text:
\section{Introduction}
This paper concerns with the motions of the non-isentropic Newtonian rotating gaseous stars, which are described by the compressible full Euler-Poisson equations:
\begin{equation}\label{FEP}
\left\{
\begin{aligned}
\rho_t+\dv_x(\rho \vu)&=0,\\
(\rho \vu)_t+\dv_x(\rho \vu\otimes\vu)+\nabla p+\rho \nabla\Phi&=0,\\
(\rho S)_t+\dv_x(\rho \vu S)&=0,\\
\Delta \Phi&=4\pi  \rho,
\end{aligned}
\right.
\end{equation}
where $\rho$, $\vu$, $p$, $\Phi$ and $S$  represent the density, velocity, pressure, gravitational potential and entropy respectively;
$(t,x)=(t,x_1,x_2,x_3)\in \mathbb{R}^+\times\mathbb{R}^3$ are the time and space variables.
The pressure $p$ is given by the equation of state,
\begin{equation}
\label{state}
p=p(\rho, S)=f(\rho)e^S
\end{equation}
and let $\Phi$ vanish as $x$ tends to infinity,
\begin{equation}
\label{B}
\Phi=-\int_{\mathbb{R}^3}|x-y|^{-1}\rho(y)\  dy=:-B\rho.
\end{equation}

There have been many studies on the compactly supported steady (i.e. time-independent) solutions  to \eqref{FEP},
which describe the equilibrium configurations of self-gravitating gaseous stars,
such as white dwarf stars \cite{Chandrasekhar1957}.
The isentropic spherically symmetric stationary ($S=0,\rho(x)=\rho(|x|), \vu=0$) solutions are often referred as the Lane-Emden solutions,
and their existence and linearly stability problem can be found in \cite{Chandrasekhar1957, Lin1997}. Recently the global nonlinear  well-posedness and stability theory have been studied extensively in \cite{Gu2012, Gu2016, Hadzic2016, Jang2008, Jang2010, Jang2014, Jang2016a,  Jang2013, Liu2016a, Luo2014,Luo2016b, Luo2016a, Makino2015, Makino2017}.

For isentropic axisymmetric ($\rho(x)=\rho(\sqrt{x_1^2+x_2^2},x_3)$) rotating solutions, namely the ones rotating along the $x_3$-axis with prescribed angular momentum,
if in the cylindrical coordinates
\begin{equation}\label{cylinder}
\eta(x)=\sqrt{x_1^2+x_2^2}~ \text{and} ~z(x)=x_3~,
\end{equation} when the angular momentum of the fluid per unit mass $\sqrt{L}$ is given, $\vu=(-\frac{x_2\sqrt{L}}{\eta^2},\frac{x_1\sqrt{L}}{\eta^2}),0)$ ($\vu (\mathbf{0})=\mathbf{0}$),
the compressible Euler-Poisson equations change to
\begin{equation}
\label{sss}
\nabla p=\rho\{\nabla B\rho+L(m_{\rho}(\eta))\eta^{-3} \mathbf{i_{\eta}}  \},
\end{equation}
where $\mathbf{i_{\eta}}$ is the unit vector in the radial direction in the cylindrical coordinates, and
\begin{equation}
\label{m}
m_{\rho}(s)=\int_{\eta(y)<s}\rho(y) \ dy.
\end{equation}
Auchmuty and Beals \cite{Auchmuty1971} established the existence of steady rotating solutions by the variational method.
They proved that when $p=f(\rho)$ and $L$ satisfy the assumption (A1) and (A3) (see Section \ref{cond}),
the minimizer of $E(\rho)$
\begin{equation*}\label{E1}
\widehat{E}_1(\rho)=\int_{\mathbb{R}^3}[A(\rho)(x)+\frac{1}{2}\rho(x)L(m_{\rho}(\eta(x)))\eta(x)^{-2}-\frac{1}{2}\rho(x)B\rho(x)] \ dx,
\end{equation*}
where
\begin{equation}\label{A}
A(s)=s\int_{0}^{s}f(t)t^{-2}\ dt,
\end{equation}
in the set $W$ (in which $\rho$ is axisymmetric, of total mass $M$ and the three integrals in $E(\rho)$ are finite) exists and is compactly supported, and satisfies \eqref{sss} where $\rho>0$.
Later, Li \cite{Li1991} studied the uniformly rotating gaseous stars with prescribed constant angular velocity.
These axisymmetric rotating solutions may have a finite number of rings (\cite{Caffarelli1980, Chanillo1994}), while for non-rotating fluids, the energy minimizers must be symmetric decreasing after translation, namely, the shapes must be balls(\cite{Lieb1987}).
Other related results about the existence, regularities and stability  of these rotating solutions  can be found in \cite{Auchmuty1991, Chanillo2012, Federbush2014, Friedman1980, Friedman1981, Jang2017, Luo2008, Luo2009,  Rein2003, Wu2016}.

Now consider the non-isentropic steady solutions. Stationary solutions are obtained by Deng etc.\cite{Deng2002}.
For rotating ones,  if $S$ is spherically symmetric (i.e. $S(x)=S(|x|)$) and some other conditions of $S$ are assumed,  Luo and Smoller \cite{Luo2004} obtained the existence theorem of spherically symmetric solutions with uniformly rotating regular velocity for $\frac{6}{5}<\gamma<2$ , and the non-existence theorem for $1<\gamma<\frac{6}{5}$.
See \cite{Deng2006} for more results in the regime.
However, these spherically symmetric solutions in \cite{Luo2004} might not be the minimizers of the physical energy.
Recently Wu \cite{Wu2015} proved the existence of variational rotating solutions to the full Euler-Poisson equations,
but had to replacing the original constraint $\int_{\mathbb{R}^3} \rho=M$ in \cite{Auchmuty1971} by $\int_{\mathbb{R}^3}f_1w=P$, where  $w=\frac{\gamma}{\gamma-1}e^{\frac{\gamma-1}{\gamma}S}\rho^{\gamma-1}$ and $f_1$ denotes a function of the angular velocity.
However, in order to study steady states of a fixed mass of a rotating gaseous star with the prescribed mass angular momentum, the fixed mass constraint has to be kept.
In this paper, we impose a different variational structure which keeps the above constraint, and prove the existence of non-isentropic rotating solutions.

\vspace{0.3cm}

\textbf{Motivations:}
To find variational solutions to \eqref{sss} and $\eqref{FEP}_3$,
the first main difficulty is to find the variational structure for equation \eqref{sss} in the non-isentropic case.
The specific internal energy $e$ satisfies the thermodynamics identity
\begin{equation*}
d e=T(\rho,S) dS+\frac{p(\rho,S)}{\rho^2} d\rho,
\end{equation*}
where $T$ is the temperature.
Thus
\begin{equation*}
e(\rho,S)=\int_0^{\rho}\frac{p(t,S)}{t^2} dt=\frac{A(\rho)}{\rho} e^S.
\end{equation*}
Considering the simplest case, define $S=S(x)$ just as  \cite{Luo2004}, then as before the physical energy $\widehat{E}_2(\rho)$ should be the sum of the internal energy, the kinetic energy and the gravitational potential energy:
\begin{equation}
\label{E2}
\begin{aligned}
\widehat{E}_2(\rho)
=&\int_{\mathbb{R}^3} A(\rho)(x)e^S(x)\ dx+\frac{1}{2}\int_{\mathbb{R}^3} \rho(x)L(m_{\rho}(\eta(x)))\eta^{-2}(x) \ dx\\
&\qquad  - \frac{1}{2}\int_{\mathbb{R}^3} \rho(x)B\rho(x)\ dx.
\end{aligned}
\end{equation}
However, \eqref{sss} can not be derived by taking the gradient of the corresponding Euler-Lagrange equations of $\widehat{E}_2(\rho)$, which means that the solutions of \eqref{sss} are not the minimizers of the physical energy functional $\widehat{E}_2(\rho)$.
Therefore, defining $S=S(x)$ may not be appropriate.

To overcome this difficulty, we define $S$ as a function of the mass variable $n$,
where $n$ could be a function of $x$ which also depends on $\rho$, just like $m=m_{\rho}(\eta(x))$ (see \eqref{n}).
This assumption is reasonable because
by $\eqref{FEP}_1$ and $\eqref{FEP}_3$,
\begin{equation*}
S_t+v\cdot\nabla S=0,
\end{equation*}
which indicates that the entropy is invariant along the particle path.
Once having found the variational structure, we could follow the method in \cite{Auchmuty1971}:
first prove the existence of minimizers of the energy functional in the class $W_{R}^b$, in which the support of the density is contained in the ball of radius $R$;
then obtain uniform bounds in  $L^{\infty}$ norm and radii of these minimizers.

Here appears another main difficulty:
there is one more term $\int_{r_b(y)>r_b(x)}A(\rho(y))\cdot e^{S(n_{b,\rho}(r_b(y)))}S'(n_{b,\rho}(r_b(y)))\ dy$ in the potential function \eqref{eprime}.
This term has to be proved to be sufficiently small for all large $|x|$.
When $S$ is continuously differentiable,
it suffices to prove $\int_{r_b(y)>r_b(x)}A(\rho(y))\ dy$ sufficiently small  for all large $|x|$.
To overcome this difficulty, we will modify the arguments in \cite{Rein2001, Luo2008}

\section{Main Results}

Throughout this paper,
the physical energy $E(\rho,n)$ is defined as the sum of the internal energy, the kinetic energy and the gravitational potential energy:
\begin{equation}
\label{energy}
\begin{aligned}
E(\rho,n)=&\int_{\mathbb{R}^3} A(\rho(x))T(n(x))\ dx+\frac{1}{2}\int_{\mathbb{R}^3} \rho(x)L(m_{\rho}(\eta(x)))\eta^{-2}(x) \ dx\\
&\qquad - \frac{1}{2}\int_{\mathbb{R}^3} \rho(x)B\rho(x)\ dx,
\end{aligned}
\end{equation}
where $\eta$, $m_{\rho}$ and $B\rho$ are defined in \eqref{cylinder}, \eqref{m} and \eqref{B} respectively.
Let \begin{equation}
\label{rb}
r_b(x)=\sqrt{\eta^2(x)+\frac{z^2(x)}{b^2}}, \qquad \text{for some}\ b>0
\end{equation}
\begin{equation}
\label{n}
n_{b,\rho}(s)=\int_{r_b(y)\leq s}\rho(y)\ dy.
\end{equation}
and  define
\begin{equation*}
E_b(\rho)=E(\rho,n_{b,\rho}).
\end{equation*}
The expression of the potential function $E_b'$ is
\begin{equation}
\label{eprime}
\begin{aligned}
E_b'(\rho)(x)=& A'(\rho(x))T\Big(n_{b,\rho}(r_b(x))\Big)+\int_{r_b(y)>r_b(x)}A(\rho(y))T'\Big(n_{b,\rho}(r_b(y))\Big)\ dy \\
&\qquad \qquad \qquad +\int_{\eta(x)}^{\infty}L(m_{\rho}(s))s^{-3}\ ds-B\rho(x).
\end{aligned}
\end{equation}
The admissible set $W^b$ is defined as
\begin{equation}
\label{w}
\begin{aligned}
W^b&=\{ \rho: \mathbb{R}^3\rightarrow \mathbb{R} | \; \rho(x)\geq0\ \text{a.e.},\ \int_{\mathbb{R}^3}\rho(x)\ dx=M,\ \rho(x)=\rho(r_b(x)) \\
&\text{and the three integrals in \eqref{energy} are all finite} \}.
\end{aligned}
\end{equation}
Here $\rho(x)=\rho(r_b(x))$ implies that $\rho$ is a constant on every ellipsoid $x_1^2+x_2^2+x_3^2/b^2=C^2$. So we call it ellipsoidal symmetry in the following.

\vspace{0.5cm}

Now assume the following conditions on $f$, $S$(entropy per mass) and $L$(the square of angular momentum per mass):
\begin{enumerate}
\item[(A1)] \label{cond} $f(s)\geq 0$, is continuously differentiable and strictly increasing for $s>0$.
$\lim\limits_{s\rightarrow0}f(s)s^{-4/3}=0$, $\lim\limits_{s\rightarrow\infty}f(s)s^{-4/3}=\infty$.
\item[(A2)]
$\lim\limits_{s\rightarrow\infty}f(s)s^{-\bar{\gamma}}=0$ for
 some constant  $\bar{\gamma}>1$
\item[(A3)]
$L(m)$ is non-negative and absolutely continuous for $0\leq m\leq M$, with $L(0)=0$.
\item[(A4)] $L(am)\geq a^{4/3}L(m)$ for $0\leq a\leq1$ and $0\leq m\leq M$.
\item[] $L'(m)\geq 0$ for $0\leq m\leq M$.
\item[(A5)] $S(n)$ is continuously differentiable for $0\leq n\leq M$ and $S(0)=0$.
\item[(A6)]  For $0\leq a\leq1$ and $0\leq n\leq M$
 \begin{equation}
 \label{conditionS}
e^{S(an)}\geq a^{2/3} e^{S(n)}, \qquad e^{S(M-aM+an)}\geq a^{2/3}e^{S(n)},
 \end{equation}
\item[(A7)] There exists $\delta_0>0$ such that $S'(n)\leq 0$ for $M-\delta_0\le n\leq M$.
\end{enumerate}

The condition (A1) and the definition of $A$ \eqref{A} imply that $A''(s)=f'(s)/s>0$ if $s>0$. Therefore $A$ is convex and $A'$ is strictly increasing.

For simplicity, in the following we denote $T:= e^S$, and thus $(A5)$ implies
that $T(n)$ is continuously differentiable for $0\leq n\leq M$ and $T(0)=1$.
Therefore, there exist positive constants $T_0$ and $T_1$ such that
\begin{equation}
\label{conditionT}
 T_1\leq T(n) \leq T_0,~ |T'(n)|\leq T_0\ \text{for} \ 0\leq n\leq M.
\end{equation}

\vspace{0.5cm}
It is noted that if $\mathbf{v}=(-x_2\sqrt{L}/\eta^2, x_1\sqrt{L}/\eta^2,0)$ and $n=n_{b,\rho}$, then entropy function $\eqref{FEP}_3$  is satisfied.

The main results of this paper can be stated as:
\begin{thm}
\label{thm1}
Assume that $\rho$ minimizes $E_b$ in $W^b$. Then $\rho\in C(\mathbb{R}^3)$. Moreover, $\rho$ is continuously differentiable where it is positive, and \eqref{sss} is satisfied there.
\end{thm}
\begin{thm}
\label{thm2}
If (A1)-(A7) are all satisfied, then there is a $\rho\in W^b$ which minimizes $E_b$. Moreover, there is a constant $R_0$ such that any such $\rho$ vanishes for $|x|\geq R_0$.
\end{thm}
\begin{thm}
\label{thm3}
For any $\xi>1$, if (A1)-(A7) are all satisfied, then there is a $(\rho,n)$ in
\begin{equation}
\label{W}
W=\{(\rho,n) \ | \ \text{there exists some} \ b\in [\frac{1}{\xi},\xi] \ \text{s.t.}\ \rho\in W^b, n=n_{b,\rho} \}.
\end{equation}
which minimizes $E(\rho,n)$.
Moreover, there is a constant $R_0$ such that any such $\rho$ vanishes for $|x|\geq R_0$.
\end{thm}

\vspace{0.5cm}
Theorem \ref{thm1} implies that the energy minimizers of $E_b$ are the non-isentropic axisymmetric rotating solutions to the full Euler-Poisson equations \eqref{FEP}.
Theorem \ref{thm2} and \ref{thm3} show that such minimizers exist in $W_b$ and $W$.
	Here are some remarks on the results and the conditions (A1)-(A7).
\begin{remark}
		Our result extends that of the isentropic case (\cite{Auchmuty1971}) to the non-isentropic case.
		However, in order to make sure that the energy minimizers are the solutions to \eqref{sss},
		here $\rho$ has to be assumed to satisfy $\rho(x)=\rho(r_b(x))$.
		(See \eqref{symmetry} in the proof of Theorem \ref{thm1}.)
		This condition is stronger than the axisymmetry ( $\rho(x)=\rho(\eta(x),z(x))$ ) (\cite{Auchmuty1971}), but weaker than the spherical symmetry (\cite{Luo2004}):
		when $b=1$, $\rho(x)=\rho(r_b(x))$ is exactly the spherical symmetry, and Theorem \ref{thm3} implies that the energy minimizers exist among a large class of ellipsoidally symmetric functions.
		The motivation to consider this ellipsoidal symmetry rather than the spherical symmetry is that the shape of steady axisymmetric rotating solutions to \eqref{sss} may not be balls (\cite{Caffarelli1980}),
		and we are trying to prove the existence of energy minimizers not only among spherically symmetric functions, but also some other axisymmetric ones.
\end{remark}
\begin{remark}
		The main difference between our result and the previous result (\cite{Wu2015}) of the existence of variational non-isentropic solutions is that, the original constraint $\int_{\mathbb{R}^3} \rho=M$ in \cite{Auchmuty1971} is kept unchanged here, while in \cite{Wu2015} it was changed  to $\int_{\mathbb{R}^3}f_1w=P$, where  $w=\frac{\gamma}{\gamma-1}e^{\frac{\gamma-1}{\gamma}S}\rho^{\gamma-1}$ and $f_1$ denotes a function of the angular velocity.
		We prefer to keep this fixed mass constraint, because with it the energy minimizers can be interpreted as steady states of a fixed mass of rotating gaseous star with the prescribed angular momentum.
\end{remark}
\begin{remark}
\begin{enumerate}
	\item[1)] The condition (A4) is imposed by Luo and Smoller in \cite{Luo2008, Luo2009}. The monotonicity condition in (A4) is called the S\"{o}lberg stability criterion(\cite{Tassoul2015}).
	\item[2)] The condition (A6) can be fulfilled by $\sup\limits_{n\in [0,M]}|S'(n)|\leq -\frac{2}{3M}$.
	\begin{equation*}
	\begin{aligned}
	&\exp\Big\{ S(an)-S(n)\Big\}
		\geq \exp\Big\{-\sup\limits_{n\in [0,M]}|S'(n)| \cdot (1-a)n\Big\}\\
		&\qquad \geq \exp\Big\{-\frac{2}{3M}(1-a) n  \Big\}
		\geq \exp\Big\{-\frac{2}{3}  (1-a) \Big\}\\
	&\qquad \geq \exp \Big\{\frac{2}{3}\ln a \cdot \frac{1-a}{-\ln a} \Big\}
		\geq \exp\Big\{\frac{2}{3}  \ln a \Big\}  =  a^{2/3}\\
	&\exp\Big\{ S(M-aM+an)-S(n)\Big\}
		\geq \exp\Big\{-\sup\limits_{n\in [0,M]}|S'(n)| \cdot (1-a)(M-n)\Big\}\\
	&\qquad \geq \exp\Big\{-\frac{2}{3M} (1-a)(M-n) \Big\}
		\geq  a^{2/3}
		\end{aligned}
		\end{equation*}
	Here the inequality $\frac{1-a}{-\ln a}\leq 1$ is used ($\frac{-\ln a}{1-a}$ is monotone decreasing on $(0,1)$ and $\lim\limits_{t\rightarrow 1^- } -\frac{\ln t}{1-t}=1$).
	So condition \eqref{conditionS} is coincident with the one in \cite{Luo2004}, under which $|S'|$ is required to be sufficiently small.
    \item[3)] The condition (A7) requires that the entropy of the gaseous star is decreasing when sufficiently near to the vacuum.
	This condition is weaker than those conditions on the entropy in \cite{Luo2004, Wu2015}
\end{enumerate}
\end{remark}

Throughout the paper, we will write $C$ for a generic constant and
$$
\int f\ dx=\int_{\mathbb{R}^3}f(x)\ dx.
$$
$\|\cdot\|_p$ denotes the norm in the $L^p$ function space:
$$
\|f\|_p:=\left(\int |f|^p \ d\mu \right)^{1/p}.
$$

The rest of the paper is arranged as the following.
We give the basic inequalities used in this paper and prove
Theorem \ref{thm1} in Section 3.
And then by analyzing an energy-minimizing sequence, the existence of the energy minimizers on $W^b$ is established in Section 4,
which is exactly Theorem \ref{thm2}.
This existence theorem is improved by enlarging the class $W^b$ of $\rho$ to $\bigcup\limits_{b\in [\frac{1}{\xi},\xi]} W^b$ in Section 5.

\section{Inequalities and Regularity of minimizers}

\begin{prop}
\label{prop1}
Assume that $\rho\in L^1\cap L^p$. If $1<p\leq \frac{3}{2}$. Then $B\rho\in L^r$ for $3<r<3p/(3-2p)$, and
\begin{equation} \label{ineq1}
\| B\rho\|_r\leq C(\|\rho\|_1^{\alpha_1}\|\rho\|_p^{1-\alpha_1}+\|\rho\|_1^{\alpha_2}\|\rho\|_p^{1-\alpha_2})
\end{equation}
where $0<\alpha_1,\alpha_2<1$.\\
If $p>\frac{3}{2}$, then $B\rho$ is bounded and continuous and satisfies \eqref{ineq1} with $r=\infty$.
\end{prop}

\begin{prop}
\label{prop2}
Suppose that $\rho \in L^1\cap L^{4/3}$. Then
\begin{equation}
\label{ineq2}
\|\nabla B\rho\|_{2}^2=|\int\rho B\rho|\leq C\int |\rho|^{4/3}(\int|\rho|)^{2/3}.
\end{equation}
\end{prop}

\begin{prop}
\label{prop3}
If $\rho \in L^1\cap L^p$ for some $p>3$, then $B\rho$ is continuously differentiable.
\end{prop}

The above three propositions are proved in \cite{Auchmuty1971} by standard H\"{o}lder inequality and estimates for the Riesz potential.

Let $B_R=\{x\in \mathbb{R}^3| |x|<R \}$.
\begin{prop}
\label{prop4}
Let $\{\rho_j\}_{j\in \mathbb{N}^*}$ be a bounded sequence in $L^{p}(\mathbb{R}^3)$, $p>3$.
Suppose that
$$
\rho_j\rightharpoonup \rho_0 \quad \text{weakly in} \ L^{p}(\mathbb{R}^3),
$$
and $\int \rho_j=M$, $j\in \mathbb{N}^*\cup \{0\}$.
Then for any $R>0$,
$$
\nabla B(\chi_{B_R}\rho_j)\rightarrow \nabla B(\chi_{B_R}\rho_0) \quad \text{strongly in }\ L^2(\mathbb{R}^3),
$$
where $\chi$ is the characteristic function.
\end{prop}
\begin{proof}
See \cite[Lemma~3.7]{Rein2001} and \cite[Lemma~3.7]{Luo2008}.
\end{proof}

\subsection{Variational Solutions}
Given $\rho\in W^b$, choose $\epsilon>0$ small enough such that $\Omega_{\epsilon}(\rho)=\{x| \ \epsilon<  \rho(x)<\epsilon^{-1}\}$ has positive measure. Define
\begin{equation}
\label{P_0}
\begin{aligned}
P_{\epsilon}(\rho)&= \{\sigma:\ \mathbb{R}^3\rightarrow\mathbb{R}|\ \sigma\in L^{\infty}(\mathbb{R}^3);\ \sigma(x)=\sigma(r_b(x)) ; \\
&\qquad \sigma=0 \ \text{a.e.} \ \text{outside}\ \Big(\Omega_\epsilon(\rho) \cup \{x|\ \rho(x)=0 \}\Big); \\
&\qquad \sigma(x)\geq 0 \ \text{if} \ \rho(x)=0;\ \sigma(x)=0\ \text{if}\ \eta(x)<\epsilon \ \text{or}\ |x|>\epsilon^{-1} \},  \\
P_0(\rho)&=\bigcup\limits_{\epsilon>0}P_{\epsilon}(\rho).
\end{aligned}
\end{equation}

\begin{lem}
\label{lem1}
Suppose that $\rho\in W^b$ and $P_0(\rho)$ is defined as above. Then
\begin{equation}
\lim\limits_{t\rightarrow0^+}\frac{E_b(\rho+t\sigma)-E_b(\sigma)}{t}=\int E'_b(\rho(x))\sigma(x) \ dx \quad \text{for} \ \forall\ \sigma \in P_0(\rho) .
\end{equation}
\end{lem}
\begin{proof}
Since the last two integrals in $\widehat{E}_1(\rho)$ and $E_b(\rho)$ are the same,
the proof of \cite[Lemma~1]{Auchmuty1971} implies that
it remains to prove that
for any $ \epsilon >0$ and any $ \sigma \in P_{\epsilon}(\rho)$,
\begin{equation}
\label{claim1}
\begin{aligned}
\lim\limits_{t\rightarrow0^+}& t^{-1}\Big[ \int A(\rho+t\sigma)T(n_{b,\rho+t\sigma}(r_b))-\int A(\rho)T(n_{b,\rho}(r_b))\Big]=\\
& \int \Big[A'(\rho(x))T(n_{b,\rho}(r_b(x)))+\int_{r_b(y)>r_b(x)}A(\rho(y))T'(n_{b,\rho}(r_b(y)))\ dy \Big]\sigma(x)\ dx
\end{aligned}
\end{equation}

Actually, noting that $A$ and $T$ are both continuously differentiable, it follows from the Mean Value Theorem that
\begin{align*}
I&:=t^{-1}\Big[ \int A(\rho+t\sigma)T(n_{b,\rho+t\sigma}(r_b))-\int A(\rho)T(b,n_\rho(r_b))\Big]\\
&=\int \left[ \frac{A(\rho+t\sigma)-A(\rho)}{t}T(n_{b,\rho+t\sigma}(r_b)) +A(\rho)\frac{T(n_{b,\rho+t\sigma}(r_b))-T(n_{b,\rho}(r_b))}{t} \right]\\
&=\int A'(\rho+\theta \sigma)\sigma T(n_{b,\rho+t\sigma}(r_b))+\int A(\rho) T'(n_{b,\rho+\theta'\sigma}(r_b))n_{b,\sigma}(r_b),
\end{align*}
where $\theta$, $\theta'$ are functions of $x$, and $0<\theta,\theta'<t$.
Since $A'$ is stricty increasing,  $|\sigma(x)|\leq k$ for some constant $k$, and $\sigma(x)=0$ if $\rho(x)>\epsilon^{-1}$ or $|x|>\epsilon^{-1}$, with $T$ satisfying the condition \eqref{conditionT}, it holds that
\begin{align*}
|A'(\rho+\theta \sigma)\sigma T(b,n_{\rho+t\sigma}(r_b))|\leq A'(\epsilon^{-1}+t\|\sigma\|_{\infty})\|\sigma\|_{\infty}T_0\\
|A(\rho) T'(n_{b,\rho+\theta'\sigma}(r_b))n_{b,\sigma}(r_b)|\leq T_0M A(\rho).
\end{align*}
Since $\sigma$ is compactly supported, $A'(\epsilon^{-1}+t\|\sigma\|_{\infty})\|\sigma\|_{\infty}T_0$ is integrable. Thus, it follows from the Lebesgue's Dominated Convergence Theorem that
$$I\rightarrow \int A'(\rho)T(n_{b,\rho}(r_b))\sigma+\int A(\rho)T'(n_{b,\rho}(r_b))n_{b,\sigma}(r_b)=: I_1+I_2 \quad \text{as}\ t\rightarrow 0^+.$$

Let
$$\tilde{\rho}(s)=\int_{r_b(y)=s} \rho(y) \ dS_y,$$
and then the fact that $\rho(x)=\rho(r_b(x))$ and $\sigma(x)=\sigma(r_b(x))$ implies the following equality
\begin{equation*}
\begin{aligned}
I_2&=\int_{0}^{\infty}\widetilde{A(\rho)}(r_b)T'(n_{b,\rho}(r_b))\int_{0}^{r_b}\tilde{\sigma}(s) \ ds \ dr_b\\
&=\int_{0}^{\infty}\tilde{\sigma}(s)\int_{s}^{\infty}\widetilde{A(\rho)}(r_b)T'(n_{b,\rho}(r_b))\ dr_b\ ds\\
&=\int \Big[\int_{r_b(y)>r_b(x)}A(\rho(y))T'\Big(n_{b,\rho}(r_b(y))\Big)\ dy \Big]\sigma(x) \ dx.
\end{aligned}
\end{equation*}
Hence, \eqref{claim1} is verified.

Therefore, this lemma is proved.
\end{proof}

\begin{lem}
\label{lem2}
Suppose that $\rho$ minimizes $E_b$ on $W^b$. Then there is a constant $\lambda$ such that
\begin{eqnarray}
&& E'_b(\rho)\geq \lambda \qquad \text{a.e.}, \label{eprime1}\\
&& E'_b(\rho)=\lambda \qquad \text{a.e.}, \ \text{where}\ \rho(x)>0. \label{eprime2}
\end{eqnarray}
\end{lem}
\begin{proof}
See \cite[Lemma~2]{Auchmuty1971}
\end{proof}

Lemma \ref{lem1} and Lemma \ref{lem2} show the Euler-Lagrange equations of the minimizers of $E_b$ on $W^b$.
Lemma \ref{lem3} will provide an inequality that is the key to run the bootstrap arguments to improve the regularities of the minimizers of $E_b$ on $W^b$.

\begin{lem}
\label{lem3}
Suppose that $\rho$ minimizes $E_b$ on $W^b$. Then there exist constants $c_1$ and $K$ such that
\begin{equation}
\label{ineq3}
\rho^{1/3}\leq c_1 B\rho
\end{equation}
a.e. on the set where $\rho\geq K$.
\end{lem}
\begin{proof}
It follows from the definition of $A$ and the assumptions on $f$ (A1) that
for any $c_2$, $A'(s)\geq c_2 s^{1/3}$ for large $s$.
By \eqref{eprime2} and the condition of $T$ \eqref{conditionT}, when $\rho>0$,
it holds that
\begin{align*}
& T_1A'(\rho)\leq A'(\rho)T(n_{b,\rho}(r_b))\\
\leq& -\int_{r_b(y)>r_b(x)}A(\rho(y))T'(n_{b,\rho}(r_b(y))) dy+ B\rho+\lambda\\
\leq& T_0 \int A(\rho(y))\ dy+B\rho+\lambda.
\end{align*}
Then inequality \eqref{ineq3} follows by choosing $c_1>c_2$ and $K$ large enough.
\end{proof}
\begin{remark}
The constants $K$ and $c_1$ in Lemma \ref{lem3} depend on $\rho$.
\end{remark}

\begin{proof}[\textbf{Proof of Theorem \ref{thm1}}]
Since $T$ is continuously differentiable, then it follows from the proof of \cite[Theorem~1]{Auchmuty1971} that
$\rho\in C(\mathbb{R}^3)$ and is continuous differentiable where it is positive.
When $\rho$ is positive, taking the gradient in \eqref{eprime2} gives that
\begin{equation*}
\begin{aligned}
& A''(\rho)T(n_{b,\rho}(r_b))\nabla \rho+A'(\rho)T'(n_{b,\rho}(r_b))\left(\int_{r_b(y)=r_b}\rho\right)\nabla r_b\\
&\qquad -\left(\int_{r_b(y)=r_b}A(\rho)
T'(n_{b,\rho}(r_b))\right)\nabla r_b=\cdots+\nabla B\rho,
\end{aligned}
\end{equation*}
where ``$\cdots$'' are terms obtained from the third term in \eqref{eprime}, which can be handled in the same way as that in \cite{Auchmuty1971} and thus are omitted here.
Note that $\rho(x)=\rho(r_b(x))$ for any $\rho\in W^b$, and hence
\begin{equation}
\label{symmetry}
\frac{\int_{r_b(y)=r_b}A(\rho)}{\int_{r_b(y)=r_b}\rho}=\frac{A(\rho)}{\rho}.
\end{equation}
By the definition of $A$ \eqref{A}, $A''(\rho)=\frac{f'(\rho)}{\rho}$ and $A'(\rho)-\frac{A(\rho)}{\rho}=\frac{f(\rho)}{\rho}$.
So it follows that
\begin{equation*}
\frac{f'(\rho)}{\rho}T(n_{b,\rho}(r_b))\nabla\rho+\frac{f(\rho)}{\rho}T'(n_{b,\rho}(r_b))\left(\int_{r_b(y)=r_b}\rho\right)\nabla r_b=\cdots+\nabla B\rho
\end{equation*}
Multiplying it with $\rho$ gives \eqref{sss}.
Therefore, the proof is finished.
\end{proof}

\section{Existence of Rotating Solutions}
In this section, the existence of the compactly supported minimizers of the energy functional $E_b$ on $W^b$ is established.
We first prove the existence of the minimizers on $W_{R}^b$  \eqref{W,R}.
Then it should be pointed out that the constrained minimizers $\rho_R$ on $W_{R}^b$ actually form a subsequence $\{\rho_{R_j}\}_{j\in\mathbb{N}^*}$
which minimize the energy functional $E_b$ on $W^b$ and has a weak limit $\rho_0$ in $L^{4/3}(\mathbb{R}^3)$.
At last, the uniform estimates of $\{\rho_{R_j}\}_{j\in\mathbb{N}^*}$ show that $\rho_0$ is the compactly supported energy minimizer on $W^b$.

\subsection{Existence of Constrained Minimizers}
Given $R>0$, denote
\begin{equation}
\label{W,R}
\begin{aligned}
W^b_{R}&=\{ \rho: \mathbb{R}^3\rightarrow \mathbb{R} | \; \rho(x)\geq0\ \text{a.e.};\ \int\rho(x)\ dx=M;\ \rho(x)=\rho(r_b(x));\\
& \
\rho(x)=0 \ \text{if}\ |x|\geq R;\ \rho(x) \leq R \ \text{a.e.}    \}.
\end{aligned}
\end{equation}

\begin{lem}
\label{lem4}
For any $R$ large enough, there exists a $\rho_R\in W^b_{R}$ which minimizes $E_b$ on $W^b_{R}$. And such $\rho_R$ is continuous if $|x|<R$. Furthermore, there exists a constant $\lambda_R$ such that
\begin{eqnarray}
&& E_b'(\rho_R)\geq \lambda_R \qquad \text{a.e.},\ \text{where}\ |x|<R,\ \rho_R(x)<R \label{eprime1r}\\
&& E_b'(\rho_R)=\lambda_R \qquad \text{a.e.}, \ \text{where}\ |x|<R,\ 0<\rho_R(x)<R. \label{eprime2r}
\end{eqnarray}
\end{lem}
\begin{proof}
Since the last two integrals in $\widehat{E}_1(\rho)$ and $E_b(\rho)$ are the same,
the proof of \cite[Theorem~$B_R$]{Auchmuty1971} implies that it suffices to prove that $\int A(\rho) T(n_{b,\rho})$ is weakly lower semi-continuous in $W^b_{R}$. Here $W^b_{R}$ is a bounded, closed and convex subset of $L^p$, $1< p\leq \infty$ and thus it is also weakly closed and weakly compact(see \cite[Theorem~6,~Chapter~10]{Lax}) .

Suppose that $\{\rho_n\}_{n=1}^{\infty}, \rho_0 \in W^b_{ R}$ and $\rho_n$ converges weakly to $\rho_0$ in $L^{p}$.
Since $\{\rho\in W^b_{ R} | \int A(\rho)T(n_{b,\rho_0}) \leq k \}$ is convex and closed,
then it is weakly closed .
Therefore, $\liminf\limits_{n\rightarrow \infty}\int A(\rho_n)T(n_{b,\rho_0})\geq\int A(\rho_0)T(n_{b,\rho_0})$.

Since $n_{b,\rho_n}\rightarrow n_{b, \rho_0}$ a.e. and these functions are continuous, monotonically increasing in $\mathbb{R}$ and bounded uniformly,
then $n_{b,\rho_n}\rightarrow n_{b, \rho_0}$ uniformly on compact subsets of $[0,\infty)$ (see \cite[Problem~13,~p.167]{Rudin}).
Therefore, $T(n_{b, \rho_n})\rightarrow T(n_{b, \rho_0})$ uniformly.

So for any $\epsilon >0$ , there exists $N(\epsilon)$ such that $\forall n>N$
\begin{equation*}
\begin{aligned}
&\inf\limits_{j>n} \int A(\rho_j)T(\rho_0) +\frac{\epsilon}{2}\geq \int A(\rho_0)T(\rho_0),\\
&\left| \int A(\rho_n)T(\rho_n)- \int A(\rho_n)T(\rho_0) \right|
\leq A(R) \int |T(\rho_n)-T(\rho_0)|<\frac{\epsilon}{2}.
\end{aligned}
\end{equation*}
Then
\begin{equation*}
\begin{aligned}
\inf\limits_{j>n} \int A(\rho_j)T(\rho_n) &\geq \inf\limits_{j>n} \int A(\rho_j)T(\rho_0) -\sup\limits_{j>n} \left| \int A(\rho_j)T(\rho_0)- \int A(\rho_j)T(\rho_j) \right|\\
&\geq A(\rho_0)T(\rho_0)-\epsilon.
\end{aligned}
\end{equation*}
This inequality gives $\liminf\limits_{n\rightarrow \infty}\int A(\rho_n)T(n_{b,\rho_n})\geq\int A(\rho_0)T(n_{b,\rho_0})$.
Hence a minimizer $\rho_R$ on $W_{R}^b$ exists.
The rest of the proof can be completed by following Lemma \ref{lem2} and the proof of Theorem \ref{thm1}.
\end{proof}

Lemma \ref{lem4} corresponds to Lemma \ref{lem1}, Lemma \ref{lem2} and Lemma \ref{lem3} for the minimizers of $E_b$ on $W^b_{R}$.
The Euler-Lagrange equations provided by Lemma \ref{lem4} will be used throughout the remaining proof of the paper.
\begin{lem}
\label{lem5}
There exist $k_0>0$ and $\pi_1>0$ such that $\|\rho_{R}\|_{4/3}\leq k_0$, when $R\geq\pi_1$.
\end{lem}
\begin{proof}
For any fixed $\pi_0>0$, we have $E_b(\rho_R)\leq E_b(\rho_{\pi_0})$, when $R\geq \pi_0$.
\eqref{ineq2} and the condition of $T$ \eqref{conditionT} imply that
\begin{equation}
\label{ineq4}
T_1\int A(\rho_R)\leq E_b(\rho_{\pi_0})+C\int \rho_R^{4/3}.
\end{equation}
With the inequality, the rest of the proof follows easily from \cite[Lemma~4]{Auchmuty1971}
\end{proof}
Lemma \ref{lem5} shows that $\rho_{R}$ has a uniform bound in $L^{4/3}$ norm.

\subsection{Uniform estimates of the constrained minimizers}
Since $\rho_R$ are uniformly bounded in $L^{4/3}(\mathbb{R}^3)$,
there exist a subsequence of $\rho_R$, i.e. $\{\rho_{R_j} \}_{j\in \mathbb{N}^*}\subset
W^b$  and $\rho_0\in L^{4/3}$ such that
\begin{equation}
\label{sequence}
\rho_{R_j} \rightharpoonup\rho_0 \ \text{weakly in } \ L^{4/3}.
\end{equation}
Since $W^b$ is a convex set, $\rho_0$ also satisfies $\rho_0(x)=\rho_0(r_b(x))$. (See \cite[Theorem~6,~Chapter~10]{Lax}).
The remaining part of this section is devoted to deriving the uniform estimates of their $L^{\infty}$ norm and radii.
We would not focus on the subsequence until Lem \ref{lem12}.

\begin{lem}
\label{lem6}
There exists $\pi_2>\pi_1$ such that $\lambda_R<C E_b(\rho_{\pi_0})+C k_0^{4/3}$ for all $R \geq \pi_2$.
\end{lem}
\begin{proof}
Since $T_1 \leq |T(n)| \leq T_0, |T'(n)|\leq T_0$, the proof of \cite[Lemma~5]{Auchmuty1971} implies that
$$
\lambda_R\leq o(R^{-1})+O(R^{-2})-O(R^{-1})
+\int_{r_b(y)>r_b(x)}A(\rho(y))T'(n_{b,\rho}(r_b(y)))\ dy.
$$
on a special subset of $B_R$.
So there exists $\pi_2>\pi_1$ such that for any $R\geq \pi_2$,
\begin{equation*}
\begin{aligned}
\lambda_R &\leq
\sup\left|\int_{r_b(y)>r_b(x)}A(\rho(y))T'(n_{b,\rho}(r_b(y)))\ dy \right|\\
&\leq T_0\int A(\rho_R) \overset{\eqref{ineq4}}{\leq} C E_b(\rho_{\pi_0})+C k_0^{4/3}.
\end{aligned}
\end{equation*}
Hence this lemma is proved.
\end{proof}

\begin{lem}
\label{lem7}
$\|\rho_R\|_{\infty}\leq k_1$, all $R$.
\end{lem}
\begin{proof}
It follows from \eqref{eprime2r} and Lemma \ref{lem6} that,
\begin{equation*}
\begin{aligned}
T_1A'(\rho_R)&\leq \max\{0,\lambda_R-\int_{r_b(y)>r_b(x)}A(\rho_R(y))T'(n_{b,\rho_R}(r_b(y)))\ dy+B\rho_R \}\\
&\leq  \max\{0,CE_b(\rho_{\pi_0})+Ck_0^{4/3}+B\rho_R \},\qquad \text{for}\ R\geq\pi_2
\end{aligned}
\end{equation*}
So as Lemma \ref{lem3}, there exist constants $C_1'$, $K'$ such that $\rho^{\gamma-1}\leq C_1'B\rho$ a.e. when $\rho_R\geq K'$.
Since $CE_b(\rho_1)+Ck_0^{4/3}$ does not depend on $\rho_R$, $C_1'$ and $K'$ do not depend on $\rho_R$ either.
Thus in the proof of Theorem \ref{thm1}, $\| \rho_R\|_{\infty}$ can be shown to be uniformly bounded when $R\geq \pi_2$.
Since $\rho_R\in W^b_{R}$, $\| \rho_R\|_{\infty}\leq \pi_2$ for $R\leq \pi_2$. So this lemma is proved.
\end{proof}
Lemma \ref{lem7} implies that
$\rho_R$ could not concentrate at some point as $R\rightarrow \infty$.

The proof of the following three lemmas are standard, see \cite[Lemma~6,~Corollary,~Lemma~7]{Auchmuty1971}.
$T$ would not change the result for $T$ is bounded.

\begin{lem}
\label{lem8}
There exists an $e<0$ such that $E_b(\rho_R)\leq e$ for all large $R$.
\end{lem}

\begin{lem}
\label{lem9}
$\|B\rho_R\|_{\infty}\geq -2e/M$ for all large $R$.
\end{lem}

\begin{lem}
\label{lem10}
There is an $\epsilon_0>0$ such that for each $R$
$$
\int_{|x-x_R|<1}\rho_R(x)\ dx\geq \epsilon_0
$$
for some $x_R$.
\end{lem}
\iffalse
\begin{proof}
Suppose that the claim is not true. Let
$$
\epsilon_R=\sup\limits_{x}\int\limits_{|y-x|<1}\rho_R(y) dy.
$$
Then for any $j\in \mathbb{N}^*$, there exists a $R_j>j$ such that $\epsilon_{R_j}\leq 1/j$.
However, \begin{align*}
B\rho_R(x)=&\int\frac{\rho_R(y)}{|y-x|}dy\\
=& \int\limits_{|y-x|<1}+\int\limits_{1\leq|y-x|<r}+\int\limits_{|y-x|\geq r}\\
=:& B_1+B_2+B_3,
\end{align*}
Lemma \ref{lem5} and Proposition \ref{prop1} imply that $B_1\leq C(\epsilon_R^{a_1}+\epsilon_R^{a_2})$, where $C$ is independent of $\rho_R$. By covering $1\leq |x-y|<r$ with cubes of length $1$ and then covering every cube with $4$ unit balls, one can show that $B_2\leq Cr^3\epsilon_R$. $B_3\leq Mr^{-1}$.
As $j\rightarrow \infty$, $\epsilon_{R_j}\rightarrow0$ and then by choosing $r$ properly for each $j$, $\|B\rho_{R_j}\|_{\infty}\rightarrow 0$. This contradicts Corollary \ref{lem9}.
So the proof of the lemma is completed.
\end{proof}
\fi

\begin{lem}
\label{lem11}
There is a constant $r_0$ such that if $x_R$ is as in Lemma \ref{lem10}, then $|x_R|+1\leq r_0$, all $R$.
\end{lem}
\begin{proof}
Claim that when $r_b(x_R)>\max\{b^{-1},1\}+1$,
\begin{equation*}
\begin{aligned}
&\int_{r_b(x)<2r_b(x_R)}\rho_R(x)\ dx \geq  \frac{\min\{b,1\}r_b(x_R)\epsilon_0}{4\pi}
\end{aligned}
\end{equation*}
Once it is proved, since $\rho_R$  is of total mass $M$, it follows that
\begin{equation*}
\begin{aligned}
\min\{b^{-1},1\}|x_R| & \leq r_b(x_R) \leq  \left(\frac{4\pi M }{\epsilon_0}+1\right) (\max\{ b^{-1},1\}+1)\\
 |x_R| &\leq  \left(\frac{4\pi M }{\epsilon_0}+1\right) (2b+b^{-1}+1)
\end{aligned}
\end{equation*}
So it suffices to prove the above claim.

Suppose that $$x=(r_b\cos\theta\cos\beta,r_b\sin\theta\cos\beta,br_b\sin\beta),$$
where $r_b$ and $\theta$ are defined as before, and $\beta\in[-\frac{\pi}{2},\frac{\pi}{2}]$ satisfies $\sin\beta=z(x)b^{-1}r_b(x)^{-1}$ and $\cos\theta=\eta(x)r_b(x)^{-1}$.
And define $r_b(x_R)$, $\theta(x_R)$ and $\beta(x_R)$ the same way as above for $x_R$.
Then $|x-x_R|<1$ implies that
\begin{equation}
\label{ineq6}
\begin{aligned}
&[r_b\cos\theta\cos\beta-r_b(x_R)\cos\theta(x_R)\cos\beta(x_R)]^2+[r_b\sin\theta\cos\beta
-\\&\qquad r_b(x_R)\sin\theta(x_R)\cos\beta(x_R)]^2
+b^2[r_b\sin\beta-r_b(x_R)\sin\beta(x_R)]^2<1.
\end{aligned}
\end{equation}
\begin{enumerate}
\item[(1)] If $b\leq1$, inequality \eqref{ineq6} gives
\begin{equation*}
\begin{aligned}
&[r_b\cos\theta\cos\beta-r_b(x_R)\cos\theta(x_R)\cos\beta(x_R)]^2+[r_b\sin\theta\cos\beta
-\\&\qquad r_b(x_R)\sin\theta(x_R)\cos\beta(x_R)]^2
+[r_b\sin\beta-r_b(x_R)\sin\beta(x_R)]^2<b^{-2}.
\end{aligned}
\end{equation*}
\begin{equation*}
r_b^2+r_b(x_R)^2-2r_br_b(x_R)\{\cos\beta\cos\beta(x_R)\cos[\theta-\theta(x_R)]+\sin\beta\sin\beta(x_R)\}<b^{-2}.
\end{equation*}
Since $\beta\in[-\frac{\pi}{2},\frac{\pi}{2}]$, it holds that
\begin{equation*}
r_b^2+r_b(x_R)^2-2r_br_b(x_R)[\cos\beta\cos\beta(x_R)+\sin\beta\sin\beta(x_R)]<b^{-2}.
\end{equation*}
Therefore,
\begin{equation*}
\begin{aligned}
& |r_b-r_b(x_R)|^2<b^{-2},\\
& \cos |\beta-\beta(x_R)|> \frac{r_b^2+r_b(x_R)^2-b^{-2}}{2r_br_b(x_R)}.
\end{aligned}
\end{equation*}
Here  $0=\frac{b^{-2}-b^{-2}}{2r_br_b(x_R)}\leq\frac{r_b^2+r_b(x_R)^2-b^{-2}}{2r_br_b(x_R)}\leq\frac{r_b^2+r_b(x_R)^2-|r_b-r_b(x_R)|^2}{2r_br_b(x_R)}=1$.
So it holds that
when $r_b(x_R)>\max\{b^{-1},1\}+1$,
\begin{equation*}
\sin|\beta-\beta(x_R)|<\sqrt{\frac{b^{-2}-[r_b-r_b(x_R)]^2}{2r_br_b(x_R)}}<\sqrt{\frac{b^{-2}}{2[r_b(x_R)-1]r_b(x_R)}}<\frac{1}{br_b(x_R)}.
\end{equation*}
Thus, Lemma \ref{lem10} yields that
\begin{align*}
\epsilon_0&\leq \iiint\limits_{\substack{|r_b-r_b(x_R)|\leq b^{-1},\\ |\beta-\beta(x_R)|\leq 2b^{-1}[r_b(x_R)]^{-1}}} \rho(y) dy
\\
&\leq
\int_{r_b(x_R)-b^{-1}}^{r_b(x_R)+b^{-1}}\int_{0}^{2\pi}\int_{\beta(x_R)-|\beta-\beta(x_R)|}^{\beta(x_R)+|\beta-\beta(x_R)|} \rho(r_b) \ br_b^2\cos\beta\ d\beta d\theta dr_b\\
&\leq 4\pi \cos [\beta(x_R)] \sin |\beta-\beta(x_R)|\int_{r_b(x_R)-b^{-1}}^{r_b(x_R)+b^{-1}}\rho(r_b)\  br_b^2\ dr_b\\
&\leq \frac{4\pi}{br_b(x_R)}\int_{r_b(x_R)-b^{-1}}^{r_b(x_R)+b^{-1}}\rho(r_b)\  br_b^2\ dr_b,
\end{align*}
and the mass in the open set  $\{x|r_b(x)<2r_b(x_R) \}$ satisfies
\begin{align*}
\iiint\limits_{r_b(y)<2r_b(x_R)} \rho(y)\ dy\geq
\int_{r_b(x_R)-b^{-1}}^{r_b(x_R)+b^{-1}}\rho(r_b) \ br_b^2  \  dr_b\geq\frac{br_b(x_R)\epsilon_0}{4\pi}.
\end{align*}
So the claim is proved when $b\leq 1$.

\item[(2)] If $b\geq1$, then the inequality \eqref{ineq6} implies that
\begin{equation*}
\begin{aligned}
&[r_b\cos\theta\cos\beta-r_b(x_R)\cos\theta(x_R)\cos\beta(x_R)]^2+[r_b\sin\theta\cos\beta
-\\&\qquad r_b(x_R)\sin\theta(x_R)\cos\beta(x_R)]^2
+[r_b\sin\beta-r_b(x_R)\sin\beta(x_R)]^2<1.
\end{aligned}.
\end{equation*}
Then remaining of the estimate is the same.
\end{enumerate}
Thus the claim is proved and Lemma \ref{lem11} follows.
\end{proof}

\vspace{0.5cm}

Up to now, we have just used conditions (A1), (A3) and (A5).
In the following, conditions (A2), (A4), (A6) and (A7) are used
and we will focus on the subsequence $\{\rho_{R_j}\}$
\begin{lem}
\label{lem12}
$\{\rho_{R_j}  \}_{j\geq1}$ is the minimizing sequence of $E_b$ on $W^b$, i.e.
$$
\lim\limits_{j\rightarrow \infty}E_b(\rho_{R_j})=\min\limits_{\rho\in W^b}E_{b}(\rho).
$$
\end{lem}
\begin{proof}
$E_b(\rho_{R_j})$ is a decreasing sequence and by \eqref{ineq2}
$$
E_b(\rho_{R_j})
\geq -\int \rho_{R_j} B\rho_{R_j} \geq -CM^{2/3}\|\rho_{R_j}\|^{4/3}_{4/3}\geq -CM^{2/3}k_0.
$$
So $\lim\limits_{j\rightarrow\infty}E_b(\rho_{R_j})$ exists,
and it suffices to prove that for any $\tau\in W^b$, $E_b(\tau)\geq \lim\limits_{j\rightarrow \infty}E_b(\rho_{R_j})$.

Take a nonnegative function $\sigma\in P_{\epsilon}(\tau)$ for some $\epsilon>0$, $\int \sigma=1$ and $\sigma(x)=0$ when $\tau(x)=0$.
Let
\begin{align*}
\tau'_{\alpha}(x)&=\begin{cases}
\tau(x) & \text{if}\ |x|<\alpha \ \text{or}\ \tau(x)<\frac{1}{2}\alpha\\
0 & \text{otherwise}
\end{cases},\\
\tau''_{\alpha}(x)&=(M-\int \tau'_{\alpha})\sigma,\\
\tau_{\alpha}&=\tau'_{\alpha}+\tau''_{\alpha}.
\end{align*}
The definition of $P_{\epsilon}(\tau)$ (see \eqref{P_0}) implies that $\sigma\in L^{\infty}(\mathbb{R}^n)$ and $\sigma\neq 0$ only if $\eta(x)\geq\epsilon$, $|x|\leq \epsilon^{-1}$ and $\tau(x)<\epsilon^{-1}$.
Hence, as $\alpha\rightarrow\infty$, if $\sigma(x)=0$, then $\tau_{\alpha}(x)$ increases to $\tau(x)$ a.e.; if $\sigma(x)>0$, then $\tau_{\alpha}(x)\rightarrow\tau(x)$ a.e. and $\tau_{\alpha}(x)$ are uniformly bounded by some constant $D$ (depending on $\|\sigma\|_{L^{\infty}}$ and $\epsilon$).

Note that $n_{b,\tau_{\alpha}}(r_b)\rightarrow n_{b,\tau}(r_b)$ and $n_{b,\tau_{\alpha}}$ are uniformly bounded by $M$, and
\begin{align*}
\int A(\tau_{\alpha})T(n_{b,\tau_{\alpha}}(r_b))
& \leq\int \Big\{A(\tau)T(n_{b,\tau}(r_b))\chi_{\{\sigma=0\}}+A(D)T_0\chi_{\{\sigma>0\}} \Big\}\\
&  \leq\int A(\tau(x))T(n_{b,\tau}(r_b(x)))\ dx+\frac{4}{3}\pi \epsilon^{-3}A(D)T_0,
\end{align*}
where $\chi$ is the characteristic function.
Therefore the Lebesgue's Dominated Convergence Theorem implies that $\int A(\tau_{\alpha})T(n_{b,\tau_{\alpha}}(r_b)) \rightarrow\int A(\tau)T(n_{b,\tau}(r_b))$ as $\alpha\rightarrow \infty$.
Similar arguments can also be applied to the other two parts of $E_b$.
Noting that $\sigma(x)=0$ when $\eta(x)<\epsilon$, it holds that
\begin{align*}
\int\tau_{\alpha}\frac{L(m_{\tau_{\alpha}}(\eta))}{\eta^2}
&\leq \int\Big\{\tau\frac{L(m_{\tau}(\eta))}{\eta^2}\chi_{\{\eta(x)<\epsilon\}}+\tau\frac{L(M)}{\epsilon^2}\chi_{\{\eta(x)\geq\epsilon,\sigma=0\}}\\
&\qquad\qquad+ D\frac{L(M)}{\epsilon^2}\chi_{\{\eta(x)\geq\epsilon,\sigma>0\}}  \Big\};\\
\int \tau_{\alpha}B\tau_{\alpha}&\leq \int (\tau+D\chi_{\{\sigma>0 \}})B(\tau+D\chi_{\{\sigma>0 \}})\\
&=\frac{1}{4\pi}\|\nabla B(\tau+D\chi_{\{\sigma>0 \}})\|_2^2\\
&\leq \Big( (\int \tau B\tau)^{1/2}+(\int D\chi_{\{\sigma>0 \}} BD\chi_{\{\sigma>0 \}})^{1/2} \Big)^2.
\end{align*}
Therefore it follows from the Lebesgue's Dominated Convergence Theorem that
$$
\lim\limits_{\alpha\rightarrow\infty}E_b(\tau_{\alpha})=E_b(\tau).
$$
Due to that  $E_b(\tau_{\alpha})\geq E_b(\rho_{\alpha})$ when $\alpha$ is large enough, it holds that $E_b(\tau)\geq\lim\limits_{j\rightarrow \infty}E_b(\rho_{R_j})$. Since $\tau$ is chosen arbitrarily, $\{\rho_{R_j} \}_{j\geq1}$ is the minimizing sequence of $E_b$ on $W^b$.
\end{proof}

\begin{lem}
\label{lem13}
Suppose that the subsequence $\{\rho_{R_j} \}_{j\geq1}$ and $\rho_0$ are given in \eqref{sequence}.
If the conditions (A4) and (A6) hold, then for any $\delta>0$, there exists a $r>2r_0>0$ such that
\begin{equation*}
\int_{|x|\geq r}\rho_{R_j}(x)\ dx\leq \delta, \qquad \text{for} \ \text{all}\ R_j.
\end{equation*}
\end{lem}
\begin{proof}
Take an arbitrary  $\rho \in W^b$.
For any $r$ satisfying $r>2r_0>0$,  define
\begin{equation*}
\rho=\rho\chi_{|x|\leq r}+\rho\chi_{|x|> r}=:\rho_1+\rho_2
\end{equation*}
where $\chi$ is the characteristic function.
Denote
$$M_i:= \int \rho_i,~a_i:=(\frac{M_i}{M})^{1/3},~\bar{\rho}_i(\cdot):=\rho_i(a_i\cdot)\qquad\qquad i=1,2.$$
Then $a_1, a_2\in [0,1]$,  $\bar{\rho}_1, \bar{\rho}_2 \in W^b$, and
$$m_{\rho_i}(\eta)=a_i^3m_{\bar{\rho}_i}(a_i^{-1}\eta),~ n_{b,\rho_i}(r_b)=a_i^3n_{b,\bar{\rho}_i}(a_i^{-1}r_b)$$
The condition (A6) implies that
\begin{align*}
\int A(\rho)T(n_{b,\rho}(r_b))
&=\int A(\rho_1)T(n_{b,\rho_1}(r_b)) + \int A(\rho_2)T(M-M_2+n_{b,\rho_2}(r_b))\\
&=\int A\Big(\bar{\rho}_1(a_1^{-1}\cdot)\Big)T\Big(a_1^3n_{b, \bar{\rho}_1}(r_b(a_1^{-1}\cdot))\Big) \\
&\qquad+ \int A\Big(\bar{\rho}_2(a_2^{-1}\cdot)\Big)T\Big(M-a_2^3 M+a_2^3n_{b, \bar{\rho}_2}(r_b(a_2^{-1}\cdot))\Big)\\
&\geq\int A\Big(\bar{\rho}_1(a_1^{-1}\cdot)\Big)a_1^2T\Big(n_{b, \bar{\rho}_1}(r_b(a_1^{-1}\cdot))\Big) \\
&\qquad+ \int A\Big(\bar{\rho}_2(a_2^{-1}\cdot)\Big)a_2^2T\Big(n_{b, \bar{\rho}_a}(r_b(a_2^{-1}\cdot))\Big)\\
&=a_1^5\int A\Big(\bar{\rho}_1\Big)T\Big(n_{b, \bar{\rho}_1}(r_b)\Big) + a_2^5\int A\Big(\bar{\rho}_2\Big)T\Big(n_{b, \bar{\rho}_a}(r_b)\Big).
\end{align*}
The condition (A4) implies the similar estimate:
\begin{equation*}
\begin{aligned}
\int\frac{\rho L\Big(m_{\rho}(\eta)\Big)}{\eta^2}
&\geq\int\frac{\rho_1 L\Big(m_{\rho_1}(\eta)\Big)}{\eta^2}+\int\frac{\rho_2 L\Big(m_{\rho_2}(\eta)\Big)}{\eta^2}\\
&\geq \int\frac{\bar{\rho}_1(a_1^{-1}\cdot) a_1^4 L\Big(m_{\bar{\rho}_1}(\eta(a_1^{-1}\cdot))\Big)}{a_1^2[\eta(a_1^{-1}\cdot)]^2}
+\int\frac{\bar{\rho}_2(a_2^{-1}\cdot) a_2^4 L\Big(m_{\bar{\rho}_2}(\eta(a_2^{-1}\cdot))\Big)}{a_2^2[\eta(a_2^{-1}\cdot)]^2}\\
&\geq a_1^5\int\frac{\bar{\rho}_1 L\Big(m_{\bar{\rho}_1}(\eta)\Big)}{\eta^2}
+a_2^5\int\frac{\bar{\rho}_2 L\Big(m_{\bar{\rho}_2}(\eta)\Big)}{\eta^2}.
\end{aligned}
\end{equation*}
\begin{equation*}
\begin{aligned}
\int \rho B\rho = a_1^5 \int \bar{\rho}_1 B\bar{\rho}_1 +a_2^5\int \bar{\rho}_1 B\bar{\rho}_1 +2\int \rho_1 B\rho_2.
\end{aligned}
\end{equation*}
Therefore, set $F_{b}=\min\limits_{\rho\in W^b}E_{b}(\rho)$ and then $F_b<0$,
\begin{equation*}
\begin{aligned}
E_b(\rho)&\geq a_1^5 E_b(\bar{\rho}_1)+ a_2^5 E_b(\bar{\rho}_2)-\int \rho_1 B\rho_2\\
&\geq  F_b-(1-a_1^5-a_2^5)F_b-\int \rho_1 B\rho_2\\
&\geq F_b-\frac{1}{C}M_1M_2 F_b-\int \rho_1 B\rho_2,
\end{aligned}
\end{equation*}
where one has used the inequality $x (1-x)\leq C [1-x^{5/3}-(1-x)^{5/3}]$ for $0\leq x\leq 1$.
\begin{equation*}
\begin{aligned}
\int \rho_1 B\rho_2&=\int(\rho \chi_{|x|\leq \frac{r}{2}}+\rho \chi_{ \frac{r}{2}<|x|\leq r}) B(\rho\chi_{|x|> r})\\
&=\iint\frac{\Big(\rho \chi_{|x|\leq \frac{r}{2}}\Big)(x)\Big(\rho\chi_{|y|> r}\Big)(y)}{|x-y|} \ dx dy+\frac{1}{4\pi}\int\nabla B(\rho \chi_{ \frac{r}{2}<|x|\leq r})\cdot \nabla B(\rho\chi_{|x|> r})\\
&\leq Cr^{-1}+C\|\nabla B(\rho \chi_{ \frac{r}{2}<|x|\leq r})\|_2 M^{1/3}\|\rho \|_{4/3}^{2/3},
\end{aligned}
\end{equation*}
where the definition of $B$ \eqref{B} and the inequality \eqref{ineq2} are used.
Then it follows that
\begin{equation*}
-F_bM_1M_2\leq Cr^{-1}+C\|\nabla B(\rho \chi_{ \frac{r}{2}<|x|\leq r})\|_2+C(E_b(\rho)-F_b)
\end{equation*}
Now applying the above inequality to the sequence $\{\rho_{R_j} \}_{j\geq1}$,
Since $r>2r_0$, one can get by Lemma \ref{lem11}  that $M_{R_j, 1}\geq \epsilon_0$.
And then
\begin{equation*}
\begin{aligned}
-F_b\epsilon_0M_{R_j,2}&\leq Cr^{-1}+C\|\nabla B(\rho_{R_j} \chi_{ \frac{r}{2}<|x|\leq r})\|_2+C(E_b(\rho_{R_j})-F_b)\\
&\leq Cr^{-1}+C\|\nabla B(\rho_{R_j} \chi_{ \frac{r}{2}<|x|\leq r})-\nabla B(\rho_0 \chi_{ \frac{r}{2}<|x|\leq r})\|_2\\
&\qquad +C\|\nabla B(\rho_0 \chi_{ \frac{r}{2}<|x|\leq r})\|_2+C(E_b(\rho_{R_j})-F_b)
\end{aligned}
\end{equation*}
Proposition \ref{prop4} and Lemma \ref{lem12} show that the second term and the forth term in the right hand side are small for $R_j$ large enough.
$\|\nabla B(\rho_0 \chi_{ \frac{r}{2}<|x|\leq r})\|_2$\ tends to $0$ as $r\rightarrow \infty$  if  $\|\nabla B\rho_0 \|_2$ is bounded.
Therefore, $M_{R_j,2}$ is uniformly small by choosing $r$ large enough and this proves the lemma.
\end{proof}

\begin{lem}
\label{lem14}
If the condition (A7) holds, then there exists a $l<0$ such that $\lambda_{R_j}\leq l<0$ for all $R_j>2r_1$ for some $r_1>2r_0>0$.
\end{lem}
\begin{proof}
By Lemma \ref{lem13}, choosing $r_1>2r_0$ large such that
\begin{equation*}
\int_{|x|\geq r_1}\rho_{R_j}(x)\ dx\leq \delta_0, \qquad \text{for} \ \text{all}\ R_j.
\end{equation*}
Hence,  (A7) gives that
\begin{equation*}
T'\Big(n_{b,\rho_{R_j}}(r_b(x))\Big) \leq 0 \qquad \text{for} \ \text{all}\ |x|\geq r_1.
\end{equation*}
Suppose that $R_j\geq \max\{\pi_1,2r_1\}$.
Set $x_{R_j}$ as in Lemma \ref{lem10}, and then $|x_{R_j}|+1\leq r_0<\frac{r_1}{2}$.
Note that $A'(\rho)=o(\rho^{1/3})$ as $\rho\rightarrow 0$.
Now for any $x$ satisfying $|x|<2r_1$, $\eta(x)>r_1$ and $\rho_{R_j}(x) \leq C r_1^{-3}$, $C$ independent of $r_1$,
\begin{equation*}
\begin{aligned}
&\qquad A'(\rho_{R_j})=o(\rho_{R_j}^{1/3})=o(r_1^{-1})\\
&\int_{r_b(y)>r_b(x)}A(\rho_{R_j}(y))T'\Big(n_{b,\rho_{R_j}}(r_b(y))\Big)\ dy \leq 0\\
&\int_{\eta(x)}^{\infty}L(m_{\rho_{R_j}}(s))s^{-3}\ ds\leq \frac{1}{2}\sup L(m) \cdot \frac{1}{4}r_1^{-2}=O(r_1^{-2})\\
&B\rho_{R_j}(x)\geq \int_{|y-x_{R_j}|<1}\frac{\rho_{R_j}(y)}{|y-x|}\ dy\geq \dfrac{\epsilon_0}{r_0+2r_1}=O(r_1^{-1}).
\end{aligned}
\end{equation*}
So it follows  the inequality \eqref{eprime1r} that
\begin{equation*}
\lambda_{R_j} \leq o(r_1^{-1})+O(r_1^{-2})-O(r_1^{-1})
\end{equation*}
Fixing $r_1$ yields Lemma \ref{lem14}.
\end{proof}

\begin{proof}[\textbf{Proof of Theorem \ref{thm2}}]
It follows from \eqref{eprime2r} and Lemma \ref{lem14} that
\begin{equation*}
\label{eprime3r}
\begin{aligned}
 A'(\rho_{R_j}(x))T(n_{b,\rho_{R_j}}(r_b(x)))&\leq
-\int_{r_b(y)>r_b(x)}A(\rho_{R_j}(y))T'(n_{b,\rho_{R_j}}(r_b(y)))\ dy\\
&\qquad\qquad+B\rho_{R_j}(x)+l.
\end{aligned}
\end{equation*}
To prove that there is a $R_0$ such that $\rho_{R_j}=0$ outside the ball $B_{R_0}$, it suffices to prove that $-\int_{r_b(y)>r_b(x)}A(\rho_{R_j}(y))T'(n_{b,\rho_{R_j}}(r_b(y)))\ dy\leq -l/2$ and $B\rho_{R_j}(x)\leq-l/2$ when $|x|>R_0$.

We first prove that $-\int_{r_b(y)>r_b(x)}A(\rho_R(y))T'(n_{b,\rho_R}(r_b(y)))\ dy\leq -l/2$ for $|x|$ large enough.
By (A1) and (A2), there exists a constant $s_c'$ such that
\begin{equation*}
A(s)\leq\begin{cases}
 Cs^{4/3} & s\leq s_c'\\
 Cs^{\bar{\gamma}} & s>s_c'
\end{cases},
\end{equation*}
where $C$ is a new constant.
Thus,
\begin{equation*}
\begin{aligned}
\int_{r_b(y)>r_b(x)} A(\rho_{R_j}(y))& T'(n_{b,\rho_{R_j}}(r_b(y)))\  dy \leq T_0\int_{r_b(y)>r_b(x)} A(\rho_{R_j})\\
 &\leq
CT_0(\|\rho_{R_j}\|_{\infty}^{1/3}+\|\rho_{R_j}\|_{\infty}^{\bar{\gamma}-1})\int_{r_b(y)>r_b(x)}\rho_{R_j},
\end{aligned}
\end{equation*}
where $C$ is a constant independent of $\rho_{R_j}$.
Then Lemma \ref{lem13} implies that there is a $r_2>2r_0>0$ such that for all $|x|\geq r_2$
$$
-\int_{r_b(y)>r_b(x)}A(\rho_{R_j}(y))T'(n_{\rho_{R_j}}(r(y))) dy\leq -l/2 \qquad \text{for all} \ j. $$

$B\rho_{R_j}(x) \leq -l/2$ follows the proof of Auchmuty and Beals \cite{Auchmuty1971}, and for completeness we present it here.
Let
$$
\epsilon_{r_3, R}=\sup\limits_{|x|>r_3/2} \int_{|y-x|<1}\rho_R(y)\ dy.
$$
A similar argument to Lemma \ref{lem11} gives that $\epsilon_{r_3,R}=O(r_3^{-1})$ as $r_1\rightarrow\infty$. Then for $|x|>r_3$ and $1<r<\frac{1}{2}r_3$, we have
\begin{align*}
B\rho_R(x)&=\int\limits_{|x-y|<1}+\int\limits_{1\leq|x-y|<r}+\int\limits_{|x-y|\geq r}\\
&\leq C(r_3^{-\alpha_1}+r_3^{-\alpha_2})+Cr_3^{-1}r^3+Mr^{-1}.
\end{align*}
Note that the shell $1<|y-x|<r$ can be covered by no more than $Cr$ balls of radius $1$.
Choosing $r$ large and then $r_3$ large, we get that $B\rho_{R_j}(x)\leq -l/2$ when $|x|\geq r_3$ for all $R_j$.

As a result, there exists a $R_0=\max\{2r_1, r_2, r_3\}$ such that $\{\rho_{R_j}\}_{j\geq1}\subset W_{R_0}^b$.
The proof of Lemma \ref{lem4} shows that $E_b$ is weakly lower semicontinuous on $W_{R_0}^b$.
Hence the weakly limit $\rho_0$ is also in $W_{R_0}^b$ and $\lim\limits_{j\rightarrow\infty}E_b(\rho_{R_j})=E_b(\rho_0)$.
So Lemma \ref{lem13} yields that $\rho_0$ is the compactly supported minimizer in $W^b$. (Actually, any $\rho_{R_j}$ satisfy $R_j\geq R_0$ is the minimizer in $W_{R_0}^b$.)
Consequently, Theorem 2 is proved.
\end{proof}
\clearpage

\section{An Improved Existence Theorem}

In this section, we will improve Theorem 2 by enlarging the admissible set $W^b$ to $\bigcup\limits_{ b\in[\frac{1}{\xi},\xi] } W^b$.
Theorem \ref{thm2} shows that for any $b>0$ there exists a $\rho_b\in W^b$ which minimizes $E_b$, and there exists a constant $R_b$ such that such $\rho_b$ vanishes for $|x| \geq R_b$.
The minimum of $E_b$ on $W^b$ can be considered as a function of $b$.
We denote it as $F_b=E_b(\rho_b)$ and thus $F_b$ represents the energy minimum for $E_b$.
To prove Theorem 3, it suffices to prove that
\begin{equation}
\label{3.3}
F_b \ \text{is continuous with respect to} \ b \ \text{in} \ [\frac{1}{\xi},\xi].
\end{equation}

\begin{cor}
\label{lem16} Fix $b_0\in [\frac{1}{\xi},\xi]$ and let $ \Gamma=\{b\in[\frac{1}{\xi},\xi]~|~ F_b\leq F_{b_0}\}$.
Then there exists a $K_0<\infty$ such that
\begin{equation*}
\max\limits_{b\in \Gamma} \{\| \rho_b\|_{4/3} \}\leq K_0
\end{equation*}
\end{cor}
\begin{proof}
Note that for any $b\in\Gamma$, $F_b=E_b(\rho_b)\leq F_{b_0}<0$. So the proof of Lemma \ref{lem5} gives the desired conclusion which completes the proof of the corollary.
\end{proof}

\begin{cor}
\label{lem17}
Define
$$
\rho^*:=\max\limits_{b\in \Gamma} \{\| \rho_b\|_{\infty} \}.
$$
Then $\rho^*<\infty.$
\end{cor}
\begin{proof}
The proof follows from  Lemma \ref{lem6} and \ref{lem7} by replacing the term $E_b(\rho_{\pi_0})$ and $k_0$ with $F_{b_0}$ and $K_0$ respectively.
\end{proof}

\begin{cor}
\label{lem18}
For any $b\in \Gamma$,
$$\|B\rho_b \|_{\infty}\geq -2F_{b_0}/M,$$
and there exists a uniform constant $R_0$ such that any $\rho_b$ obtained in Theorem \ref{thm1} vanishes for $|x|\geq R_0$.
\end{cor}
\begin{proof}
A similar argument to Lemma \ref{lem9} gives the lower bounds of $\|B\rho_b\|_{\infty}$.
With this and $b\in\Gamma\subset[\frac{1}{\xi},\xi]$, the proof of Theorem 2 shows that $R_0$ can be uniformly bounded.
\end{proof}

\begin{proof}[\textbf{Proof of Theorem \ref{thm3}}]
Now  we will prove the continuity of $F_b$.
Fix $b\in\Gamma\subset[\frac{1}{\xi},\xi]$.
Assume that $\rho\in W^b$ and $\bar{b}\in[\frac{1}{\xi},\xi]$,
and define $a=b/\bar{b}$,
$\bar{\rho}=a\rho(x')$, where $x=(x_1,x_2,x_3)$ and $x'=(x_1,x_2,ax_3)$.
Thus, $\rho(x)=\rho(r_b(x))$, $\bar{\rho}(x)=\bar{\rho}(r_{\bar{b}}(x))$ and $\bar{\rho}\in W^{\bar{b}}$.
Denote
\begin{align*}
E_{b,1}&:=\int A(\rho(x))T\Big(n_{b,\rho}(r_b(x))\Big) dx;\\
E_{\bar{b},1}&:=\int A(\bar{\rho}(x))T\Big(n_{\bar{b},\bar{\rho}}(r_{\bar{b}}(x))\Big)dx
=\int A(a\rho(x'))T\Big(n_{b,\rho}(r_b(x'))\Big) a^{-1}dx';\\
E_{b,2}&:=\int \rho(x)L\Big(m_{\rho}(\eta(x))\Big)\eta^{-2}(x)dx;\\
E_{\bar{b},2}&:=\int \bar{\rho}(x)L\Big(m_{\bar{\rho}}(\eta(x))\Big)\eta^{-2}(x) dx\\
&=\int a\rho(x')L\Big(m_{\rho}(\eta(x'))\Big)\eta^{-2}(x') a^{-1} dx'=E_{b,2};
\end{align*}
\begin{align*}
E_{b,3}&:=\iint \dfrac{\rho(x)\rho(y)}{|x-y|} dx dy
=\int\dfrac{\rho(x')\rho(y')}{|x'-y'|}dx'dy';\\
E_{\bar{b},3}&:=\iint \dfrac{\bar{\rho}(x)\bar{\rho}(y)}{|x-y|} dx dy=\iint \dfrac{\rho(x')\rho(y')}{|x-y|}dx'dy'.
\end{align*}
Note that $A'(s)\leq A'(\rho^*)$ when  $s\in[0,\rho^*]$ and $|\frac{1}{|x-y|}-\frac{1}{|x'-y'|}|\leq C_1\frac{1}{|x'-y'|}$, where $C_1$ is a continuous function of $a$.

Now set $\rho=\rho_b$. It holds that
\begin{equation}
\label{3.4}
\begin{aligned}
|E_{\bar{b},3}-E_{b,3}|& \leq C_1 E_{b,3}\\
&\leq C_1'M^{2/3}K_0^{4/3}\\
|E_{\bar{b},1}-E_{b,1}|& \leq T_0\int C_2 |A(a\rho(x))-A(\rho(x))| dx\\
& \leq C_2A'(\rho^*) T_0 \int |a-1|\rho^* dx\\
&\leq C_2'A'(\rho^*)T_0 \rho^* (\frac{4}{3}\pi R_0^3),
\end{aligned}
\end{equation}
here $C_1, C_1', C_2$ and $C_2'$ are all continuous functions of $a$.
This implies that for any $\epsilon>0$, there exists $\delta>0$ such that any $\bar{b}$ satisfying $|b-\bar{b}|<\delta$,
$$
F_{\bar{b}}-F_{b}= F_{\bar{b}}-E_b(\rho_b)\leq E_{\bar{b}}(\bar{\rho})-E_{b}(\rho)<\epsilon.
$$
Note that the terms in the right hand side of equation \eqref{3.4} do not depend on $\rho_b$, but depend only on $a$.
So let $\bar{\rho}=\rho_{\bar{b}}$.
Then a similar argument shows that
$$F_{\bar{b}}-F_{b}\geq E_{\bar{b}}(\bar{\rho})-E_{b}(\rho)>-\epsilon.$$
Therefore the continuity of $F_b$ on $\Gamma$ is proved and thus Theorem \ref{thm3} is proved.
\end{proof}

\begin{remark}
If the following claim holds,
then the condition $b\in [\frac{1}{\xi},\xi]$ in \eqref{W} can be removed.

\begin{enumerate}
\item[]
\begin{tabular}[t]{|p{10cm}}
  The maximum $\|\int_{R^3}\frac{\rho(y)}{|x-y|} dy\|_{\infty} $
  subject to the constraints of $\rho$
    \begin{equation*}
    \ 0\leq \rho(y)\leq \rho^*, \ \rho(x)=\rho(r_b(x)),\ \int \rho(y) dy =M.
    \end{equation*}
  is attained by $x=0$, $\rho=\rho^* \chi_{B_0}$
  , where $B_0$ is an ellipsoid with center $(0,0,0)$ and semi-principal axes of length $r_b^*$, $r_b^*$ and $br_b^*$, and $\frac{4}{3}\pi b(r_b^*)^3\rho^*=M$.
\end{tabular}
\end{enumerate}

Indeed a direct calculation using changing variables gives the maximum $I^*$:
        \begin{equation*}
		I^*=\int_{R^3}\frac{\rho^*\chi_{B_0}(y)}{|y|} dy=\begin{cases}
		\left(\dfrac{9\pi M^2\rho^*}{2} \right)^{1/3}\dfrac{b^{1/3}}{\sqrt{b^2-1}}\ln \Big(\sqrt{b^2-1}+b\Big) & \text{if}\ b>1\\
		\left(\dfrac{9\pi M^2\rho^*}{2} \right)^{1/3}& \text{if}\ b=1\\
		\left(\dfrac{9\pi M^2\rho^*}{2} \right)^{1/3}\dfrac{b^{1/3}}{\sqrt{1-b^2}}\arcsin\left(\sqrt{1-b^2} \right) & \text{if}\ b<1
		\end{cases}.
		\end{equation*}
$I^*$ tends to $0$ as $b$ tends to $0$ or $\infty$, which contradicts to Corollary \ref{lem18}.
Therefore, the bound $[\frac{1}{\xi},\xi]$ for $b$ is naturally satisfied if the above claim holds.
\end{remark}

\vspace{0.8cm}
\textbf{Acknowledgement} The author would like to express gratitude to Professor Zhouping Xin for his constructive suggestions on this paper.
\clearpage

\bibliographystyle{amsplain}


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PDF Download The Motive: Why So Many Leaders Abdicate Their Most Important Responsibilities Full Online
Arlean31Acuff
Leaders,
Shay,
Liam,
Lencioni,
Abdicate,
Responsibilities,
Fable,
Readers,
Onlinedescription
[PDF]Download The Motive: Why So Many Leaders Abdicate Their Most Important Responsibilities Read Online Details
Details Product: Shay was still angry but shrugged nonchalantly as if to say, it's not that big of a deal. So, what am I wrong about?You're not going to want to hear this, but I have to tell you anyway. Liam paused before finishing. You might be working hard, but you're not doing it for the company.What the hell does that mean? Shay wanted to know.Knowing that his adversary might punch him for what he was about to say, Liam responded. You're doing it for yourself.New York Times bestselling author Patrick Lencioni has written a dozen books that focus on how leaders can build teams and lead organizations. In The Motive, he shifts his attention toward helping them understand the importance of why they're leading in the first place.In what may be his most compelling fable to date, Lencioni introduces readers to a pair of new characters, Shay Davis and Liam Alcott, rival leaders of two very different organizations in the same industry. With sharp dialogue and unexpected plot twists, he takes us on a journey that culminates in a resolution that is as unpredictable as it is enlightening.Beyond the fictional story, Lencioni provides a straightforward summary of the lessons from the fable, combining a clear explanation of his theory with practical advice and a simple self-assessment. In doing so, he teaches his readers how to avoid the most dangerous temptation that derails leaders and hurts the people they are meant to serve.
PDF Download The Motive: Why So Many Leaders
Abdicate Their Most Important Responsibilities
Description PDF Download The Motive: Why So Many
Leaders Abdicate Their Most Important
Responsibilities Full Online
Shay was still angry but shrugged nonchalantly as if to say, it's not that big of a deal.
So, what am I wrong about?You're not going to want to hear this, but I have to tell
you anyway. Liam paused before finishing. You might be working hard, but you're not
doing it for the company.What the hell does that mean? Shay wanted to
know.Knowing that his adversary might punch him for what he was about to say,
Liam responded. You're doing it for yourself.New York Times bestselling author Patrick
Lencioni has written a dozen books that focus on how leaders can build teams and
lead organizations. In The Motive, he shifts his attention toward helping them
understand the importance of why they're leading in the first place.In what may be
his most compelling fable to date, Lencioni introduces readers to a pair of new
characters, Shay Davis and Liam Alcott, rival leaders of two very different
organizations in the same industry. With sharp dialogue and unexpected plot twists,
he takes us on a journey that culminates in a resolution that is as unpredictable as it
is enlightening.Beyond the fictional story, Lencioni provides a straightforward
summary of the lessons from the fable, combining a clear explanation of his theory
with practical advice and a simple self-assessment. In doing so, he teaches his
readers how to avoid the most dangerous temptation that derails leaders and hurts
the people they are meant to serve.
Page 2 and 3: PDF Download The Motive: Why So Man
Magazine: PDF Download The Motive: Why So Many Leaders Abdicate Their Most Important Responsibilities Full Online

redpajama_set_name: RedPajamaCommonCrawl


My new project I Love this City | Collection which celebrates individual cities across the globe will launch on 6th June at the iconic The Strawberry, Newcastle. The collection features my city-inspired Biro drawings and includes a range of high quality eco-friendly t-shirts, posters and limited edition prints. Ceramic designs will join the collection in the not too distant future. Products from the collection will be available to purchase from my online store soon.
Kick-starting the project in my adopted backyard of North East England, I have produced original drawings which celebrate the cities of Newcastle and Sunderland. Sourcing some of the best manufacturers and artisans in the UK to make the products, luxurious 100% organic brushed cotton t-shirts, ethically sourced, digitally printed using eco-friendly inks are available in men's, women's and children's fits. Posters are printed on 200gsm satin paper and limited edition prints are crafted by one of the best print makers in the UK.
Every city I visit, no matter how small or big has beauty, accomplishment and always spirit in abundance. I extract personal and empathetic perspectives from my city experiences, which are inclusive of both citizens and visitors, and aim to create artworks that represent each city's individual spirit.
City drawings are inspired by heraldic coats of arms and woven with imagery depicting treasured landmarks, references to football or sports clubs where appropriate, associated animal mascots, as well as references to heritage industries and exciting facts I uncover. Making each artwork involves weeks of research and several months to create each drawing. Using Biro as a medium is painstaking as there is little room for error!
Staying true to my art practice which explores loss generated by human destruction, I Love this City | Collection also communicates, through the inclusion of animal imagery, that humanity is entwined with the animal kingdom. With the growing global realisation that we need our wildlife, I hope this project will also raise environmental awareness at a local and global level.
Being dual hearted, I'm both a townie and a country girl with an appreciation for metropoles and wildernesses! Having lived in and explored many cities taking in the good, the bad and the awe-inspiring, I forever find myself throwing back my arms in my mind (or in an empty street - I'm shy!) and saying "I love this city!". I got a job in London several years back as a Guard / Emergency Tube Driver to fund my art practice. More specifically, I chose to be a tube driver because I wanted to experience being at the beating heart of London. In my mind London is an entity. I saw life at its best and at its most desperate. I was in awe of London, its history, its community, its beauty and its innovation and I was proud to be a small cog in its wheel. Likewise with great pride I love my hometown Edinburgh, my new home city, Newcastle and the North East.
There's so much more I want to achieve with this project. By producing city artworks that celebrate the positive achievements of civilisation, as well as highlighting the importance of the environment, I aim to support environmental organisations associated with each city, region or country through online limited edition print sales, as well as taking this project to schools in my local area.
Newcastle and Sunderland t-shirts, posters and limited edition prints will be in store soon and are also available for retail. Travel plans are afoot to make artwork for the next city that will join the collection, so watch this space. With so much to celebrate in any city, no city is exempt.


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McCandlish is Scottish surname (and rarely also a given name), derived from Scottish Gaelic  (among other spellings, including  and ), meaning 'son of Cuindlis', an Old Irish name of uncertain meaning that dates to the 8th century.  Some spelling variants include McAndlish, McCanalish, McCandelich, McCandelish, McCandish, McCandlash, McCandleis, and McCandleish, among others. The name is closely related to McCandless, found in Scotland and especially Northern Ireland (as well as the Gaelic diaspora). McCandlish has also sometimes been attested in Northern Ireland.  It is etymologically but probably not familially related to west Irish , 'descendant of Cuindlis'. The earliest form of the given name can be traced back to an abbot from the 8th century called Cuindles.

Surname
 Benjamin McCandlish (1886–1975), United States Navy flag officer; 36th naval governor of Guam
 Edward Gerstell McCandlish (1887–1946), American illustrator, mapmaker, toymaker, and author of the Bunny Tots series of children's books (1920s); perhaps best known for illustrating Laboulaye's Fairy Book
 John Edward Chalmers McCandlish  (1901–1974), British Army major-general
 John MacGregor McCandlish (1821–1901), Scottish lawyer; first president of the Faculty of Actuaries
 Mackey McCandlish, animator and co-creator of Blahbalicious
 Margherita McCandlish  Wood (1892–1954), former first lady of Guam, wife of Benjamin
 William McCandlish, British dog breeder; see List of Chairmen of the Kennel Club

Given name
 McCandlish Phillips (1927–2013), American journalist and evangelist

See also
 Candlish, a derived Scottish surname
 McCandless (surname), a more common Scots-Irish form of the name, primarily found in Northern Ireland
 Ó Cuindlis, a west Irish surname dating to the 14th century; anglicized as Conlisk, Cundlish, Quinlist, and several other variants

References

Surnames
Surnames of Scottish origin

redpajama_set_name: RedPajamaWikipedia


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PAS-Advantages
Receive our Press Releases
Akari Therapeutics Plc and XL-protein GmbH Sign License Agreement to Develop a Long Acting Version of Coversin Using PASylation® Technology
14. June 2016 /in News /by web28627236
NEW YORK, LONDON, and FREISING, Germany, April 14, 2016 – Akari Therapeutics (NASDAQ: AKTX), an emerging growth, development-stage biopharmaceutical company, and XL-protein, a privately owned biopharmaceutical company, announced today that they have entered into a License, Development and Commercialization Agreement. This partnership is focused on developing a second generation, longer acting version of Coversin.
Under this collaboration agreement, XL-protein will apply its proprietary PASylation® technology for drug half-life extension to Coversin. Coversin is a second-generation complement inhibitor that acts on complement component-C5, preventing release of C5a and formation of C5b-9 (also known as the membrane attack complex or MAC).
XL-protein has previously demonstrated that it can manufacture recombinant PASylated Coversin. An initial study conducted in a mouse model indicates that PASylated Coversin administered by subcutaneous injection remains fully active, with the high C5 binding activity of Coversin retained. In this study, it was found that PASylation of Coversin extended the plasma half-life by over 50 fold.
"XL-protein's PASylation technology provides an elegant approach to extending the half-life of Coversin, which we hope to demonstrate in future studies will reduce the frequency of dosing," said Miles Nunn, Chief Scientific Officer of Akari Therapeutics. "Our current plan is to investigate the relative performance of PASylated Coversin, administered by the subcutaneous route, in animal models of disease. If successful, we expect to progress PASylated Coversin to the clinic.""
"We are delighted to enter into the partnership with Akari Therapeutics; the data from the initial study indicate that PASylation could lead to a longer acting, less frequently dosed, subcutaneous version of Coversin", commented Claus Schalper, Chief Executive Officer of XL-protein. "XL-protein successfully continues to add collaborations with renowned partners that can leverage our best-in-class half-life extension technology."
Under the terms of the agreement, XL-protein will receive an upfront payment as well as payments for achievement of preclinical, clinical, regulatory and commercial milestones. Furthermore, XL-protein will receive royalties on sales from marketed compounds resulting from the collaboration. Further financial terms have not been disclosed.
About XL-protein GmbH:
XL-protein is a German biotech company commercializing the ground-breaking PASylation® technology, which enables the design of biopharmaceuticals with extended plasma half-life and enhanced action. With its strong proprietary technology position XL-protein focuses at the preclinical as well as clinical development of PASylated proteins in various disease areas. The company is located at Freising, Germany, in the neighbourhoods of Munich International Airport and the Technical University of Munich.
(www.xl-protein.com)
About Akari Therapeutics Plc:
Akari is a clinical-stage biopharmaceutical company focused on the development and commercialization of innovative therapeutics to treat orphan autoimmune and inflammatory diseases. Akari's lead drug, Coversin is a second-generation complement inhibitor that acts on complement component-C5, preventing release of C5a and formation of C5b-9 (also known as the membrane attack complex or MAC). C5 inhibition is growing in importance in a range of rare autoimmune diseases related to dysregulation of the complement component of the immune system, including Paroxysmal Nocturnal Hemoglobinuria (PNH), atypical Hemolytic Uremic Syndrome (aHUS), and Guillain Barré syndrome (GBS).
(www.akaritx.com)
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Ajinomoto and XL-protein forge Strategic Alliance to develop PASylated therapeutics applying the Corynex® platform 18. December 2018
Horizon Pharma plc and XL-protein GmbH have entered into a collaboration agreement on a potential next-generation biologic for uncontrolled gout using PASylation® technology 7. January 2018
AdAlta and XL-protein execute commercial license agreement 13. November 2017
Partnering contact:
XL-protein GmbH
Claus Schalper
85354 Freising
Phone: +49 (0) 8161 53730-90
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Social Exclusion: bullying, stigma, and being different.
As members of the human species we are, by our very nature, social animals. We evolved living in groups, our hunter gatherer lifestyle meant that we were seldom, if ever, alone. Because of this we tend to go into a physical and mental decline if we are denied contact with our fellow humans. A strong, supportive social network of family and friends is one of the main protectors against stress.
It should therefore come as no surprise, that research has demonstrated strong emotional responses to both being excluded from the group, or being forced to exclude others from the group. This could be something as simple as being briefly involved in a game with strangers in the park, only for them to suddenly just carry on as if you were not there, to the deliberate ostracism that occurs when bullies get to work in schools, or whole societies make artificial divisions among people. These emotional responses can also come into play where groups are socially excluded or isolated as part of a deliberate policy, or unintentionally, when forces such as institutional racism come into play.
There are very good reasons why solitary confinement has been used as a punishment or form of torture for thousands of years.
In schools this sense of exclusion or social isolation can occur for many reasons. It can be produced by the deliberate ignoring, or scapegoating of a child by teachers or other students. Powerful figures in the student community may coerce others to join in and exclude others as well. This sense of exclusion can be brought about in many ways. This might be achieved through spreading rumours, calling the other person names or deliberately setting out to embarrass them. Something that the current love affair with social media makes even easier than ever before.
This deliberate social exclusion affects the normal development of children and adolescents in many areas of their lives, compromising both their physical and mental health. Social connectedness is important at any age, but this is especially so during our childhood development.
In adolescents social exclusion has particular effects on the brain's emotion regulating circuits, with the anterior cingulate cortex and other brain centres that have a role in emotional regulation, such as the ventrolateral prefrontal cortex, being particularly affected. The anterior cingulate cortex reacts to peer rejection by becoming more active – leading to strong emotional responses, while other areas, such as the ventrolateral prefrontal cortex, that play a role in with how we regulate, and control our emotions become less active. This leads to strong emotional reactions which are exacerbated because they are only poorly controlled. This kind of response by the brain's circuitry is particularly noticeable in adolescents, and highlights the significant role that social inclusion and a strong, supportive peer group have on normal adolescent social and personal development.
Adolescence is a time where reliance on our family is gradually superseded by reliance on our peer group. While we learn how to be a human being from being part of our family and our culture, it is more in relationship to our peer group that we develop our sense of identity. This is further compounded by the state of flux concerning what is in, or what is out, that surrounds many teenage social groupings.
Research undertaken at Edith Cowan University in 2009, gave an estimate of 1 in 6 children reporting that they had experienced this kind of social exclusion. The researchers felt that this was likely to be an understimate given the difficulties involved in measuring social exclusion because it is often covert. Social exclusion has an impact on the physical, emotional and mental health of individuals to whom it has happened. It also interferes with the ability to form enduring , close relationships.
So what are the main signs that a child or adolescent is socially excluded?
Lower immune function with the associated tendency to get any virus or other illness that is going around.
Poor sleep quality, both in relation to the anxiety that is present and to a tendency to lie awake, ruminating on what is happening.
Where chronic bullying is occurring, there is also a strong tendency to develop low self worth and self esteem. Young people start to think that there must be something wrong with them, and that they are responsible for the situation in which they find themselves.
Poor emotional regulation can lead to severe anxiety, depression and aggressive outbursts.
Our social nature means that we need to be connected to other people, and to feel loved and valued for ourselves. This needs us to develop lasting, positive reltionships, a skill that is impaired in this group of individuals. We need to help our children form secure relationships within the family, amongst their peers and in the wider community. Even one good relationship can be protective. From an early age, young people become aware of social rejection and can explain why it is wrong not to include other people in what is going on in the various communities to which they belong. It is important to help our children understand the importance of including others. Talking about how other people might feel in such situations and discussing how to make stronger social ties can help.
Should we find ourself in the position of having a child that has been socially excluded, for whatever reason, there are things that we can do, as parents, to help.
First of all we need to ensure that our children know that we are available when they want to talk. When they choose to do so, we need to be open to what they say and calm in our responses. Children worry about upsetting their parents so it is important to be engaged with them and not judgemental or show too much distress.
Be responsive to what the child says. Make sure that they know that they are allowed to feel safe, and that we will help bring this about by talking to the school or doing whatever else is necessary. With adolescents, ask what they would like us to do to help, and make a joint plan of what to do if things do not improve or change. Negotiating when we will take over as parents but allow enough time for them to address the issue in their own way first.
Make sure that they feel valued, apreciated and loved. Highlight their gifts and abilities. Engage other family members and friends to create a "circle of security" around them.
Make sure that the home is a safe haven. This is especially important in this modern era of social media where bullying no longer stops at the school gates. We may need to negotiate reduced access to the internet, setting parental controls or privacy settings, helping to block people as necessary and perhaps helping to get new e-mail adresses and phone numbers.
Help with emotional regulations. Talk about how they may be feeling, help them to use positive self talk and thinking to manage the feelings and negative thoughts. Help them to use their positive talents as a way to cope.
Encourage our childrens' healthy friendships and if necessary suggest that they not see people who are contributing to their difficulties. Find clubs that they can join where they can meet people with similar interests, encourage play dates and sleep overs.
Use the "high five" principle. One thing for each finger! Five people, places or activities that they can use to help feel safe if they are being excluded. A teacher, the library, a friend in another class, perhaps even helping their own teacher tidy up.
Help our child to develop a sense of boundaries around other people's behaviour. "This is not right, I would like you to stop.", "This is bullying. It is wrong and you should stop.", alongside this kind of action we must make sure our children also know where they can seek support and safety.
Whatever the reason that this kind of social exclusion occurs, be it from bullying at school, racial prejudice or perhaps belonging to a minority group, or even being old and infirm and unable to get out and join in, it needs to be addressed both by the individual and their supporting environment. The wider social issues need to be addressed in the wider community. Intolerance is a widespread community issue that requires education, and postive action to help eliminate it.
Strong, supportive communities can provide the support we need to allow ourselves and our children to flourish. Because they help to add a sense of belonging and cohesion to our lives they can help to counteract the negative effects of bullying and other forms of discrimination.
So let us not forget that as human beings we are social creatures that rely upon our communities for our very survival, and do what we can to help those whose communities have let them down.
Categories: Adolescence, Bullying, Living happy, prevention, Resilience | Permalink.


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Catalog Academic & College Policies A-Z CSCU & Campus Policies Systemwide Transfer Appeals Committee
The purpose of the Transfer Appeals Committee is to:
Provide a review means for a student, transferring from one community college to another within the Connecticut system of community colleges, who believes the receiving college has not allowed adequate credit for a course or courses
Provide an established procedure which shall ensure the fulfillment of Board of Regents policy regarding transfer of credit
The Transfer Appeals Committee will consist of a chairperson (the Academic Officer of the Community College system) and two members (deans selected from a pool of academic deans and deans of students).
Neither dean serving for a particular review shall be from either of the colleges involved in the appeal. The pool shall consist of four academic deans selected annually by the Council of Academic Deans and four deans of students selected annually by the Council of Deans of Students. No dean shall serve more than three consecutive years. After the initial selection of deans, the executive officer will hold a drawing to determine that initially four deans shall hold a two-year service period in order to assure accumulation of some experience by a portion of the available deans.
When a community college student transferring from one of the twelve community colleges to another concludes, after exhausting other means of review, that he or she is not being allowed sufficient credit for a course or courses he or she wishes transferred, the student may use this procedure as a final review and decision.
Before invoking this procedure the student shall have:
appealed the decision of the evaluator of transfer credit at the receiving college to the appropriate dean at the college, and if dissatisfied with the decision then
made an appeal of the decision of the dean to the president of the receiving college.
If the student receives an unsatisfactory decision at the college level, he or she may then appeal the decision of the president to the Academic Officer for the Community Colleges who shall convene the Transfer Appeals Committee for purposes of considering the appeal.
The Transfer Appeals Committee shall make a recommendation to the executive officer who shall render a final decision.
When a tie vote occurs by the two deans, the chair of the Transfer Appeals Committee shall break the tie by casting a vote.>
A report shall be made to the Academic Policies Committee of the Board in the case of such appeals reviewed and acted upon by the Transfer Appeals Committee and the executive officer.
Experiential Learning Credits
In this area, the College awards credit for learning or competency acquired through non-collegiate experience when that competency is equivalent to courses which the College offers. This is consonant with board policy on non-traditional learning. The student may apply for credit under one or more of the following procedures, given in the preferred order of selection:
Credit for programs by non-collegiate organizations
credits for life experience in lieu of:
field work requirements, or cooperative work experience requirements
College Level Examination Program (CLEP)
This testing program, administered by the College Entrance Examination Board, is designed to measure college level learning acquired outside the college classroom. Two types of CLEP examinations are offered:
The General Examinations provide a comprehensive measure of undergraduate achievement in five basic areas of liberal arts education: English Composition, Humanities, Mathematics, Natural Sciences, and Social Science/History.
The Subject Examinations measure achievement in specified undergraduate courses. These tests measure the understanding of fundamental facts and concepts, as well as the ability to apply such understanding to the solution of problems and the interpretation of materials.
For additional information, contact the office of the Dean of Students.
American College Testing Proficiency Examinations (PEP)
The ACT Proficiency Examination Program consists of 47 examinations designed for the recognition of learning acquired outside the classroom. The PEP examinations are developed under the auspices of the Regents of the State University of New York and are used to fulfill degree requirements of the Regents External Degree Program.
PEP Examinations are available in the areas of arts and sciences, business, criminal justice, education, health and nursing. Each examination is designed to measure knowledge and competencies a person might gain in non-traditional ways.
Military Subject Standardized Tests (SSTs)
The Office on Education Credit of the American Council on Education (ACE) evaluates USAFI and DANTES military subject standardized tests and makes academic credit recommendations as an advisory service to those interested in applying for or granting credit for educational tests taken in the armed services. These college-level exams, offered only to military personnel, show general educational development. MxCC will grant degree or certificate credits for equivalent courses or electives based on ACE recommendations.
For additional information, contact the Admissions Office.
Advanced Placement Program (APP)
The Advanced Placement Program of the College Entrance Examination Board allows advanced high school students to take freshman-level courses in the latter days of their secondary schooling and receive credit for them in college. End-of-year Advanced Placement Examinations consist generally of two parts: objective and free response. Composite grades are assigned on a 5-point scale.
For further information contact the Admissions Office.
Challenge Examinations
At the discretion of the appropriate division chairperson and the Dean of Academic Affairs, a student may take a challenge examination in subject areas offered at MxCC not included in CLEP or PEP. The student must first apply to the dean and the division chairperson. If approved and a faculty member agrees to administer the exam, the instructor will prepare an examination that is presented to the department for approval. Upon such approval, the faculty member will administer the exam.
Credit will be granted with the grade of "Pass" only and will be treated in the same manner as transfer credit.
An Extension Fund Academic Evaluation Fee of $15.00, as approved by the Board of Regents, will be charged for preparation, administration, and evaluation of each challenge examination.
Credit for Prior Learning through Portfolio Development
Students who plan to apply for such credit must enroll in a four-credit course entitled SPT0 122: Special Topics–Portfolio Development. The student develops a portfolio in which he or she describes the learning acquired through prior experiences, specifies learning outcomes, provides appropriate documentation, and requests college credit for that learning. An assessment committee reviews and evaluates the portfolio and then determines how many credits the student should receive. The credits gained through this evaluation process are applicable towards an associate degree at MxCC.
No credit shall be awarded via portfolio review outside of the subject areas encompassed by the approved curricula of the institution. No more than 50 percent of the credits required for a degree can be satisfied with credits awarded for prior experiential learning. For additional information contact the Dean of Academic Affairs at 860-343-5866.
Criminal justice students employed in criminal justice occupations may, with permission of the coordinator of the Criminal Justice program, enroll in CJS*298 -Special Topics in Criminal Justice-Portfolio Development. This three credit course provides students with the opportunity to receive college level credit for life/work experience.
Credit for Programs or Courses Offered by Non-Collegiate Organizations
Military Service School Courses and Military Occupational Specialties
MxCC gives credit for military training, experiences, and military occupational specialty (MOS) qualifications. The College follows the credit recommendations in the "Guide to the Evaluation of Educational Experiences in the Armed Services" developed by the Office on Education Credit of the American Council on Education.
Credit Recommended by The National Guide to Credit Recommendations of Non-collegiate Courses
MxCC grants credit for courses listed in "The National Guide to Credit Recommendations for Non-collegiate Courses" published by the American Council of Education. The ACE guide reports credit recommendations for approximately 1,000 courses offered by 80 organizations nationally, including business and industry, labor unions, government agencies, and professional and voluntary associations. The recommendations are based on evaluations conducted as part of ACE's Program on Non-collegiateSponsored Instruction by teams of faculty experts.
Credit Recommended by A Guide to Educational Programs in Non-collegiate Organizations
MxCC grants credit for courses listed in "A Guide to Educational Programs in Non-collegiate Organizations" published by the University of the State of New York Office on Non-collegiate Sponsored Instruction. (The Program on Non-collegiate Sponsored Instruction is conducted in New York State and other northeastern states by the State University of New York. The purpose of the program is to review formal educational programs and courses sponsored by non-collegiate organizations and to make appropriate credit recommendations. The 1977 edition of the Guide reports on 761 courses conducted by 67 organizations.)


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Billy Corgan had to Push for Final Deletion
By Amish Patel - September 2, 2016
New TNA President Billy Corgan was recently interviewed on Vince Russo's The Brand. Corgan talked about having to push for the infamous Final Deletion segment involving Matt and Jeff Hardy:
"The biggest thing I had to do was push our creative, the office, Dixie, to take a chance, to go outside of the box which I know you were a big fan of. The reaction to that first angle, with Jeff going through the table, was pretty 50-50. A lot of people were howling, saying this may be the best-worst thing they've ever seen in wrestling. The day that came out I was inside the company, on my e-mail saying – We've got to do a bigger version than we just did – we have lightning in a bottle. And, there were people inside the company that were afraid to move in that direction. I pushed, and pushed and pushed for what was the Final Deletion. And now you have fans on other products chanting, "Delete, Delete, Delete."
(All transcribed quotes courtesy of Wrestling Inc.)
Tags: Billy Corgan