Existence and Asymptotic Behaviour of Solutions of Differential and Integral Equations in Some Function Spaces 2016View this Special Issue
Research Article | Open Access
Shengjun Li, Yuming Zhu, "Periodic Orbits of Radially Symmetric Keplerian-Like Systems with a Singularity", Journal of Function Spaces, vol. 2016, Article ID 7134135, 11 pages, 2016. https://doi.org/10.1155/2016/7134135
Periodic Orbits of Radially Symmetric Keplerian-Like Systems with a Singularity
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.
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 , 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 , 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 , 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
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.  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  the existence of infinitely many periodic solutions to (10).
When is semilinear at , del Pino et al.  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 , 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 . 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 .
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.
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 . 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  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 , 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 ,
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
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 , 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.
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 .
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.
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 function