Periodic solutions of radially symmetric systems with a singularity
© Li et al.; licensee Springer. 2013
Received: 30 November 2012
Accepted: 13 April 2013
Published: 29 April 2013
In this paper, we study the existence of infinitely many periodic solutions to planar radially symmetric systems with certain strong repulsive singularities near the origin and with some semilinear growth near infinity. The proof of the main result relies on topological degree theory. Recent results in the literature are generalized and complemented.
where is T-periodic in the time variable t for some and satisfies the -Carathéodory condition. Setting , may be singular at , we therefore look for non-collision solutions, i.e., solutions which never attain the singularity.
Roughly speaking, system (1.1) is singular at 0 means that becomes unbounded when . We say that (1.1) is of repulsive type (attractive type) if (respectively ) when .
which describes the motion of a particle subjected to the gravitational attraction of a sun that lies at the origin. If we take (), (1.1) may be used to model Rutherford’s scattering of α particles by heavy atomic nuclei.
The question about the existence of non-collision periodic orbits for scalar equations and dynamical systems with singularities has attracted much attention of many researchers over many years [1–10]. There are two main lines of research in this area. The first one is the variational approach [11–13]. Usually, the proof requires some strong force condition, which was first introduced with this name by Gordon in , although the idea goes back at least to Poincaré . Gordon’s result, later improved by Capozzi, Greco and Salvatore , is stated as follows.
Theorem 1.1 Let and the following assumptions hold.
for every t and .
has a periodic solution with a minimal period kT.
the strong force condition corresponds to the case .
Besides the variational approach, topological methods have been widely applied, starting with the pioneering paper of Lazer and Solimini . In particular, some classical tools have been used to study singular differential equations and dynamical systems in the literature, including the degree theory [18–23], the method of upper and lower solutions [24, 25], Schauder’s fixed point theorem [26–28], some fixed point theorems in cones for completely continuous operators [29–32] and a nonlinear Leray-Schauder alternative principle [33–36]. Contrasting with the variational setting, the strong force condition plays here a different role linked to repulsive singularities. A counterexample in the paper of Lazer and Solimini  shows that a strong force assumption (unboundedness of the potential near the singularity) is necessary in some sense for the existence of positive periodic solutions in the scalar case.
However, compared with the case of strong singularities, the study of the existence of periodic solutions under the presence of weak singularities by topological methods is more recent and the number of references is much smaller. Several existence results can be found in [7, 26, 28].
As mentioned above, this paper is mainly motivated by the recent papers [19, 20]. The aim of this paper is to show that the topological degree theorem can be applied to the periodic problem. We prove the existence of large-amplitude periodic solutions whose minimal period is an integer multiple of T.
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 will state and prove the main results.
where μ is the (scalar) angular momentum of . Recall that μ is constant in time along any solution. In the following, when considering a solution of (2.2), we will always implicitly assume that and .
Let X be a Banach space of functions such that with continuous immersions, and set .
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 .
3 Main results
with the periodic boundary condition (PC): , , or with the antiperiodic boundary condition (): , . We use to denote all the eigenvalues of (3.1) with the Dirichlet boundary condition (DC): .
The following are the standard results for eigenvalues. See, e.g., reference .
where (as ), such that λ is an eigenvalue of (3.1)-(PC) if and only if or with n is even; and λ is an eigenvalue of (3.1)-() if and only if or with n is odd.
for any .
where denotes the translation of : .
Now we present our main result.
Theorem 3.1 Let the following assumptions hold.
In order to apply Theorem 2.1, we consider the T-periodic problem (2.4).
Lemma 3.2 Suppose that satisfies () and ϕ, Φ satisfy (). Then Eq. (2.4) has at least one positive T-periodic solution.
where . We need to find a priori estimates for the possible positive T-periodic solutions of (3.4).
Note that satisfies the conditions () uniformly with respect to . Moreover, for each , satisfies (3.2) with and . We will prove that satisfy (3.3) 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.
Hence (3.5) holds. □
Thus defined above satisfy (3.3) uniformly in .
In the obtention of a priori estimates for all possible positive solutions to (3.4)-(PC), we simply prove this for all possible positive solutions to (2.4)-(PC), because , satisfy (3.3) and also (3.2) uniformly in .
for all .
This completes the proof. □
for some .
Thus , there exist such that .
for all t and . We assert that for some . Otherwise, assume that for all t.
where the fact is used.
This is a contradiction.
Now it follows from (3.9) that and , a contradiction to the positiveness of . We have proved that for some and for some . Thus the intermediate value theorem implies that (3.6) holds. □
for all t and .
where , are positive constants.
Thus is obtained.
where , . □
Next, the positive lower estimates for are obtained from the condition ().
As , for . Therefore, the function has an inverse denoted by ξ.
if . Thus we know from (3.13) that for some constant . □
Thus (3.4), with , has at least one solution in Ω, which is a positive T-periodic solution of (2.4). By Theorem 2.1, the proof of Theorem 3.1 is thus completed.
The authors express their thanks to the referees for their valuable comments and suggestions. This work is supported by the National Natural Science Foundation of China (Grant No. 11161017), Hainan Natural Science Foundation (Grant No. 113001).
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