, there is such that Integrate (36) from to ; we get Thus ; there exists such that
Take some constant . From there is large enough such that for all and . We assert that for some . Otherwise, assume that for all .
Let For , let ; moreover, write as ; then satisfies the following differential equation: Integrate (55) from to ; we have Multiplying (55) by and integrating, we get where the fact that is used.
Note that for some , ; thus . We assert that . On the contrary, assume that . Now by (57), the first Dirichlet eigenvalueSoOn the other hand, , This is a contradiction, which shows that ; thus . Now it follows from (56) that and ; this contradicts the fact that is a positive solution. We have proven that for some and for some . Thus the intermediate value theorem implies that (49) holds.

Lemma 10. Assume that of the scalar differential equation ; then

Proof. By the results for eigenvalues in , we havefor all .
Then, by the theory of linear second-order differential operators [30], the eigenvalues of with Dirichlet boundary conditions form a sequence which tends to , and the corresponding eigenfunctions are an orthonormal base of . Hence, given and , we can write This completes the proof.

Lemma 11. Under the assumption in Lemma 9, there exist , such that any positive -periodic solution of (36) satisfies

Proof. Notice the inequality (51). By (16), letting , there will be some such that for all . Hence, one has some such that for all and .
Multiplying (36) by and then integrating over , we get Note from Lemma 9 that there exists which satisfies . Let ; then . Thus The other terms in (68) by the Hölder inequality can be estimated as follows:Thus (68) reads as where are positive constants.
On the other hand, using Lemma 10, and we get from (71) that Consequently, for some . By (71), one has for some From these, for any , Thus is obtained.
In order to prove (65), we write (36) as As , thus . Since , there exists such that . Therefore, where , .

Next, the positive lower estimates for are obtained from the strong force condition .

Lemma 12. Under the assumption in Lemma 9, there exists a constant such that any positive solution of (36) satisfies

Proof. From (50), we fix some such that for all and all . Assume now that By Lemma 9, . Let be the first time instant such that Then, for any , we have . Hence, for , As for . Therefore, the function has an inverse, denoted by .
Now multiplying (36) by and integrating over , we get for some , where the results from Lemma 11 are used. By (50), if . Thus we know from (82) that for some constant

Let us denote by the set of -periodic -functions with the usual norm.

Lemma 13. Given with , there is a continuum in , connecting with , whose elements are solutions of equation

Proof. Obviously, if and is a -periodic solution of (36), then also satisfies (83). Let us define the following operators: It is clear that (83) is equivalent to the operator equation Since is invertible, thus we have Define and let the open bounded subset in be By Lemmas 11 and 12, (86) has no solutions on Since is a compact operator, by Lemma 4, the result will be proved if we can show that the degree is nonzero for some
In order to compute the degree, we consider (36). By Lemmas 11 and 12, the degree has to be the same for every . Therefore, we consider (36) with , which is the equation which is equivalent to the system where
It is easy to know that has a unique zero and the determinant of Jacobian matrix satisfies . By Lemma 5, the Leray-Schauder degree of is equal to the Brouwer degree of ; that is,and the proof is completed.

We can deduce from Lemma 11 that there is a connected set , contained in , which connects with , for every , whose element is the solution of (83).

Lemma 14. For some constants , there exists such that if with , then

Proof. For , there are some and small constant such that for all . Let be a -periodic solution of (83). Notice inequality (66) and the boundary condition , by the Wirtinger inequality; we have So For , let be as in Lemma 8. Set .
Let be an element of , with . By Lemma 8 and (94), , for every . Integrating (83) from to , we have Therefore, we obtain It follows from the above argument that we can take some constant , and the proof is finished.

Define the functionIt is clear that is continuous from to ; Lemma 14 shows that its image is an interval. The following lemma is necessarily the same as Lemma in [1]. The following proof is only for completeness.

Lemma 15. For every , there are , verifying system (32), for which and for every

Proof. Given , there are , such that Obviously, is -periodic satisfying the first equation in (32) and defining it also satisfies the second equation in (32). Moreover

For every , let be a solution of system (32). Then it follows from Lemma 15 that

In particular, if for some integer , then is periodic with minimal period and rotates exactly once around the origin in the period time . Hence, for every integer , we have such a -periodic solution, which we denote by . Let be its polar coordinates and let be its angular momentum. By the above construction, satisfy system (32), , and Assume is a bounded subsequence, with for some , using Lemma 11 with ; there exists a constant such that , and hence for every , in contradiction with (103), so Moreover, by (105) and (94), with , we have The proof of Theorem 6 is finished.

4. Periodic Solutions with a Small Angular Momentum

In this section, we establish the periodic orbits of (1) with a small angular momentum. Since some parts of the proof are in the same line of that of Theorem 6, we will outline the proof with the emphasis on the difference.

Let ; (33) can be written as the -periodic problem And let be a Banach space of functions such that , with continuous immersions, and set

Define the following two operators: Taking not belonging to the spectrum of , (107) can be translated to the fixed problem

We will say that a set is uniformly positively bounded below if there is a constant such that for every In order to prove the main result of this paper, we need the following theorem, which has been proved in [4].

Lemma 16. Let be an open bounded subset of , uniformly positively bounded below. Assume that there is no solution of (107), on the boundary , and that

Then, there exists a such that, for every integer , system (1) has a periodic solution with minimal period , which makes exactly one revolution around the origin in the period time . The function is -periodic and when restricted to , it belongs to . Moreover, if denotes the angular momentum associated with , then

The main result of this section reads as follows.

Theorem 17. Assume that and are satisfied. Then there exists such that, for every integer , system (1) has a periodic solution with minimal period , which makes exactly one revolution around the origin in the period time Moreover, there exists a constant such that

Proof. We consider the -periodic problem (107). Using the same technique in Section 3, we can prove that there exists a constant such that if is a -periodic solution of (107), then for every . Note that is independent of .
DefineObviously, is an open subset of and (107) has no solutions on . Again by the homotopy invariance of degree and Lemma 5, Thus, by Lemma 16, the proof of Theorem 17 is thus completed.

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grant no. 11461016), the Scientific Research Foundation of Hainan University (kyqd1544), and Hainan Natural Science Foundation (Grants nos. 20167246 and 20161007).

References

A. Fonda and R. Toader, “Periodic orbits of radially symmetric Keplerian-like systems: a topological degree approach,” Journal of Differential Equations, vol. 244, no. 12, pp. 3235–3264, 2008.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
A. Fonda and R. Toader, “Radially symmetric systems with a singularity and asymptotically linear growth,” Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 7, pp. 2485–2496, 2011.
View at: Publisher Site | Google Scholar | MathSciNet
A. Fonda and R. Toader, “Periodic orbits of radially symmetric systems with a singularity: the repulsive case,” Advanced Nonlinear Studies, vol. 11, no. 4, pp. 853–874, 2011.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
A. Fonda and R. Toader, “Periodic solutions of radially symmetric perturbations of Newtonian systems,” Proceedings of the American Mathematical Society, vol. 140, no. 4, pp. 1331–1341, 2012.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
A. Fonda, R. Toader, and F. Zanolin, “Periodic solutions of singular radially symmetric systems with superlinear growth,” Annali di Matematica Pura ed Applicata, vol. 191, no. 2, pp. 181–204, 2012.
View at: Publisher Site | Google Scholar | MathSciNet
A. Fonda and A. J. Ureña, “Periodic, subharmonic, and quasi-periodic oscillations under the action of a central force,” Discrete and Continuous Dynamical Systems A, vol. 29, no. 1, pp. 169–192, 2011.
View at: Publisher Site | Google Scholar | MathSciNet
A. C. Lazer and S. Solimini, “On periodic solutions of nonlinear differential equations with singularities,” Proceedings of the American Mathematical Society, vol. 99, no. 1, pp. 109–114, 1987.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
D. Jiang, J. Chu, and M. Zhang, “Multiplicity of positive periodic solutions to superlinear repulsive singular equations,” Journal of Differential Equations, vol. 211, no. 2, pp. 282–302, 2005.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
S. Li, F. Liao, and H. Zhu, “Multiplicity of positive solutions to second-order singular differential equations with a parameter,” Boundary Value Problems, vol. 2014, article 115, 2014.
View at: Publisher Site | Google Scholar | MathSciNet
S. Li, F. Liao, and W. Xing, “Periodic solutions of Liénard differential equations with singularity,” Electronic Journal of Differential Equations, vol. 151, pp. 1–12, 2015.
View at: Google Scholar
S. Li, F. Liao, and H. Zhu, “Periodic solutions of second order non-autonomous differential systems,” Fixed Point Theory, vol. 15, no. 2, pp. 487–494, 2014.
View at: Google Scholar | MathSciNet
F. Wang and Y. Cui, “On the existence of solutions for singular boundary value problem of third-order differential equations,” Mathematica Slovaca, vol. 60, no. 4, pp. 485–494, 2010.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
F. Wang, F. Zhang, and Y. Yu, “Existence of positive solutions of Neumann boundary value problem via a convex functional compression-expansion fixed point theorem,” Fixed Point Theory, vol. 11, no. 2, pp. 395–400, 2010.
View at: Google Scholar | MathSciNet
H. Wang, “On the number of positive solutions of nonlinear systems,” Journal of Mathematical Analysis and Applications, vol. 281, no. 1, pp. 287–306, 2003.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
H. Wang, “Positive periodic solutions of singular systems with a parameter,” Journal of Differential Equations, vol. 249, no. 12, pp. 2986–3002, 2010.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
Y. Wang and Y.-H. Xia, “The existence of almost periodic solutions of a certain nonlinear system,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 2, pp. 1060–1072, 2011.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
W. B. Gordon, “Conservative dynamical systems involving strong forces,” Transactions of the American Mathematical Society, vol. 204, pp. 113–135, 1975.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
P. Majer and S. Terracini, “Periodic solutions to some problems of n-body type,” Archive for Rational Mechanics and Analysis, vol. 124, no. 4, pp. 381–404, 1993.
View at: Publisher Site | Google Scholar | MathSciNet
S. Zhang, “Multiple closed orbits of fixed energy for $N$ -body-type problems with gravitational potentials,” Journal of Mathematical Analysis and Applications, vol. 208, no. 2, pp. 462–475, 1997.
View at: Publisher Site | Google Scholar | MathSciNet
A. Fonda, R. F. Manásevich, and F. Zanolin, “Subharmonic solutions for some second-order differential equations with singularities,” SIAM Journal on Mathematical Analysis, vol. 24, no. 5, pp. 1294–1311, 1993.
View at: Publisher Site | Google Scholar | MathSciNet
M. A. del Pino and R. F. Manásevich, “Infinitely many T-periodic solutions for a problem arising in nonlinear elasticity,” Journal of Differential Equations, vol. 103, no. 2, pp. 260–277, 1993.
View at: Publisher Site | Google Scholar | MathSciNet
M. A. del Pino, R. F. Manásevich, and A. Montero, “T-periodic solutions for some second order differential equations with singularities,” Proceedings of the Royal Society of Edinburgh A, vol. 120, no. 3-4, pp. 231–243, 1992.
View at: Publisher Site | Google Scholar
P. Yan and M. Zhang, “Higher order non-resonance for differential equations with singularities,” Mathematical Methods in the Applied Sciences, vol. 26, no. 12, pp. 1067–1074, 2003.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
M. Zhang, “A relationship between the periodic and the Dirichlet BVPs of singular differential equations,” Proceedings of the Royal Society of Edinburgh. Section A. Mathematics, vol. 128, no. 5, pp. 1099–1114, 1998.
View at: Publisher Site | Google Scholar | MathSciNet
W. Magnus and S. Winkler, Hill's Equation, Dover, New York, NY, USA, 1979.
View at: MathSciNet
M. Zhang, “The rotation number approach to eigenvalues of the one-dimensional p-Laplacian with periodic potentials,” Journal of the London Mathematical Society, vol. 64, no. 1, pp. 125–143, 2001.
View at: Publisher Site | Google Scholar | MathSciNet
E. Zeidler, Nonlinear Functional Analysis and Its Applications, vol. 1, Springer, New York, NY, USA, 1986.
View at: Publisher Site | MathSciNet
A. Capietto, J. Mawhin, and F. Zanolin, “Continuation theorems for periodic perturbations of autonomous systems,” Transactions of the American Mathematical Society, vol. 329, no. 1, pp. 41–72, 1992.
View at: Publisher Site | Google Scholar | MathSciNet
J. Mawhin, “Topological degree methods in nonlinear boundary value problems,” CBMS Regional Conference Series in Mathematics, vol. 40, 122 pages, 1979.
View at: Google Scholar
W. N. Everitt, M. K. Kwong, and A. Zettl, “Oscillation of eigenfunctions of weighted regular Sturm-Liouville problems,” Journal of the London Mathematical Society, vol. 27, no. 1, pp. 106–120, 1983.
View at: Publisher Site | Google Scholar | MathSciNet

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Copyright © 2016 Shengjun Li and Yuming Zhu. 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.

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