\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2014 (2014), No. 128, pp. 1--11.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2014 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2014/128\hfil Periodic orbits with small angular momentum]
{Periodic orbits with small angular momentum for singular radially symmetric systems}

\author[F.-F. Liao \hfil EJDE-2014/128\hfilneg]
{Fang-Fang Liao}  % in alphabetical order

\address{Fang-Fang Liao \newline
Department of Mathematics, College of Science,
Hohai University, Nanjing 210098, China}
\email{liaofangfang8178@sina.com, hohai\_deds@126.com}

\thanks{Submitted February 7, 2014. Published June 6, 2014.}
\subjclass[2000]{34C25}
\keywords{Periodic solutions; singular radially symmetric systems;
\hfill\break\indent topological degree theory}

\begin{abstract}
 We study radially symmetric systems with a repulsive singularity,
 and show the existence of periodic orbits with small angular momentum.
 Our proofs are based on the use of topological degree theory.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks

\section{Introduction}

During the previous years, Fonda and his coworkers have studied the periodic,
subharmonic and quasi-periodic orbits for the radially symmetric system
\begin{equation} \label{skl}
\ddot{x}+f(t,|x|)\frac{x}{|x|}=0,\quad x\in\mathbb{R}^2\setminus\{0\},
\end{equation}
in a systematic way, where $f$ is a real function, $T$-periodic in $t$ and is
defined on $\mathbb{R}\times(0,+\infty)$, so that 0 may be a singularity. See
\cite{ft08, ft-na, ft-ans, ft-pams, ftz, fu}.

Here we recall one result proved by Fonda and Toader for the  system
\begin{equation} \label{simple}
\ddot{x}+c(t)\frac{x}{|x|^{\omega+1}}=e(t)\frac{x}{|x|},\quad
x\in \mathbb{R}^{2}\setminus\{0\},
\end{equation}
where $\omega>0$ and $c,e\in L_{\rm loc}^{1}(\mathbb{R})$ are $T$-periodic.

\begin{corollary}\cite[Corollary 1.1]{ft-ans}\label{pams-l}
Assume that $\omega\geq 1$, $\bar{e}=\frac{1}{T}\int_0^Te(t)dt<0$ and
\[
c_1\leq c(t)\leq c_2<0,\quad \text{for a.e. }t\in\mathbb{R},
\]
for some negative constants $c_1$ and $c_2$.
Then there exists $k_1\geq 1$ such that for
every integer $k\geq k_1$, equation \eqref{simple} has
a periodic solution $x_{k}(t)$ with minimal period $kT$, which makes exactly one
revolution around the origin in the period time $kT$. Moreover, there exists
a constant $C>0$ such that, for every $k\geq k_1$,
\begin{gather*}
\frac{1}{C}<|x_k(t)|<C,\quad \text{for every } t\in \mathbb{R},
\lim_{k\to \infty}\mu_k=0,
\end{gather*}
where $\mu_k$ denotes the angular momentum associated to $x_k$.
\end{corollary}

On the other hand, Chu, Li and Siegmund in the recent paper \cite{cls}
studied the system
\begin{equation} \label{cxx}
\ddot{x}+l^2x=f(t,|x|)\frac{x}{|x|} , \quad x\in \mathbb{R}^{2}\setminus\{0\},
\end{equation}
where $0<l<\pi/T$ and the function $f:\mathbb{R}\times (0,+\infty)\to \mathbb{R}$
is continuous, $T$-periodic in the time variable $t$. Based on topological
degree theory, they proved the following result.

\begin{theorem}\cite[Main result]{cls}
Assume that the following condition is satisfied
\begin{itemize}
 \item[(H)] There exist continuous $T$-periodic
functions $b,\hat{b}\geq 0$ with positive mean and a constant
$\alpha>0$ such that
\[
\frac{b(t)}{r^{\alpha}}\leq f(t,r)\leq \frac{\hat{b}(t)}{r^{\alpha}},
\quad \text{for all } t\in\mathbb{R} \text{ and } r>0.
\]
\end{itemize}
Then system \eqref{cxx} has two distinct families of periodic orbits
with the following distinct behavior: one rotates around the origin with
small angular momentum, and the other one rotates around the origin with
large angular momentum and large amplitude.
\end{theorem}

Motivated by those works mentioned above, in this work, we
study the system
\begin{equation} \label{xfk}
\ddot{x}+l^2x=(f(t,|x|)+e(t))\frac{x}{|x|} , \quad
x\in \mathbb{R}^{2}\setminus\{0\},
\end{equation}
where $0<l<\pi/T$ is a constant, $e\in C(\mathbb{R}/T\mathbb{Z},\mathbb{R})$ and the function
 $f:\mathbb{R}\times (0,+\infty)\to \mathbb{R}$ is continuous, $T$-periodic in
the time variable $t$ with period $T>0$. Therefore, $f(t,r)$ may be singular
at $r=0$. We look for solutions $x(t)\in\mathbb{R}^2$ which never attain the
singularity in the sense that
 \begin{equation} \label{sold}
x(t)\neq 0,\quad \text{for every }t\in\mathbb{R}.
\end{equation}
We write the solutions of \eqref{xfk} in polar coordinates
\[
x(t)=\rho(t)(\cos\varphi(t),\sin\varphi(t)).
\]
Then we have the collisionless orbits if $\rho(t)>0$ for every $t$.
Moreover, equation \eqref{xfk} is equivalent to the  system
\begin{equation}\label{equi}
\begin{gathered}
\ddot{\rho}+l^2\rho=\frac{\mu^2}{\rho^3}+f(t,\rho)+e(t),\\
\rho^2\dot{\varphi}=\mu.
\end{gathered}
\end{equation}
In our analysis, we will use such an equivalent system.

Finally we remark that the existence of periodic solutions for
singular differential equations has been studied by many researchers.
Usually, the proof is based on either
a variational approach \cite{hs, rt} or topological methods,
starting with the pioneering paper of Lazer and
Solimini \cite{ls}. From then on,  the method of upper and lower
solutions \cite{rtv}, degree theory \cite{z06}, fixed
point theorems \cite{fw, t03, t04} and a
nonlinear alternative principle of Leray-Schauder type \cite{ctz, jcz}
have been widely applied.

Throughout this paper, for a function $\psi\in C[0,T]$, we use the
symbols $\psi_*=\min_{t}\psi(t)$ and $\psi^*=\max_{t}\psi(t)$.

\section{Preliminaries and main results}

 Let us define the function $\gamma:\mathbb{R}\to\mathbb{R}$ by
\[
\gamma(t)=\int_0^TG(t,s)e(s)ds,
\]
which is the unique $T$-periodic solution of
\begin{equation} \label{ke}
\ddot{x}+l^2x=e(t),
\end{equation}
where
\begin{equation}\label{2.4}
G(t,s)=\begin{cases}
\frac{\sin l(t-s)+\sin l(T-t+s)}{2l(1-\cos lT)} ,& 0\leq s \leq t \leq T ,\\[4pt]
\frac{\sin l(s-t)+\sin l(T-s+t)}{2l(1-\cos lT)} ,& 0\leq t \leq s \leq T,\\
\end{cases}
\end{equation}
is the Green function. See \cite{fw}.
Since we  assume that $l\in (0,\pi/T)$, we know that $G(t,s)>0$
for $ 0\leq s,t\leq T$, and we denote
\[
m=\min_{0\leq s,t\leq T}G(t,s),\quad
M=\max_{0\leq s,t\leq T}G(t,s), \quad
\sigma=m/M.
\]
A direct calculation shows that
$\sigma=\cos(lT/2)\in(0,1)$.
Note that, since both $x(t)=\int_0^TG(t,s)ds$ and $x(t)\equiv\frac{1}{l^2}$ are
$T$-periodic solution of \eqref{ke} with $e(t)\equiv1$, by uniqueness, we obtain
\[
\int_0^TG(t,s)ds=\frac{1}{l^2}.
\]
The above facts have been used in \cite{fw}.
The main result of this paper reads as follows.

\begin{theorem}\label{main}
Let $l\in (0,\pi/T)$. Assume that the following conditions are satisfied:
\begin{itemize}
\item[(H1)] for each constant $L>0$, there
exists a continuous nonnegative function $\phi_{L}$ with positive mean such that
$f(t,r)\geq\phi_{L}(t)$ for all $t\in [0,T]$ and $r\in (0, L]$.

\item[(H2)] there exist continuous, non-negative functions $g(r)$
and $k(t)$ such that
\[
0\leq f(t,r)\leq k(t)g(r), \quad \text{for all }
(t,r)\in[0,T]\times(0,+\infty),
\]
where $g(r)>0$ is non-increasing.

 \item[(H3)] there exists a positive number
$R>0$ large enough and a positive constant $\alpha$ such that
\[
K^{*}g(\sigma R+\gamma_*)+\sigma^{-\alpha}\Phi_*<R,
\]
where
\[
K(t)=\int_{0}^T G(t,s)k(s)ds,\quad
\Phi(t)=\int_{0}^T G(t,s)\phi_{R+\gamma^*}(s)ds.
\]
\end{itemize}
Then for any function $e$ with $\gamma_*\geq0$, there exists $K\geq 1$
such that for every integer $k\geq K$, system \eqref{xfk} has a periodic solution
$x_k(t)$ with minimal period $kT$, which makes exactly one revolution around the
origin in the period time $kT$. Moreover, $\lim_{k\to\infty}\mu_k=0$ and
there exists a constant $C>0$ such that
\[
\frac{1}{C}<|x_k(t)|<C,\quad \text{for every $t\in\mathbb{R}$ and every $k\geq K$}.
\]
\end{theorem}

As an application of Theorem \ref{main}, we consider the system
\begin{equation} \label{exa}
\ddot{x}+l^{2}x= \frac{c(t)x}{|x|^{\omega+1}}+e(t)\frac{x}{|x|},
\quad x\in \mathbb{R}^{2}\setminus\{0\}.
\end{equation}

\begin{corollary}
Assume that $0<l<\pi/T$, $\omega>0$, and $c$ is a continuous nonnegative 
$T$-periodic function with positive mean. Then for any $T$-periodic 
function $e\in L_{\rm loc}^{1}(\mathbb{R})$ with $\gamma_*\geq0$, system \eqref{exa}
has a family of periodic orbits $\{x_k\}$ with angular momentum $\{\mu_k\}$ 
satisfying $\lim_{k\to\infty}\mu_k=0$.
\end{corollary}

\begin{proof}
Let us take 
\[
f(t,r)=\frac{c(t)}{r^{\omega}},r>0.
\]
Then (H1) and (H2) hold by taking
\[
\phi_{L}(t)=\frac{c(t)}{L^\omega},\quad k(t)=c(t),\quad 
g(r)=\frac{1}{r^{\omega}}.
\]
Let 
\[
C^*=\sup_{t\in[0,T]}\int_{0}^T G(t,s)c(s)ds,\quad
C_*=\inf_{t\in[0,T]}\int_{0}^T G(t,s)c(s)ds>0.
\]
Note that
\begin{align*}
\Phi_*&= \inf_{t\in[0,T]}\int_{0}^T G(t,s)\phi_{R+\gamma^*}(s)ds\\
&= \inf_{t\in[0,T]}\int_{0}^T G(t,s)\frac{c(s)}{(R+\gamma^*)^{\omega}}ds\\
&= \frac{C_*}{(R+\gamma^*)^{\omega}}.
\end{align*}
Then condition (H3) becomes
\[
\frac{C^*}{(\sigma R+\gamma_*)^\omega}
+\frac{C_*}{\sigma^{\alpha}(R+\gamma^*)^{\omega}}<R,
\]
for some positive constant $R$, which is obvious since $\omega>0$. 
Now the proof is finished by applying
Theorem \ref{main}. 
\end{proof}

\begin{remark}\rm
 We remark that Corollary 1.1 and Corollary 2.2 are different. In fact,
Corollary 2.2 is applicable to the case of a strong singularity as
well as the case of a weak singularity because we only need $\omega>0$, 
while we required that $\omega\geq 1$ in Corollary 1.1. 
Moreover, condition imposed on $e$ in Corollary 1.1 and that in 
Corollary 2.2 are also different.
\end{remark}

\section{Proof of Theorem \ref{main}}

We will consider the equivalent system \eqref{equi}. If $x$ is
a $T$-radially periodic solution of \eqref{xfk}, then $\rho$ must be 
a $T$-periodic solution of the first equation of \eqref{equi}. 
Thus we consider the boundary value problem
\begin{equation}\label{rhop}
\begin{gathered}
\ddot{\rho}+l^{2}\rho=f(t,\rho)+\frac{\mu^{2}}{\rho^{3}}+e(t),\\
\rho(0)=\rho(T),\quad \dot{\rho}(0)=\dot{\rho}(T).
\end{gathered} 
\end{equation}
To prove that \eqref{rhop} has a $T$-periodic solution, we first consider
\begin{equation}\label{rhopp}
\begin{gathered}
\ddot{\rho}+l^{2}\rho=f(t,\rho+\gamma)+\frac{\mu^{2}}{(\rho+\gamma)^{3}},\\
\rho(0)=\rho(T),\quad \dot{\rho}(0)=\dot{\rho}(T),
\end{gathered}
\end{equation} 
and show that \eqref{rhopp} has a $T$-periodic
solution $\rho$ satisfying $\rho(t)+\gamma(t)>0$ for $t\in[0,T]$. 
If this is true, it is easy to see that
$\tilde{\rho}(t)=\rho(t)+\gamma(t)$ will be a positive $T$-periodic solution of
\eqref{rhop}, since
\begin{align*} 
\ddot{\tilde{\rho}}+l^2\tilde{\rho}
&= \ddot{\rho}+l^2\rho+\ddot{\gamma}+l^2\gamma\\
&=  f(t,\rho+\gamma)+\frac{\mu^2}{(\rho+\gamma)^3}+e(t)\\
&=  f(t,\tilde{\rho})+\frac{\mu^2}{\tilde{\rho}^3}+e(t).
\end{align*}

We consider first \eqref{rhopp} for the case $\mu=0$, which corresponds to 
the boundary value problem
\begin{equation}\label{rhop0}
\begin{gathered}
\ddot{\rho}+l^{2}\rho=f(t,\rho+\gamma),\\
\rho(0)=\rho(T),\quad \dot{\rho}(0)=\dot{\rho}(T).\\
\end{gathered} 
\end{equation}

We will prove several preliminary results. To do this, let us define
the truncated functions $f_{n}:\mathbb{R}\times\mathbb{R}\to \mathbb{R}$ by
\[
f_{n}(t,\rho)= \begin{cases}
 f(t,\rho) ,& \text{if } \rho\geq 1/n ,\\
f(t,1/n) ,& \text{if } \rho \leq  1/n,
 \end{cases}
\]
and consider the family of boundary-value problems
\begin{equation}\label{fne}
\begin{gathered}
\ddot{\rho}+l^{2}\rho=f_n(t,\rho+\gamma)+\frac{l^2}{n},\\
\rho(0)=\rho(T),\quad \dot{\rho}(0)=\dot{\rho}(T).
\end{gathered} 
\end{equation}
To use the topological degree theory, we also consider the homotopy equations
\begin{equation}\label{hfne}
\begin{gathered}
\ddot{\rho}+l^2\rho =f^{\lambda}_{n}(t,\rho+\gamma)+\frac{l^2}{n},\\
\rho(0)=\rho(T),\quad \dot{\rho}(0)=\dot{\rho}(T),
\end{gathered}
\end{equation}
where $\lambda\in[0,1]$ and
\[
f^{\lambda}_{n}(t,\rho)=\lambda f_{n}(t,\rho)+(1-\lambda)\frac{c}{\rho^{\alpha}},
\]
 where $\alpha>0$ is fixed and the constant $c$ is chosen as
\[
c=\frac{\Phi_*R^{\alpha}}{\omega_*}.
\]

\begin{lemma}[{\cite[Lemma 3.2]{cls}}] \label{minmax}
Assume that $\rho$ is a $T$-periodic solution of \eqref{hfne}, then
\[
\min_{t\in\mathbb{R}} \rho(t) \geq \sigma \|\rho\|,
\]
 for every $\lambda \in [0,1]$ and $n\in\mathbb{N}$.
\end{lemma}

\begin{lemma}\label{KK}
There exists a constant $C>0$ such that if $\lambda\in[0,1]$ and $\rho$ 
is a $T$-periodic solution of \eqref{hfne}, then
\[
\frac{1}{C}<\rho(t)<C,\quad |\dot{\rho}(t)|<C
\]
for every $t\in[0,T]$ and $n\geq n_0$ with some $n_0\in\mathbb{N}$.
\end{lemma}

\begin{proof}
Since (H3) holds, we can choose
$n_{0}\in \{1,2,\dots \}$ such that
$\frac{1}{n_{0}}<\sigma R+\gamma_*$ and 
\[
K^{*}g(\sigma R+\gamma_*)+\sigma^{-\alpha}\Phi_*+\frac{1}{n_0}<R.
\]
Let
$N_{0}=\{n_{0},n_{0}+1,\dots\}$. Fix $n\in N_0$.

We claim that any possible solution $\rho(t)$ of \eqref{hfne} must satisfy 
$\rho(t)<R$, i.e., $\|\rho\|< R$.
Otherwise, assume that $\rho$ is a solution
of \eqref{hfne} such that $\|\rho\|\geq R$. By Lemma \ref{minmax},
\[
\rho(t)\geq \sigma\|\rho\| \geq \sigma R.
\]
On the other hand, using the fact
\[ 
\rho(t)=\int^T _0 G(t,s)f^{\lambda}_{n}(s,\rho(s)+\gamma(s))ds+\frac{1}{n},
\]
we have
$\rho(t)\geq 1/n$ for all $t\in\mathbb{R}$. Since
$\gamma_*\geq 0$, we have
\[
f^{\lambda}_{n}(t,\rho(t)+\gamma(t))=f^{\lambda}(t,\rho(t)+\gamma(t))
=\lambda f(t,\rho)+(1-\lambda)\frac{c}{\rho^{\alpha}}.
\]
Using conditions (H2) and (H3), we have
 \begin{align*}
 \rho(t)
&= \int^T _0 G(t,s)f^{\lambda}_{n}(s,\rho(s)+\gamma(s))ds+\frac{1}{n}\\
&= \int^T _0 G(t,s)\Big(\lambda f(s,\rho(s)+\gamma(s))+(1-\lambda)
 \frac{c}{\rho^{\alpha}(s)}\Big)ds+\frac{1}{n}\\
&\leq  \int^T _0 G(t,s)\Big(\lambda k(s)g(\rho(s)+\gamma(s))+(1-\lambda)
 \frac{c}{\rho^{\alpha}(s)}\Big)ds+\frac{1}{n}\\
&\leq  \int^T _0 G(t,s)\Big(k(s)g(\sigma R+\gamma_*)
 +\frac{c}{(\sigma R)^{\alpha}}\Big)ds+\frac{1}{n}\\
&\leq  g(\sigma R+\gamma_*)K^*+\frac{c\omega^*}{(\sigma R)^{\alpha}}+\frac{1}{n_0}\\
&=  K^{*}g(\sigma R+\gamma_*)+\sigma^{-\alpha}\Phi_*+\frac{1}{n_0}<R,
\end{align*}
which is a contradiction, and the claim is proved.

On the other hand, using condition (H1), we have
\begin{align*}
\rho(t)
&= \int^T _0 G(t,s)f^{\lambda}_{n}(t,\rho(s)+\gamma(s))ds+\frac{1}{n}\\
&\geq  \int^T _0 G(t,s)\Big(\lambda\phi_{R+\gamma^*}(s)
 +(1-\lambda)\frac{c}{\rho^{\alpha}(s)}\Big)ds\\
&\geq  \lambda\Phi_*+\frac{(1-\lambda)c\omega_*}{R^{\alpha}}\\
&\geq  \Phi_*=:\delta>0.
\end{align*}
Next we prove that $|\dot{\rho}(t)|\leq M$ for some constant $M>0$. 
To this end, by the boundary condition $\rho(0)=\rho(T)$, we have
$\dot{\rho}(t_{0})=0$ for some $t_{0}\in[0,T]$.
Integrating \eqref{hfne} from $0$ to $T$, we obtain
\[
l^2\int^T _0\rho(t)dt=\int^T _0 \Big(f^{\lambda}_{n}(t,\rho(t)
+\gamma(t))+\frac{l^2}{n}\Big)dt.
\]
Then
\begin{align*}
\|\dot{\rho}\|
&=  \max _{0\leq t \leq T}|\dot{\rho}(t)|= \max _{0\leq t \leq T} |\int^{t}_{t_{0}} 
 \ddot{\rho}(s)ds |\\
&= \max _{0\leq t \leq T}|\int^{t}_{t_{0}}\Big(f^{\lambda}_{n}(s,\rho(s)+\gamma(s))
 +\frac{l^2}{n}-l^2\rho(s)\Big)ds\\
 &\leq  \int^T _{0}\Big(f^{\lambda}_{n}(s,\rho(s)+\gamma(s))
 +\frac{l^2}{n}+l^2\rho(s)\Big)ds\\
 &=  2l^2\int^T _0 \rho(t)dt \\
 &< 2l^2RT=M.
\end{align*}
The proof is complete when we take $C=\max\{R,M,1/\delta\}$.
\end{proof}

Now let us define the  operators:
\begin{gather*}
L:D(L)\subset C^{1}[0,T]\to L^{1}(0,T), \\
D(L)=\{\rho\in W^{2,1}(0,T):\rho(0)=\rho(T),\dot{\rho(0)}=\dot{\rho(T)}\},\\
L\rho=\ddot{\rho}+l^2\rho, \\
N_n^\lambda:[0,\infty)\times C^{1}[0,T]\to L^{1}(0,T), \\
N_n^\lambda(\mu,\rho)(t)=f^\lambda_{n}(t,\rho(t)+\gamma(t))
 +\frac{\mu^{2}}{(\rho(t)+\gamma(t))^3}+\frac{l^2}{n}.
\end{gather*}
It is clear that the equation
\begin{equation} \label{scc}
\ddot{\rho}+l^2\rho=f^\lambda_{n}(t,\rho(t)+\gamma(t))
 +\frac{\mu^{2}}{(\rho(t)+\gamma(t))^3}+\frac{l^2}{n}
\end{equation} 
is equivalent to the operator equation
\[
L\rho=N_n^\lambda(\mu,\rho).
\]
Since $L$ is invertible,  we have
\begin{equation}\label{LN}
\rho-L^{-1}N_n^\lambda(\mu,\rho)=0.
\end{equation}
Define 
\[
\Omega=\{\rho\in C^{1}[0,T]:\frac{1}{C}<\rho(t)<C 
\text{ and $|\dot{\rho}(t)|<C$ for every }  t\in[0,T] \big\},
\]
where $C$ is the constant given by Lemma \ref{KK}. Obviously, $\Omega$ 
is an open subset of $C^1[0,T]$.
By Lemma \ref{KK}, \[\rho-L^{-1}N_n^\lambda(0,\rho)=0\] has no solutions 
$\rho$ on $\partial\Omega$.

\begin{lemma}\label{um}
There exist a constant $M>0$ such that for each $\mu\in[0,M]$,
there exists no solution of \eqref{LN} on the boundary of $\Omega$.
\end{lemma}

\begin{proof}
On the contrary, assume that there exist sequences $\{\mu_m\}$ and $\{\rho_m\}$ 
such that
\[
\mu_m\to 0\quad (m\to+\infty), \quad
\rho_m\in\partial\Omega,\quad \rho_m=L^{-1}N_n^\lambda(\mu_m,\rho_m).
\]
Since $\{\rho_m\}$ is uniformly bounded, $\{N_n^\lambda(\mu_m,\rho_m)\}$ 
is bounded in $L^1(0,T)$. Since
\[
L^{-1}:L^1(0,T)\to C^1[0,T]
\] 
is a compact operator, there exists a subsequence $\{\rho_{m_i}\}$
of $\{\rho_m\}$ such that 
\[
L^{-1}N_{n}^\lambda(\mu_{m_i},\rho_{m_i})\to\bar{\rho},\quad i\to+\infty,
\] 
for some $\bar{\rho}$.
Hence 
\[
\rho_{m_i}\to\bar{\rho},\quad i\to+\infty.
\] 
Since $\partial\Omega$ is closed, we have
$\bar{\rho}\in\partial\Omega$. Moreover,
\[
\bar{\rho}=L^{-1}N_n^\lambda(0,\bar{\rho}),
\]
which implies that $\bar{\rho}$ solves \eqref{hfne}. It is a contradiction
since \eqref{hfne} has no solution on $\partial\Omega$.
\end{proof}

\begin{lemma} 
For $\mu\in[0,M]$ and $n\in N_0$, there exists a positive $T$-periodic solution
of equation 
\begin{equation} \label{xux}
\ddot{\rho}+l^2\rho=f_n(t,\rho+\gamma)+\frac{\mu^{2}}{(\rho+\gamma)^{3}}
+\frac{l^2}{n}.
\end{equation}
\end{lemma}

\begin{proof}
We need to compute the degree for \eqref{xux}. By Lemma \ref{um}, the degree 
is the same for equations
\begin{equation} \label{xkfn}
\ddot{\rho}+l^2\rho=f^\lambda_n(t,\rho+\gamma)+\frac{\mu^{2}}{(\rho+\gamma)^{3}}
+\frac{l^2}{n},
\end{equation} 
for every $\lambda\in[0,1]$. Therefore, we consider the equation \eqref{xkfn}
with $\lambda=0$, that is the equation
\[
\ddot{\rho}+l^2\rho =\frac{c}{\rho^{\alpha}}
+\frac{\mu^{2}}{(\rho+\gamma)^{3}}+\frac{l^2}{n},
\]
which is equivalent to the system
\[
\dot{Y}=F_n(Y),
\]
with $Y=(\rho,u)$ and 
\[
F_n(Y)=\Big(u,-l^2\rho+\frac{\mu^2}{(\rho+\gamma)^{3}}
+\frac{c}{\rho^{\alpha}}+\frac{l^2}{n}\Big).
\]

It is easy to know that $F_n$ has a unique zero $(\rho_{0},u_{0})$ and the 
determinant of the Jacobian matrix satisfies
$|J_{F_n}(\rho_{0},u_{0})|>0$. By \cite[Theorem 1]{cmz}, the Leray-Schauder 
degree of $I-L^{-1}N_n^\lambda(\mu,\cdot)$ is equal to the Brouwer degree 
of $F_n$, i.e.,
\begin{align*}
d_{L}(I-L^{-1}N_n^\lambda(\mu,\cdot,),\Omega,0)
=d_{B}(F_n,(\frac{1}{C},C)\times(-C,C))=1,
\end{align*}
which implies that \eqref{xux}
has a $T$-periodic solution $\rho_n$ for any fixed $n\in\mathbb{N}$.
\end{proof}

Up to now, for each $n\in N_0$, we have proved that the system
\begin{equation} \label{news}
\begin{gathered}
\ddot{\rho}+l^2\rho =f_{n}(t,\rho+\gamma)+\frac{\mu^{2}}{(\rho+\gamma)^{3}}
+\frac{l^2}{n},\\
(\rho+\gamma)^2\dot{\varphi}=\mu,
\end{gathered}
\end{equation}
has a solution $(\mu_n,\rho_n)$.
By Lemma \ref{KK}, $\{ \rho_{n}\}$ is a bounded and equi-continuous family on 
$[0,T]$. The Arzela-Ascoli Theorem guarantees that $\{ \rho_{n}\}$ has a 
subsequence $\{ \rho_{n_{i}}\}_{i\in \mathbb{N}}$,
converging uniformly on $[0,T]$ to a function $\rho\in C[0,T]$. 
Without loss of generality, we can assume
that $\mu_{n_i}\to \mu$ as $i\to\infty$. Moreover $\rho_{n_{i}}$ satisfies
the integral equation
\begin{align*}
\rho_{n_{i}}(t)=\int^T _0 G(t,s)f(s,\rho_{n_{i}}(s)+\gamma(s))ds 
+\frac{\mu_{n_i}^2}{(\rho_{n_i}(t)+\gamma(t))^3}+\frac{1}{n_{i}}.
\end{align*}
Let $i\to \infty$, we arrive at
\[
\rho(t)=\int^T _0 G(t,s)f(s,\rho(s)+\gamma(s))ds
 +\frac{\mu^2}{(\rho(t)+\gamma(t))^3}.
\]
Therefore, we have constructed the solution $(\rho,\mu)$ of the system 
\begin{equation}\label{newso}
\begin{gathered}
\ddot{\rho}+l^2\rho =f(t,\rho+\gamma)+\frac{\mu^{2}}{(\rho+\gamma)^{3}},\\
(\rho+\gamma)^2\dot{\varphi}=\mu.
\end{gathered}
\end{equation}

By the global continuation principle of Leray-Schauder \cite[Theorem 14. C]{zei}, 
there is a connected set $\mathcal{C}$, contained in
$[0,M]\times \Omega$, which connects $\{0\}\times \Omega$ with 
$\{M\}\times  \Omega$, whose elements $(\mu,\rho)$ are solutions of
system \eqref{newso}. As a result, we have also obtained a connected set 
$\tilde{\mathcal{C}}$, contained in
$[0,M]\times \tilde{\Omega}$, which connects $\{0\}\times \tilde{\Omega}$ with 
$\{M\}\times \tilde{\Omega}$, whose elements $(\mu,\tilde{\rho})$ are solutions
 of system \eqref{equi}, where $\tilde{\rho}=\rho+\gamma$ and $\tilde{\Omega}$ 
is deduced from $\Omega$ with the constant $C$ being changed as another constant 
$\tilde{C}$.

Let us define the function $\Phi:\tilde{\mathcal{C}}\to\mathbb{R}$ by
\[
\Phi(\mu,\tilde{\rho})\mapsto \int^T _0 \frac{\mu}{\tilde{\rho}^2(t)}dt.
\]
It is continuous and defined on a compact and connected domain, so its image
is a compact interval. Since $\Phi(0,\tilde{\rho})=0$ and $\Phi$ is not 
identically zero, this interval
is of the type $[0,\bar{\theta}]$ for some $\bar{\theta}>0$.

\begin{lemma}\label{rhopsi}
 For every $\theta \in ( 0,\bar{\theta}]$, there exist $(\mu,\tilde{\rho},\varphi)$
 satisfying system \eqref{equi}, for which $(\mu,\tilde{\rho})\in\mathcal{C}$ and
\[
\tilde{\rho}(t+T)=\tilde{\rho}(t),\quad \varphi(t+T)=\varphi(t)+\theta,
\]
for every $t\in\mathbb{R}$.
\end{lemma}

\begin{proof}
Given $\theta\in(0,\bar{\theta}]$, there exist 
$(\mu,\tilde{\rho})\in\tilde{\mathcal{C}}$, such that
\[
\Phi(\mu,\tilde{\rho})=\theta.
\]
Obviously, the first equation in \eqref{equi} is satisfied and 
$\tilde{\rho}$ is $T$-perodic. Define
\[
\varphi(t)=\int^{t}_0 \frac{\mu}{\tilde{\rho}^{2}(s)}ds,
\]
which satisfies the second equation in \eqref{equi}. Moreover,
\[
\varphi(t+T)-\varphi(t)=\int^{t+T}_t \frac{\mu}{\tilde{\rho}^{2}(s)}ds
=\int^T _0 \frac{\mu}{\tilde{\rho}^{2}(s)}ds=\theta.
\]
\end{proof}

Now we are in a position to complete the proof of Theorem \ref{main}.
For every $\theta\in(0,\bar{\theta}]$, the solution of system \eqref{equi} found in
Lemma \ref{rhopsi} provides a solution of \eqref{xfk} such that
\[
x(t+T)=x(t)e^{i\theta},\quad \text{for every }t\in\mathbb{R}.
\]

In particular, if $\theta=2\pi/k$ for some integer $k\geq 1$, then $x(t)$ is
 periodic with minimal period $kT$
and rotates exactly once around the origin in the period time $kT$ since 
$\varphi(t+kT)=\varphi(t)+2\pi$. Hence, for every integer
$k\geq 2\pi/\bar{\theta}$, we have such a $kT$-periodic solution, which we denote 
by $x_{k}(t)$. Let $(\tilde{\rho}_{k}(t),\varphi_{k}(t))$ be its
polar coordinates, and $\mu_{k}$ be its angular momentum. By the above construction, 
$(\mu_{k},\tilde{\rho}_{k},\varphi_{k})$ satisfy system \eqref{equi},
 $(\mu_{k},\tilde{\rho}_{k})\in\mathcal{C}$, and
\begin{equation}\label{muk}
\int^T _0 \frac{\mu_{k}}{\tilde{\rho}_{k}^{2}(t)}dt=\frac{2\pi}{k}.
\end{equation}
Since $\rho_k\in\Omega$, we have
\[
\frac{2\pi}{k}=\int^T _0 \frac{\mu_{k}}{\tilde{\rho}_{k}^{2}(t)}dt
>\frac{\mu_kT}{C^{2}},
\]
which implies that $\lim_{k\to\infty}\mu_k=0$. The proof is thus finished.

\subsection*{Acknowledgments} 
The author would like to show her appreciation to the anonymous
referee for the valuable suggestions and comments,
which improve the original version of this article.


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\end{document}