Abstract
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.
MSC: 34C25.
Keywords:
periodic solution; singular systems; topological degree1 Introduction
In this work, we are concerned with the existence of positive periodic solutions for the following radically symmetric system:
where
Roughly speaking, system (1.1) is singular at 0 means that
Such a type of singular systems appears in many problems of applications. Such as,
if we take
which describes the motion of a particle subjected to the gravitational attraction
of a sun that lies at the origin. If we take
The question about the existence of noncollision periodic orbits for scalar equations and dynamical systems with singularities has attracted much attention of many researchers over many years [110]. There are two main lines of research in this area. The first one is the variational approach [1113]. Usually, the proof requires some strong force condition, which was first introduced with this name by Gordon in [14], although the idea goes back at least to Poincaré [15]. Gordon’s result, later improved by Capozzi, Greco and Salvatore [16], is stated as follows.
Theorem 1.1Let
(
(
for everytand
(
for every
has a periodic solution with a minimal periodkT.
The strong force conditions (
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 [17]. In particular, some classical tools have been used to study singular differential equations and dynamical systems in the literature, including the degree theory [1823], the method of upper and lower solutions [24,25], Schauder’s fixed point theorem [2628], some fixed point theorems in cones for completely continuous operators [2932] and a nonlinear LeraySchauder alternative principle [3336]. 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 [17] 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 largeamplitude 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.
2 Preliminaries
In this section, we present some results which will be applied in Section 3. We may write the solutions of (1.1) in polar coordinates as follows:
Eq. (1.1) is then equivalent to the system
where μ is the (scalar) angular momentum of
If x is a Tradially periodic, then r must be Tperiodic. We will prove the existence of a Tperiodic solution r of the first equation in (2.2). We thus consider the boundary value problem
Let
Let X be a Banach space of functions such that
Define the following two operators:
and
Taking
We will say that a set
Theorem 2.1Let Ω be an open bounded subset ofX, uniformly positively bounded below. Assume that there is no solution of (2.4) on the boundary∂Ω, and that
Then, there exists a
3 Main results
First we introduce some known results on eigenvalues. Let
with the periodic boundary condition (PC):
The following are the standard results for eigenvalues. See, e.g., reference [37].
(
where
(
for any
(
where
Now we present our main result.
Theorem 3.1Let the following assumptions hold.
(
for alltand all
and
(
uniformly int. Moreover,
Then Eq. (2.4) has aTperiodic solution, and there exists a
and if
In order to apply Theorem 2.1, we consider the Tperiodic problem (2.4).
Lemma 3.2Suppose that
Now we begin by showing that Lemma 3.2 holds, and use topological degree theory. To this end, we deform (2.4) to a simpler singular autonomous equation
where a for some positive constant satisfies
where
Note that
This follows from the convexity of the first eigenvalues with respect to potentials.
Lemma 3.3Given
Proof Put
For (3.5), applying
for all
Hence (3.5) holds. □
Applying Lemma 3.3 to
Thus
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
Lemma 3.4Assume that
Proof By the results for eigenvalues in (
for all
Then, by the theory of linear secondorder differential operators [38], the eigenvalues of
and
This completes the proof. □
Lemma 3.5Under the assumptions as in Theorem 3.1, there exist
for some
Proof Let
Integrating (2.4) from 0 to T, we get
Thus
Take some constant
for all t and
Let
Moreover, write r as
Integrating (3.8) from 0 to T, we have
Multiplying (3.8) by
where the fact
Note that
So,
On the other hand,
This is a contradiction.
Now it follows from (3.9) that
Lemma 3.6There exist
Proof From (
for all t and
Multiplying (2.4) by r and then integrating over
Note from Lemma 3.5 that there exists
The other terms in (3.11) by the Hölder inequality can be estimated as follows:
Thus (3.11) reads as
where
On the other hand, using Lemma 3.4,
we get from (3.12) that
Consequently,
Thus
As
where
Next, the positive lower estimates for
Lemma 3.7There exists a constant
Proof From (
for all t and all
By Lemma 3.5,
As
Now multiplying (2.4) by
for some
if
Now we give the proof of Lemma 3.2. Consider the homotopy equation (3.4), we can get a priori estimates as in Lemmas 3.5, 3.6, 3.7. That is, any positive Tperiodic solution of (3.4) satisfies
for some positive constants
By the homotopy invariance of degree and the result of Capietto, Mawhin and Zanolin [39],
Thus (3.4), with
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors read and approved the final manuscript.
Acknowledgements
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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