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.


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  ∈ ((R/Z) × (0, ∞), R) may be singular at the origin.See [1][2][3][4][5][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.
And in [2], they proved the case of solutions with small angular moment.

Theorem 2. Let the following two assumptions hold.
(A 1 ) There are an integer  and two constants ,  for which (5) Then, there exists  2 ≥ 1 such that, for every integer  ≥  2 , (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  > 0 such that, for every  ≥  2 , 1  <       ()     < , for every  ∈ R, and if   denotes the angular momentum associated with   () then lim →∞   = 0. 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 lim →0 +  (, ) = +∞, uniformly in  ∈ R.
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][9][10][11][12][13][14][15][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][18][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  ∈ (R + , R) and  ∈ (R, R) are -periodic satisfying the following strong force condition at  = 0: 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  = 0 that states that there are positive constants ,   ,  such that  ≥ 1 and for 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  = 0 and the following nonresonance conditions at  = +∞: there exists  ∈ N and a small constant  > 0 such that for all  and all  ≫ 1.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 secondorder 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.
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.

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  ∈  1 (R).Consider the scalar eigenvalue problems of with the periodic boundary condition, or with the antiperiodic boundary condition We use   1 () <   2 () < ⋅ ⋅ ⋅ <    () < ⋅ ⋅ ⋅ 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].
(E 4 ) The eigenvalues   () and   () can be recovered from the Dirichlet eigenvalues in the following way.
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 14]).Let the operator  : [ 1 ,  2 ] ×  →  be compact, where  is a bounded open set in the Banach space .Then equation has a continuum C of solutions in R ×  which connects the set { 1 } ×  with the set { 2 } × , if the following conditions are satisfied: To state the second preliminary result, we recall some notation and terminology from [28].Define  : dom  ⊂  → ,  =   , a Fredholm mapping of index zero, with dom  = { ∈  : (⋅) is absolutely continuous}, where the Banach spaces ,  are given as with their usual norms.Let  0 be the Nemitzky operator from  to  induced by the map  0 ; that is,  0 : (⋅) →  0 ((⋅)).Consider the equation Lemma 5 (see [28,Theorem 1]).Let Ω ⊂  be a bounded open subset and assume that there is no (⋅) ∈   Ω such that   =  0 ().Then where deg  , deg  denote the Schauder degree and the Brouwer degree, respectively.
We refer the reader to [29] for more details about degree theory.

Periodic Solutions with a Large Angular Momentum
We look for solutions () ∈ R 2 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 () > 0 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  ≥ 0 and  > 0.
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.
Let us denote by  1  the set of -periodic  1 -functions with the usual norm.Lemma 13.Given ,  with  ≤  ≤ , there is a continuum C , in [, ] ×  1   , connecting {} ×  1  with {} ×  1  , 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: In order to compute the degree, we consider (36).By Lemmas 11 and 12, the degree has to be the same for every  ∈ [0, 1].Therefore, we consider (36) with  = 0, which is the equation which is equivalent to the system where  = (, ).
It is easy to know that  has a unique zero ( 0 ,  0 ) and the determinant of Jacobian matrix satisfies |  ( 0 ,  0 )| > 0. By Lemma 5, the Leray-Schauder degree of  −  −1 (, ) 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 C, contained in [, +∞] × Therefore, we obtain It follows from the above argument that we can take some constant  ∈ (0, √ 2 For every  ∈ (0, ], let () be a solution of system (32).Then it follows from Lemma 15 that In particular, if  = 2/ for some integer  ≥ 1, then () is periodic with minimal period  and rotates exactly once around the origin in the period time .Hence, for every integer  ≥ 2/, 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), (  ,   ) ∈ C, and Assume (   ) is a bounded subsequence, with (   ) ∈ [, ] for some , using Lemma 11 with  = 1; there exists a constant  3 > 0 such that ‖   ‖ <  3 , and hence The proof of Theorem 6 is finished.

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  = 0; (33) can be written as the -periodic problem We will say that a set Ω ⊆  is uniformly positively bounded below if there is a constant  > 0 such that min  ≥  for every  ∈ Ω.In order to prove the main result of this paper, we need the following theorem, which has been proved in [4].
Then, there exists a  3 ≥ 1 such that, for every integer  ≥  3 , 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 [0, ], it belongs to Ω.Moreover, if   denotes the angular momentum associated with   (), then lim →∞   = 0. (111) The main result of this section reads as follows.
Theorem 17. Assume that ( 1 ) and ( 2 ) are satisfied.Then there exists  2 ≥ 1 such that, for every integer  ≥  (112) Proof.We consider the -periodic problem (107).Using the same technique in Section 3, we can prove that there exists a constant C > 0 such that if  is a -periodic solution of (107), then (115) Thus, by Lemma 16, the proof of Theorem 17 is thus completed.

Theorem 3 .
Let the following assumptions hold.