Open Access

Periodic solutions of radially symmetric systems with a singularity

Boundary Value Problems20132013:110

DOI: 10.1186/1687-2770-2013-110

Received: 30 November 2012

Accepted: 13 April 2013

Published: 29 April 2013

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 degree

1 Introduction

In this work, we are concerned with the existence of positive periodic solutions for the following radically symmetric system:
x ¨ + f ( t , | x | ) x | x | = 0 , x R 2 { 0 } , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_Equ1_HTML.gif
(1.1)

where f : R × ( 0 , + ) R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq1_HTML.gif is T-periodic in the time variable t for some T > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq2_HTML.gif and satisfies the L 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq3_HTML.gif-Carathéodory condition. Setting r ( t ) = | x ( t ) | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq4_HTML.gif, f ( t , r ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq5_HTML.gif may be singular at r = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq6_HTML.gif, 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 f ( t , r ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq5_HTML.gif becomes unbounded when r 0 + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq7_HTML.gif. We say that (1.1) is of repulsive type (attractive type) if f ( t , r ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq8_HTML.gif (respectively f ( t , r ) + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq9_HTML.gif) when r 0 + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq7_HTML.gif.

Such a type of singular systems appears in many problems of applications. Such as, if we take f ( t , r ) = c / r 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq10_HTML.gif ( c > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq11_HTML.gif), it is the famous Newtonian equation
x ¨ + c x | x | 3 = 0 , x R 2 { 0 } , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_Equa_HTML.gif

which describes the motion of a particle subjected to the gravitational attraction of a sun that lies at the origin. If we take f ( t , r ) = c / r 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq10_HTML.gif ( c < 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq12_HTML.gif), (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 [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 1.1 Let x ( t ) R 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq13_HTML.gif and the following assumptions hold.

( A 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq14_HTML.gif) The function V is T-periodic in t, differentiable in x 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq15_HTML.gif with continuous gradient, and such that
lim x 0 V ( t , x ) = . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_Equb_HTML.gif
( A 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq16_HTML.gif) There exist v [ 0 , 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq17_HTML.gif and positive constants c 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq18_HTML.gif, c 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq19_HTML.gif such that
V ( t , x ) c 1 | x | v + c 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_Equc_HTML.gif

for every t and x 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq15_HTML.gif.

( A 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq20_HTML.gif) There are a C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq21_HTML.gif-function U : R 2 { 0 } R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq22_HTML.gif, a neighborhood N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq23_HTML.gif of 0 and a positive constant c 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq24_HTML.gif such that
lim x 0 U ( x ) = and V ( t , x ) | U ( x ) | 2 c 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_Equd_HTML.gif
for every x N { 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq25_HTML.gif, then, for every integer k 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq26_HTML.gif, the system
x ¨ + V ( t , x ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_Eque_HTML.gif

has a periodic solution with a minimal period kT.

The strong force conditions ( A 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq16_HTML.gif), ( A 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq20_HTML.gif) guarantee that the minimization procedure does not lead to a collision orbit. This similar condition has been widely used for a voiding collisions in the singularity case. For example, if we consider the system
x ¨ = 1 | x | α + f ( t ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_Equf_HTML.gif

the strong force condition corresponds to the case α 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq27_HTML.gif.

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 Leray-Schauder 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 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.

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:
x ( t ) = r ( t ) ( cos φ ( t ) , sin φ ( t ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_Equ2_HTML.gif
(2.1)
Eq. (1.1) is then equivalent to the system
{ r ¨ + f ( t , r ) μ 2 r 3 = 0 , r 2 φ ˙ = μ , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_Equ3_HTML.gif
(2.2)

where μ is the (scalar) angular momentum of x ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq28_HTML.gif. 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 μ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq29_HTML.gif and r > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq30_HTML.gif.

If x is a T-radially periodic, then r must be T-periodic. We will prove the existence of a T-periodic solution r of the first equation in (2.2). We thus consider the boundary value problem
{ r ¨ + f ( t , r ) = μ 2 r 3 , r ( 0 ) = r ( T ) , r ˙ ( 0 ) = r ˙ ( T ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_Equ4_HTML.gif
(2.3)
Let μ = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq31_HTML.gif, (2.3) can be written as the T-periodic problem
r ¨ + f ( t , r ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_Equ5_HTML.gif
(2.4)

Let X be a Banach space of functions such that C 1 ( [ 0 , T ] ) X C ( [ 0 , T ] ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq32_HTML.gif with continuous immersions, and set X = { r X : min r > 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq33_HTML.gif.

Define the following two operators:
D ( L ) = { r W 2 , 1 ( 0 , T ) : r ( 0 ) = r ( T ) , r ˙ ( 0 ) = r ˙ ( T ) } , L : D ( L ) X L 1 ( 0 , T ) , L r = r ¨ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_Equg_HTML.gif
and
N : X L 1 ( 0 , T ) , ( N r ) ( t ) = f ( t , r ( t ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_Equh_HTML.gif
Taking σ R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq34_HTML.gif not belonging to the spectrum of L, (2.4) can be translated to the fixed problem
r = ( L σ I ) 1 ( N σ I ) r . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_Equi_HTML.gif

We will say that a set Ω X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq35_HTML.gif is uniformly positively bounded below if there is a constant δ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq36_HTML.gif such that min r δ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq37_HTML.gif for every r Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq38_HTML.gif. In order to prove the main result of this paper, we need the following theorem, which has been proved in [18].

Theorem 2.1 Let Ω be an open bounded subset of X, uniformly positively bounded below. Assume that there is no solution of (2.4) on the boundary Ω, and that
deg ( I ( L σ I ) 1 ( N σ I ) , Ω , 0 ) 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_Equj_HTML.gif
Then, there exists a k 1 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq39_HTML.gif such that, for every integer k k 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq40_HTML.gif, system (1.1) has a periodic solution x k ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq41_HTML.gif with a minimal period kT, which makes exactly one revolution around the origin in the period time kT. The function | x k ( t ) | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq42_HTML.gif is T-periodic and, when restricted to [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq43_HTML.gif, it belongs to Ω. Moreover, if μ k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq44_HTML.gif denotes the angular momentum associated to x k ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq41_HTML.gif, then
lim k μ k = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_Equk_HTML.gif

3 Main results

First we introduce some known results on eigenvalues. Let q ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq45_HTML.gif be a T-periodic potential such that q L 1 ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq46_HTML.gif. Consider the eigenvalue problems of
x + ( λ + q ( t ) ) x = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_Equ6_HTML.gif
(3.1)

with the periodic boundary condition (PC): x ( 0 ) = x ( T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq47_HTML.gif, x ( 0 ) = x ( T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq48_HTML.gif, or with the antiperiodic boundary condition ( A P C https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq49_HTML.gif): x ( 0 ) = x ( T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq50_HTML.gif, x ( 0 ) = x ( T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq51_HTML.gif. We use λ 1 D ( q ) < λ 2 D ( q ) < < λ n D ( q ) < https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq52_HTML.gif to denote all the eigenvalues of (3.1) with the Dirichlet boundary condition (DC): x ( 0 ) = x ( T ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq53_HTML.gif.

The following are the standard results for eigenvalues. See, e.g., reference [37].

( E 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq54_HTML.gif) With respect to the periodic and anti-periodic eigenvalues, there exist sequences
< λ ¯ 0 ( q ) < λ ̲ 1 ( q ) λ ¯ 1 ( q ) < λ ̲ 2 ( q ) λ ¯ 2 ( q ) < < λ ̲ n ( q ) λ ¯ n ( q ) < , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_Equl_HTML.gif

where λ ̲ n ( q ) , λ ¯ n ( q ) + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq55_HTML.gif (as n + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq56_HTML.gif), such that λ is an eigenvalue of (3.1)-(PC) if and only if λ = λ ̲ n ( q ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq57_HTML.gif or λ ¯ n ( q ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq58_HTML.gif with n is even; and λ is an eigenvalue of (3.1)-( A P C https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq49_HTML.gif) if and only if λ = λ ̲ n ( q ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq59_HTML.gif or λ ¯ n ( q ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq58_HTML.gif with n is odd.

( E 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq60_HTML.gif) The comparison results hold for all of these eigenvalues. If q 1 q 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq61_HTML.gif, then
λ ̲ n ( q 1 ) λ ̲ n ( q 2 ) , λ ¯ n ( q 1 ) λ ¯ n ( q 2 ) , λ n D ( q 1 ) λ n D ( q 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_Equm_HTML.gif

for any n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq62_HTML.gif.

( E 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq63_HTML.gif) The eigenvalues λ ̲ n ( q ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq64_HTML.gif and λ ¯ n ( q ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq58_HTML.gif can be recovered from the Dirichlet eigenvalues in the following way. For any n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq62_HTML.gif,
λ ̲ n ( q ) = min { λ n D ( q t 0 ) : t 0 R } , λ ¯ n ( q ) = max { λ n D ( q t 0 ) : t 0 R } , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_Equn_HTML.gif

where q t 0 ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq65_HTML.gif denotes the translation of q ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq45_HTML.gif: q t 0 ( t ) q ( t + t 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq66_HTML.gif.

Now we present our main result.

Theorem 3.1 Let the following assumptions hold.

( H 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq67_HTML.gif) There exist a constant R 0 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq68_HTML.gif and a function f 0 C ( ( 0 , ) , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq69_HTML.gif such that
f ( t , r ) f 0 ( r ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_Equo_HTML.gif
for all t and all 0 < r R 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq70_HTML.gif, where f 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq71_HTML.gif satisfies
lim r 0 + f 0 ( r ) = + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_Equp_HTML.gif
and
lim r 0 + 1 r f 0 ( r ) d r = . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_Equq_HTML.gif
( H 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq72_HTML.gif) There exist positive T-periodic continuous functions ϕ, Φ such that
ϕ ( t ) lim inf r + f ( t , r ) r lim sup r + f ( t , r ) r Φ ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_Equ7_HTML.gif
(3.2)
uniformly in t. Moreover,
λ ̲ 1 ( Φ ) > 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_Equ8_HTML.gif
(3.3)
Then Eq. (2.4) has a T-periodic solution, and there exists a k 1 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq39_HTML.gif such that, for every integer k k 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq40_HTML.gif, Eq. (1.1) has a periodic solution with a minimal period kT, which makes exactly one revolution around the origin in the period time kT. Moreover, there exists a constant C > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq73_HTML.gif (independent of μ and k) such that
1 C < | x k ( t ) | < C for every t R and every k k 1 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_Equr_HTML.gif
and if μ k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq44_HTML.gif denotes the angular momentum associated to x k ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq41_HTML.gif, then
lim k μ k = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_Equs_HTML.gif

In order to apply Theorem 2.1, we consider the T-periodic problem (2.4).

Lemma 3.2 Suppose that f ( t , r ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq5_HTML.gif satisfies ( H 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq74_HTML.gif) and ϕ, Φ satisfy ( H 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq72_HTML.gif). Then Eq. (2.4) has at least one positive T-periodic solution.

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
r + a r = 1 r , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_Equt_HTML.gif
where a for some positive constant satisfies 0 < a < ( π / T ) 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq75_HTML.gif for all t. Consider the following homotopy equation:
r + f ( t , r ; τ ) = 0 , τ [ 0 , 1 ] , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_Equ9_HTML.gif
(3.4)

where f ( t , r ; τ ) = τ f ( t , r ) + ( 1 τ ) ( a r 1 r ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq76_HTML.gif. We need to find a priori estimates for the possible positive T-periodic solutions of (3.4).

Note that f ( t , r ; τ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq77_HTML.gif satisfies the conditions ( H 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq67_HTML.gif) uniformly with respect to τ [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq78_HTML.gif. Moreover, for each τ [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq79_HTML.gif, f ( t , r ; τ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq77_HTML.gif satisfies (3.2) with ϕ = ϕ τ = τ ϕ ( t ) + ( 1 τ ) a https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq80_HTML.gif and Φ = Φ τ = τ Φ ( t ) + ( 1 τ ) a https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq81_HTML.gif. We will prove that Φ τ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq82_HTML.gif satisfy (3.3) uniformly in τ [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq78_HTML.gif. The usual L p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq83_HTML.gif-norm is denoted by p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq84_HTML.gif, and the supremum norm of C [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq85_HTML.gif is denoted by https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq86_HTML.gif.

This follows from the convexity of the first eigenvalues with respect to potentials.

Lemma 3.3 Given q 0 , q 1 L 1 ( 0 , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq87_HTML.gif. Then, for all τ [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq78_HTML.gif,
λ ̲ 1 ( τ q 1 + ( 1 τ ) q 0 ) τ λ ̲ 1 ( q 1 ) + ( 1 τ ) λ ̲ 1 ( q 0 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_Equ10_HTML.gif
(3.5)
Proof Put q τ = τ q 1 + ( 1 τ ) q 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq88_HTML.gif, τ [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq78_HTML.gif. Then
λ 1 D ( q τ ) = inf φ H 0 1 ( 0 , T ) φ 2 = 1 0 T ( φ 2 ( t ) q τ ( t ) φ 2 ( t ) ) d t = inf φ H 0 1 ( 0 , T ) φ 2 = 1 ( τ 0 T ( φ 2 ( t ) q 1 ( t ) φ 2 ( t ) ) d t + ( 1 τ ) 0 T ( φ 2 ( t ) q 0 ( t ) φ 2 ( t ) ) d t ) τ inf φ H 0 1 ( 0 , T ) φ 2 = 1 0 T ( φ 2 ( t ) q 1 ( t ) φ 2 ( t ) ) d t + ( 1 τ ) inf φ H 0 1 ( 0 , T ) φ 2 = 1 0 T ( φ 2 ( t ) q 0 ( t ) φ 2 ( t ) ) d t = τ λ 1 D ( q 1 ) + ( 1 τ ) λ 1 D ( q 0 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_Equu_HTML.gif
For (3.5), applying λ 1 D ( q τ ) τ λ 1 D ( q 1 ) + ( 1 τ ) λ 1 D ( q 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq89_HTML.gif to q i = q i , t 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq90_HTML.gif, where t 0 [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq91_HTML.gif, we have
λ 1 D ( q τ , t 0 ) τ λ 1 D ( q 1 , t 0 ) + ( 1 τ ) λ 1 D ( q 0 , t 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_Equv_HTML.gif
for all t 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq92_HTML.gif. Thus
λ ̲ 1 ( q τ ) = min t 0 λ 1 D ( q τ , t 0 ) min t 0 ( τ λ 1 D ( q 1 , t 0 ) + ( 1 τ ) λ 1 D ( q 0 , t 0 ) ) min t 0 τ λ 1 D ( q 1 , t 0 ) + ( 1 τ ) min t 0 λ 1 D ( q 0 , t 0 ) = τ λ ̲ 1 ( q 1 ) + ( 1 τ ) λ ̲ 1 ( q 0 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_Equw_HTML.gif

Hence (3.5) holds. □

Applying Lemma 3.3 to q 1 = Φ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq93_HTML.gif and q 0 = a https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq94_HTML.gif, we have
λ ̲ 1 ( Φ τ ) τ λ ̲ 1 ( Φ ) + ( 1 τ ) λ ̲ 1 ( a ) min ( λ ̲ 1 ( Φ ) , λ ̲ 1 ( a ) ) > 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_Equx_HTML.gif

Thus Φ τ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq82_HTML.gif defined above satisfy (3.3) uniformly in τ [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq79_HTML.gif.

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 ϕ τ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq95_HTML.gif, Φ τ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq96_HTML.gif satisfy (3.3) and also (3.2) uniformly in τ [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq78_HTML.gif.

Lemma 3.4 Assume that λ ̲ 1 ( Φ ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq97_HTML.gif of the equation y + ( λ + Φ ( t ) ) y = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq98_HTML.gif, then
y 2 2 0 T Φ ( t + t 0 ) y 2 ( t ) d t + λ 1 D ( Φ t 0 ) 0 T y 2 ( t ) d t . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_Equy_HTML.gif
Proof By the results for eigenvalues in ( E 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq63_HTML.gif), we have
λ 1 D ( Φ t 0 ) λ ̲ 1 ( Φ ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_Equz_HTML.gif

for all t 0 R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq99_HTML.gif.

Then, by the theory of linear second-order differential operators [38], the eigenvalues of y + ( λ + Φ ( t + t 0 ) ) y = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq100_HTML.gif with Dirichlet boundary conditions form a sequence λ 1 D ( Φ t 0 ) < λ 2 D ( Φ t 0 ) < https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq101_HTML.gif which tends to +∞, and the corresponding eigenfunctions ψ 1 , ψ 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq102_HTML.gif are an orthonormal base of L 2 ( 0 , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq103_HTML.gif. Hence, given c i R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq104_HTML.gif and y H 0 1 ( 0 , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq105_HTML.gif, we can write
y ( t ) = i 1 c i ψ i ( t ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_Equaa_HTML.gif
and
0 T ( ( y ( t ) ) 2 Φ ( t + t 0 ) y 2 ( t ) ) d t = i 1 c i 2 0 T ( ( ψ i ( t ) ) 2 Φ ( t + t 0 ) ψ i 2 ( t ) ) d t = i 1 c i 2 λ i D ( Φ t 0 ) 0 T ψ i 2 ( t ) d t λ 1 D ( Φ t 0 ) 0 T y 2 ( t ) d t . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_Equab_HTML.gif

This completes the proof. □

Lemma 3.5 Under the assumptions as in Theorem  3.1, there exist C 2 > C 1 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq106_HTML.gif such that any positive T-periodic solution r ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq107_HTML.gif of (2.4)-(PC) satisfies
C 1 < r ( t 0 ) < C 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_Equ11_HTML.gif
(3.6)

for some t 0 [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq91_HTML.gif.

Proof Let r ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq107_HTML.gif be a positive T-periodic solution of (2.4)-(PC). By ( H 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq67_HTML.gif), there is C 1 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq108_HTML.gif such that
f ( t , s ) < 0 for all  0 < s < C 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_Equac_HTML.gif
Integrating (2.4) from 0 to T, we get
0 T r ( t ) d t + 0 T f ( t , r ( t ) ) d t = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_Equad_HTML.gif

Thus 0 T f ( t , r ( t ) ) d t = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq109_HTML.gif, there exist t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq110_HTML.gif such that r ( t ) > C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq111_HTML.gif.

Take some constant ε 0 ( 0 , min { ϕ ¯ , λ ̲ 1 ( Φ ) } ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq112_HTML.gif, where ϕ ¯ = 1 T 0 T ϕ ( t ) d t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq113_HTML.gif is the average of ϕ ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq114_HTML.gif. From ( H 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq72_HTML.gif) there is C 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq115_HTML.gif ( > C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq116_HTML.gif) large enough such that
ϕ ( t ) ε 0 f ( t , s ) s Φ ( t ) + ε 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_Equ12_HTML.gif
(3.7)

for all t and s C 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq117_HTML.gif. We assert that r ( t ) < C 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq118_HTML.gif for some t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq119_HTML.gif. Otherwise, assume that r ( t ) C 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq120_HTML.gif for all t.

Let
p ( t ) = f ( t , r ( t ) ) r ( t ) ( ϕ ( t ) ε 0 , Φ ( t ) + ε 0 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_Equae_HTML.gif
Moreover, write r as r = r ˜ + r ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq121_HTML.gif, then r ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq122_HTML.gif satisfies the following differential equation:
r ˜ + p ( t ) r ˜ + p ( t ) r ¯ = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_Equ13_HTML.gif
(3.8)
Integrating (3.8) from 0 to T, we have
0 T p ( t ) r ˜ ( t ) d t = r ¯ 0 T p ( t ) d t . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_Equ14_HTML.gif
(3.9)
Multiplying (3.8) by r ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq122_HTML.gif and integrating, we get
r ˜ 2 2 = 0 T p ( t ) r ˜ 2 ( t ) d t + r ¯ 0 T p ( t ) r ˜ ( t ) d t = 0 T p ( t ) r ˜ 2 ( t ) d t r ¯ 2 ( t ) 0 T p ( t ) d t 0 T p ( t ) r ˜ 2 ( t ) d t , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_Equ15_HTML.gif
(3.10)

where the fact 1 T 0 T p ( t ) d t > ϕ ¯ ε 0 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq123_HTML.gif is used.

Note that r ˜ ( t 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq124_HTML.gif for some t 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq92_HTML.gif, r ˜ ( t 0 + T ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq125_HTML.gif, so r ˜ ( t ) H 0 1 ( t 0 , t 0 + T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq126_HTML.gif. We assert that r ˜ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq127_HTML.gif. On the contrary, assume that r ˜ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq128_HTML.gif. Now, by (3.10), the first Dirichlet eigenvalue
λ 1 D ( p | [ t 0 , t 0 + T ] ) = inf φ H 0 1 ( t 0 , t 0 + T ) φ 0 t 0 t 0 + T ( φ 2 ( t ) p ( t ) φ 2 ( t ) ) d t t 0 t 0 + T φ 2 ( t ) d t 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_Equaf_HTML.gif
So,
λ ̲ 1 ( p ) = min { λ 1 D ( p ) } 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_Equag_HTML.gif
On the other hand, p ( t ) < Φ ( t ) + ε 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq129_HTML.gif,
λ ̲ 1 ( p ) λ ̲ 1 ( Φ + ε 0 ) = λ ̲ 1 ( Φ ) ε 0 > 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_Equah_HTML.gif

This is a contradiction.

Now it follows from (3.9) that r ¯ = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq130_HTML.gif and r 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq131_HTML.gif, a contradiction to the positiveness of r ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq107_HTML.gif. We have proved that r ( t ) > C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq111_HTML.gif for some t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq110_HTML.gif and r ( t ) < C 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq118_HTML.gif for some t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq132_HTML.gif. Thus the intermediate value theorem implies that (3.6) holds. □

Lemma 3.6 There exist C 3 > C 2 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq133_HTML.gif, C 4 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq134_HTML.gif such that any positive T-periodic solution r ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq107_HTML.gif of (2.4)-(PC) satisfies
r < C 3 , r < C 4 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_Equai_HTML.gif
Proof From ( H 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq72_HTML.gif) and (3.7), we know that there is h 0 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq135_HTML.gif such that
f ( t , s ) ( Φ ( t ) + ε 0 ) s + h 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_Equaj_HTML.gif

for all t and s > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq136_HTML.gif.

Multiplying (2.4) by r and then integrating over [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq43_HTML.gif, we get
r 2 2 = 0 T f ( t , r ( t ) ) r ( t ) d t 0 T ( ( Φ ( t ) + ε 0 ) r ( t ) + h 0 ) r ( t ) d t = 0 T Φ ( t ) r 2 ( t ) d t + ε 0 r 2 2 + h 0 r 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_Equ16_HTML.gif
(3.11)
Note from Lemma 3.5 that there exists t 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq92_HTML.gif satisfying C 1 < r ( t 0 ) < C 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq137_HTML.gif. Let u ( t ) = r ( t + t 0 ) r ( t 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq138_HTML.gif, then u H 0 1 ( 0 , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq139_HTML.gif. Thus
0 T Φ ( t ) r 2 ( t ) d t = 0 T Φ ( t + t 0 ) r 2 ( t + t 0 ) d t = 0 T Φ ( t + t 0 ) ( r 2 ( t 0 ) + 2 r ( t 0 ) u ( t ) + u 2 ( t ) ) d t C 2 2 Φ 1 + 2 C 2 Φ 2 u 2 + 0 T Φ ( t + t 0 ) u 2 ( t ) d t . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_Equak_HTML.gif
The other terms in (3.11) by the Hölder inequality can be estimated as follows:
ε 0 r 2 2 ε 0 ( T C 2 2 + 2 C 2 T 1 2 u 2 + u 2 2 ) , h 0 r 1 h 0 ( T C 2 + T 1 2 u 2 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_Equal_HTML.gif
Thus (3.11) reads as
u 2 2 A 0 + B 0 u 2 + ε 0 u 2 2 + 0 T Φ ( t + t 0 ) u 2 ( t ) d t , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_Equ17_HTML.gif
(3.12)

where A 0 = ε 0 T C 2 2 + h 0 T C 2 + C 2 2 Φ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq140_HTML.gif, B 0 = 2 ε 0 C 2 T 1 2 + h 0 T 1 2 + 2 C 2 Φ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq141_HTML.gif are positive constants.

On the other hand, using Lemma 3.4,
λ ̲ 1 ( Φ ( t ) ) u 2 2 λ 1 D ( Φ t 0 ) u 2 2 0 T ( u 2 ( t ) Φ ( t + t 0 ) u 2 ( t ) ) d t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_Equam_HTML.gif
we get from (3.12) that
( ε 0 λ ̲ 1 ( Φ ( t ) ) ) u 2 2 + B 0 u 2 + A 0 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_Equan_HTML.gif
Consequently, u 2 < A 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq142_HTML.gif for some A 1 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq143_HTML.gif. By (3.12), one has r 2 = u 2 < A 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq144_HTML.gif for some A 2 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq145_HTML.gif. From these, for any t [ t 0 , t 0 + T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq146_HTML.gif,
| r ( t ) | | r ( t 0 ) | + | t 0 t r ( t ) d t | C 2 + T 1 2 r 2 C 2 + T 1 2 A 2 : = C 3 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_Equao_HTML.gif

Thus r < C 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq147_HTML.gif is obtained.

As 0 T f ( t , r ( t ) ) d t = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq109_HTML.gif, thus f ( t , r ( t ) ) 1 = 2 f + ( t , r ( t ) ) 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq148_HTML.gif. Since r ( 0 ) = r ( T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq149_HTML.gif, there exists t 1 [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq150_HTML.gif such that r ( t 1 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq151_HTML.gif. Therefore
r = max 0 t T | r ( t ) | = max 0 t T | t 1 t r ( s ) d s | 0 T | f ( s , r ( s ) ) | d s = 2 0 T | f + ( s , r ( s ) ) | d s 2 0 T | ( Φ + ( s ) + ε 0 ) r ( s ) + h 0 | d s 2 ( ( Φ + 1 + T ε 0 ) C 3 + h 0 T ) : = C 4 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_Equap_HTML.gif

where Φ + ( t ) = max { Φ ( t ) , 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq152_HTML.gif, f + ( t , r ( t ) ) = max { f ( t , r ( t ) ) , 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq153_HTML.gif. □

Next, the positive lower estimates for m = min t [ 0 , T ] r ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq154_HTML.gif are obtained from the condition ( H 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq67_HTML.gif).

Lemma 3.7 There exists a constant C 5 ( 0 , C 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq155_HTML.gif such that any positive solution r ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq107_HTML.gif of (2.4)-(PC) satisfies
r ( t ) > C 5 for all t . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_Equaq_HTML.gif
Proof From ( H 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq67_HTML.gif), we fix some B 1 ( 0 , C 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq156_HTML.gif such that
f ( t , s ) < C 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_Equar_HTML.gif
for all t and all 0 < s B 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq157_HTML.gif. Assume now that
m = min t [ 0 , T ] r ( t ) = r ( t 2 ) < B 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_Equas_HTML.gif
By Lemma 3.5, max t r ( t ) > C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq158_HTML.gif. Let t 3 > t 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq159_HTML.gif be the first time instant such that r ( t ) = B 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq160_HTML.gif. Then, for any t [ t 2 , t 3 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq161_HTML.gif, we have r ( t ) B 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq162_HTML.gif. Hence, for t [ t 2 , t 3 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq161_HTML.gif,
r ( t ) = f ( t , r ( t ) ) > C 4 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_Equat_HTML.gif

As r ( t 2 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq163_HTML.gif, r ( t ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq164_HTML.gif for t ( t 2 , t 3 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq165_HTML.gif. Therefore, the function r : [ t 2 , t 3 ] R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq166_HTML.gif has an inverse denoted by ξ.

Now multiplying (2.4) by r ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq167_HTML.gif and integrating over [ t 2 , t 3 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq168_HTML.gif, we get
m B 1 f ( ξ ( r ) , r ) d r = t 2 t 3 f ( t , r ( t ) ) r ( t ) d t = t 2 t 3 r ( t ) r ( t ) d t = 1 2 ( r ( t 3 ) ) 2 B 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_Equau_HTML.gif
for some B 2 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq169_HTML.gif, where the results from Lemma 3.6 are used. By ( H 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq67_HTML.gif),
m B 1 f ( ξ ( r ) , r ) d r m B 1 f 0 ( r ) d r + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_Equ18_HTML.gif
(3.13)

if m 0 + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq170_HTML.gif. Thus we know from (3.13) that m > C 5 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq171_HTML.gif for some constant C 5 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq172_HTML.gif. □

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 T-periodic solution of (3.4) satisfies
C 5 < r ( t ) < C 3 , r < C 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_Equav_HTML.gif
for some positive constants C 5 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq173_HTML.gif, C 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq174_HTML.gif, C 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq175_HTML.gif. Define C = max { 1 / C 5 , C 3 , C 4 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq176_HTML.gif and let the open bounded in X be
Ω = { r X : 1 C < r ( t ) < C  and  | r ( t ) | < C  for all  t [ 0 , T ] } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_Equaw_HTML.gif
By the homotopy invariance of degree and the result of Capietto, Mawhin and Zanolin [39],
deg ( I ( L σ I ) 1 ( N σ I ) , Ω , 0 ) = deg ( a r 1 / r , Ω R , 0 ) = 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_Equax_HTML.gif

Thus (3.4), with τ = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-110/MediaObjects/13661_2012_Article_361_IEq177_HTML.gif, 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.

Declarations

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).

Authors’ Affiliations

(1)
Department of Mathematics, Hainan University
(2)
College of Science, Hohai University
(3)
Nanjing College of Information Technology

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