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\markboth{\hfil Uniqueness for radial minimizers  \hfil EJDE--2002/43}
{EJDE--2002/43\hfil Yutian Lei \hfil}
\begin{document}

\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc  Electronic Journal of Differential Equations},
Vol. {\bf 2002}(2002), No. 43, pp. 1--11. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp  ejde.math.swt.edu  (login: ftp)}
 \vspace{\bigskipamount} \\
 %
 Uniqueness for radial Ginzburg-Landau type minimizers
 %
\thanks{ {\em Mathematics Subject Classifications:} 35J70, 49K20.
\hfil\break\indent
{\em Key words:} radial minimizer, uniqueness, Ginzburg-Landau type functional.
\hfil\break\indent
\copyright 2002 Southwest Texas State University. \hfil\break\indent
Submitted February 4, 2002. Published May 21, 2002.
} }
\date{}
%
\author{Yutian Lei}
\maketitle


\begin{abstract}
 We prove the uniqueness of radial minimizers of a
 Ginzburg-Landau type functional. We present also
 an analysis of the  the location of the zeros of
 the radial minimizer.
\end{abstract}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\numberwithin{equation}{section}


\section{Introduction }

For $n \geq 2$, let $B=\{x \in \mathbb{R}^n;|x|<1\}$ and $\partial B$
its boundary.
On this domain, we find minimizers to the Ginzburg-Landau-type
functional
$$
E_\varepsilon(u,B)=
\frac{1}{p}\int_B|\nabla u|^p
+\frac{1}{4\varepsilon^p}\int_B(1-|u|^2)^2,\quad (p\geq n)
$$
on the class of functions
$$
W=\{u(x)=f(r)\frac{x}{|x|} \in W^{1,p}(B,\mathbb{R}^n);
f(1)=1,  r=|x|\}.
$$
Such minimizer is denoted by $u_{\varepsilon}$ and is
called a {\it radial minimizer}.

Many authors have studied the existence, uniqueness and asymptotic behaviour
of $u_{\varepsilon}$ as $\varepsilon \to 0$.
For $p=n=2$, studies of asymptotic behaviour can be found in \cite{b1,s1},
studies of uniqueness in \cite{h2}, and other related topics in
\cite{b2,b3,f1,m1}.
For $p=n>2$ and $p>n=2$, the asymptotic behaviour was studied in
\cite{h1}  and \cite{l2}, respectively.
However, uniqueness was not mentioned there.
In this paper, we prove the following results for  $p\geq n$.

\begin{theorem} \label{thm1.1}
Assume $u_\varepsilon$is a radial minimizer  of
$E_\varepsilon(u,B)$. Then for any given $\eta \in (0,1/2)$
there exists a positive constant $h=h(\eta)$ such that
$$
Z_\varepsilon=\{x \in B;
|u_\varepsilon(x)|<1-\eta\} \subset B(0,h \varepsilon)
=\{x \in \mathbb{R}^n;|x| <h\varepsilon\}.
$$
\end{theorem}

\begin{theorem} \label{thm1.2}
The  radial minimizer of $E_{\varepsilon}(u,B)$ on $W$ is unique.
\end{theorem}

This article is organized as follows:
In \S2, we present some basic properties of minimizers.
In \S3, we prove  Theorem 1.1 which implies, in particular,
that the zeros of $u_\varepsilon$ are contained in $B(0,h\varepsilon)$.
We conclude with the proof of Theorem 1.2 in \S4.

\section{ Preliminaries}

For a function $u(x)=f(r)\frac{x}{|x|}$, written in polar coordinates,
we have
\begin{gather*}
|\nabla u|=(f_r^2+(n-1)r^{-2}f^2)^{1/2},
\quad \int_B|u|^p=|S^{n-1}|
\int_0^1r^{n-1}|f|^p \,dr, \\
\int_B|\nabla u|^p =|S^{n-1}| \int_0^1r^{n-1}
(f_r^2+(n-1)r^{-2}f^2)^{p/2} \,dr.
\end{gather*}
It is easily seen that $f(r)\frac{x}{|x|} \in W^{1,p}(B,\mathbb{R}^n)$ 
implies
$f(r)r^{\frac{n-1}{p}-1}$ and $f_r(r)r^{\frac{n-1}{p}}$ are in
$L^p(0,1)$.
Conversely, if $f(r)$ is in $W_{{\rm loc}}^{1,p}(0,1]$, and
$f(r)r^{\frac{n-1}{p}-1}$, $f_r(r)r^{\frac{n-1}{p}}$ are in $L^p(0,1)$, then
$f(r)\frac{x}{|x|} \in W^{1,p}(B,\mathbb{R}^n)$. Thus if we denote
$$
V=\{f \in W_{{\rm loc}}^{1,p}(0,1];r^{\frac{n-1}{p}}f_r\in L^p(0,1),
r^{(n-1-p)/p}f\in L^p(0,1),
f(1)=1\},
$$
then
$V=\{f(r);u(x)=f(r)\frac{x}{|x|} \in W\}$.

\begin{proposition} \label{prop2.1}
The set $V$ defined  above is a subset of $\{f
\in C[0,1]; f(0)=0\}$.
\end{proposition}

\paragraph{Proof.}
Let $f \in V$. If $p>n$, let $h(r)=f(r^{\frac{p-1}{p-n}})$.
Then
\begin{eqnarray*}
\int_0^1|h'(r)|^p\,dr
&=&(\frac{p-1}{p-n})^p\int_0^1|f'(r^{
\frac{p-1}{p-n}})|^p r^{\frac{p(n-1)}{p-n}}\,dr \\
&=&(\frac{p-1}{p-n})^{p-1}
\int_0^1s^{n-1}|f'(s)|^p\,ds<\infty
\end{eqnarray*}
by noting $f_s(s)s^{(n-1)/p} \in L^p(0,1)$.

If $p=n$, let $h(r)=f(r^x)$ with $x>1$ to be determined later.
Then for any $y \in (1,p)$,
\begin{eqnarray*}
\int_0^1|h'(r)|^ydr&=&x^y\int_0^1|f'(r^x)|^yr^{(x-1)y}dr\\
&=&x^{y-1}\int_0^1|f'(s)|^ys^{(x-1)(y-1)/x}ds,
\end{eqnarray*}
where $s=r^x$. Choose $x,y$ such that
$(1-\frac{1}{x})(1-\frac{1}{y})=\frac{n-1}{n}$.
Hence
\begin{eqnarray*}
\int_0^1|h'(r)|^ydr&=&
x^{y-1}\int_0^1|f'(s)|^ys^{y(n-1)/n}ds\\
&\leq& x^{y-1}(\int_0^1|f'(s)|^ns^{n-1}ds)^{y/n}<\infty.
\end{eqnarray*}
Using an interpolation inequality and Young inequality,
$\|h\|_{W^{1,y}((0,1),R)}<\infty$
which implies that $h(r) \in C[0,1]$ and hence $f(r)\in C[0,1]$.

Suppose $f(0)>0$, then $f(r) \geq s>0$ for $r \in [0,t)$ with $t>0$
small enough since $f \in C[0,1]$. We have
$$
\int_0^1 r^{n-1-p}f^p \,dr
\geq s^p \int_0^t r^{n-1-p} \,dr=\infty,
$$
which contradicts $r^{(n-1)/p-1}f
\in L^p(0,1)$. Therefore $f(0)=0$ and the
proof is complete.

\begin{proposition} \label{prop2.2}
The functional
$E_\varepsilon(u,B)$
achieves its minimum on W by a function
$u_\varepsilon(x)=f_\varepsilon(r)\frac{x}{|x|}$.
\end{proposition}

\paragraph{Proof.}
Note that $W^{1,p}(B,\mathbb{R}^n)$ is a reflexive Banach space
and $E_\varepsilon(u,B)$ is weakly
lower-semicontinuous. To prove the existence
of minimizers of $E_\varepsilon(u,B)$
in W, it suffices to verify that W is
a weakly closed subset of $W^{1,p}(B,\mathbb{R}^n)$. Clearly W is a convex subset
of $W^{1,p}(B,\mathbb{R}^n)$. Now we prove that $W$ is a closed subset of
$W^{1,p}(B,\mathbb{R}^n)$.
Let $u_k=f_k(r)\frac{x}{|x|} \in W$ and
$$
\lim_{k \to \infty}u_k=u, \quad\mbox{in }W^{1,p}(B,\mathbb{R}^n).
$$
By the embedding theorem there exists
a subsequence $u_k=f_k(r)\frac{x}{|x|}$ such that
$$
\lim_{k \to \infty}f_k=f, \quad\mbox{in }C(0,1]
$$
and $u =f(r)\frac{x}{|x|}$.
Combining this with $f_k(1)=1$,
we see that $f(1)=1$.
Thus $u \in W$.

\begin{proposition} \label{prop2.3}
The minimizer $u_{\varepsilon}$
is a weak radial solution of
\begin{gather}
-\mathop{\rm div}(|\nabla u|^{p-2}\nabla u)=\frac{1}{\varepsilon^p}u(1-|u|^2),
\quad \mbox{on } B, \label{e2.1}\\
u|_{\partial B}=x. \label{e2.2}
\end{gather}
\end{proposition}

\paragraph{Proof.}
Denote $u_{\varepsilon}$ by $u$. For any $t \in [0,1)$and $
\phi=f(r)\frac{x}{|x|} \in C_0^{\infty}(B,\mathbb{R}^n)$,
we have $u+t\phi \in W$ as long
as $t$ is small sufficiently. Since $u$ is a minimizer
we obtain
$$
\frac{dE_{\varepsilon}(u+t\phi,B)}{dt}|_{t=0}=0,
$$
namely,
\begin{align*}
0=&\frac{d}{dt}|_{t=0}\int_B
(\frac{1}{p}|\nabla (u+t\phi)|^p
+\frac{1}{4\varepsilon^p}(1-|u|^2)^2)dx\\
=&\int_B|\nabla u|^{p-2} \nabla u \nabla \phi dx
-\frac{1}{\varepsilon^p}
\int_Bu\phi (1-|u|^2) dx.
\end{align*}
By a limit process we see that the test function $\phi$ can be any
member of $\{\phi=f(r)\frac{x}{|x|}
\in W^{1,p}(B,\mathbb{R}^n); \phi|_{\partial B}=0\}$.

As in \cite[Lemma 2.2]{h1}, we also have the following statement.

\begin{proposition} \label{prop2.4}
Let $u_\varepsilon$ be a weak radial solution of (\ref{e2.1})-(\ref{e2.2}).
Then $|u_\varepsilon| \leq 1$ on $\overline{B}$.
\end{proposition}

\paragraph{Proof.}
Taking $\phi=u-\frac{u}{|u|}\min(1,|u|)$.
Let $B_+=\{x \in B;|u|>1,a.e. \mbox{ on }B\}$. Noting
$$
\nabla \phi=0,a.e.\quad on \quad B\setminus B_+;
\quad \nabla \phi=\nabla u(1-\frac{1}{|u|})
+\frac{u(u\nabla u)}{|u|^3},a.e.\quad on \quad B_+,
$$
we have
$$
\int_{B_+}|\nabla u|^p(1-\frac{1}{|u|})
+\int_{B_+}|\nabla u|^{p-2}\frac{(u\nabla u)^2}{|u|^3}
+\frac{1}{\varepsilon^p}\int_{B_+}|u|(|u|^2-1)(|u|-1)=0.
$$
This implies that $|B_+|=0$.
Thus $|u_{\varepsilon}|\leq 1$.

\begin{proposition} \label{prop2.5}
Assume $u_{\varepsilon}$ is a weak radial solution of (\ref{e2.1})-(ref{e2.2}).
Then there exist positive constants $C_1,\rho$ which are
independent of $\varepsilon$, such that
\begin{gather}
\|\nabla u_{\varepsilon}(x)\|_{L(B(x,\rho\varepsilon/8))}\leq
C_1\varepsilon^{-1}, \quad\mbox{if}\quad x \in B(0,1-\rho \varepsilon),
\label{e2.3} \\
|u_{\varepsilon}(x)|\geq \frac{10}{11}, \quad\mbox{if}\quad x \in
\overline{B}\setminus B(0,1-2\rho\varepsilon).
\label{e2.4}
\end{gather}
\end{proposition}

\paragraph{Proof.}
Let $y=x\varepsilon^{-1}$ in (\ref{e2.1}) and denote
$v(y)=u(x)$, $B_{\varepsilon}=B(0,\varepsilon^{-1})$.
Then
\begin{equation}
\int_{B_{\varepsilon}}|\nabla v|^{p-2}\nabla v\nabla \phi
=\int_{B_{\varepsilon}}v (1-|v|^2)\phi,
\quad \phi \in W_0^{1,p}(B_{\varepsilon},\mathbb{R}^n).
\label{e2.5}
\end{equation}
This implies that $v(y)$ is a weak solution of
(\ref{e2.5}). By using the standard discuss of the Holder
continuity of weak solution of (\ref{e2.5}) on the boundary
(for example see Theorem 1.1 and Line 19-21 of Page 104 in \cite{c1})
we can see that for any $y_0 \in \partial B_{\varepsilon}$ and $y
\in B(y_0,\rho_0)$ (where $\rho_0>0$ is a constant independent of
$\varepsilon$), there exist
positive constants $C=C(\rho_0)$ and $\alpha \in
(0,1)$, both independent of $\varepsilon$, such that
$$
|v(y)-v(y_0)|\leq C(\rho_0)|y-y_0|^{\alpha}.
$$
Choose $\rho>0$ sufficiently small such that
\begin{equation}
y \in B(y_0,2\rho) \subset B(y_0,\rho_0),
\quad \mbox{and}\quad
C(\rho_0)|y-y_0|^{\alpha}\leq \frac{1}{11},
\label{e2.6}
\end{equation}
then
$$
|v(y)| \geq |v(y_0)|-C(\rho_0)|y-y_0|^{\alpha}
=1-C(\rho_0)|y-y_0|^{\alpha} \geq \frac{10}{11}.
$$
Let $x=y\varepsilon$. Thus
$|u_{\varepsilon}(x)|\geq 10/11$, if $x \in B(x_0,2\rho \varepsilon)$,
where $ x_0 \in \partial B$. This implies (\ref{e2.4}).
Taking $\phi=v \zeta^p, \zeta \in C_0^{\infty}
(B_{\varepsilon},R)$ in (\ref{e2.5}), we obtain
$$
\int_{B_{\varepsilon}}|\nabla v|^p\zeta^p
\leq p\int_{B_{\varepsilon}}|\nabla v|^{p-1}\zeta^{p-1}
|\nabla \zeta||v|
+\int_{B_{\varepsilon}}|v|^2(1-|v|^2)\zeta^p.
$$
For $\rho$ as in (\ref{e2.6}),
setting $y \in B(0,\varepsilon^{-1}-\rho)$,
$B(y,\rho/2) \subset B_{\varepsilon}$,
$$
\zeta=\begin{cases}1 &\mbox{in } B(y,\rho/4),\\
      0 & \mbox{in } B_{\varepsilon}\setminus B(y,\rho/2)
  \end{cases}
 $$
and $|\nabla \zeta| \leq  C(\rho)$, we have
$$
\int_{B(y,\rho/2)}|\nabla v|^p\zeta^p
\leq C(\rho)\int_{B(y,\rho/2)}|\nabla v|^{p-1}\zeta^{p-1}+C(\rho).
$$
Using  Holder's inequality, we can derive
$\int_{B(y,\rho/4)}|\nabla v|^p \leq C(\rho)$.
Combining this with the Tolksdroff' theorem in \cite{t1} yields
$$
\|\nabla v \|_{L^{\infty}(B(y,\rho/8))}^p
\leq C(\rho)\int_{B(y,\rho/4)}(1+|\nabla v|)^p
\leq C(\rho)
$$
which implies
$$
\|\nabla u\|_{L^{\infty}(B(x,\varepsilon \rho
/8))} \leq C(\rho)\varepsilon^{-1}.
$$

\begin{proposition} \label{prop2.6}
Let $u_\varepsilon$ be a radial
minimizer of $E_\varepsilon(u,B)$. Then
\begin{gather}
E_\varepsilon(u_\varepsilon,B) \leq
C\varepsilon^{n-p}+C, \quad when \quad p>n,
\label{e2.7}\\
E_\varepsilon(u_\varepsilon,B) \leq
\frac{1}{n}(n-1)^{n/2}|S^{n-1}||\ln\varepsilon|+C
\quad when \quad p=n,
\label{e2.8}
\end{gather}
with a constant $C$ independent of
$\varepsilon \in (0,1)$.
\end{proposition}

\paragraph{Proof.}
Denote
$$
I(\varepsilon,R)=\min\Big\{\int_{B(0,R)}
[\frac{1}{p}|\nabla u|^p
+\frac{1}{\varepsilon^p}(1-|u|^2)^2]; u \in W_R\Big\},
$$
where
$$
W_R=\big\{u(x)=f(r)\frac{x}{|x|}
\in W^{1,p}(B(0,R),\mathbb{R}^n);
r=|x|, f(R)=1\big\}.
$$
Then
\begin{equation}
\begin{aligned}
I(\varepsilon,1)=&E_{\varepsilon}(u_{\varepsilon},B)\\
=&\frac{1}{p}\int_B|\nabla u_{\varepsilon}|^pdx
+\frac{1}{4\varepsilon^p}\int_B(1-|u_{\varepsilon}|^2)^2dx\\
=&\varepsilon^{n-p}[\frac{1}{p}
\int_{B(0,\varepsilon^{-1})}|\nabla u_{\varepsilon}|^pdy
+\frac{1}{4}\int_{B(0,\varepsilon^{-1})}
(1-|u_{\varepsilon}|^2)^2dy]\\
=&\varepsilon^{n-p}I(1,\varepsilon^{-1}).
\end{aligned} \label{e2.9}
\end{equation}
Let $u_1$ be a solution of $I(1,1)$ and define
$$
u_2=\begin{cases} u_1, &\mbox{if }0<|x|<1; \\
     \frac{x}{|x|}, & \mbox{if } 1 \leq |x|\leq \varepsilon^{-1}.
    \end{cases}
$$
Thus $u_2 \in W_{\varepsilon^{-1}}$ and when $p>n$,
\begin{align*}
I(1,\varepsilon^{-1})\leq &\frac{1}{p}
\int_{B(0,\varepsilon^{-1})}|\nabla u_2|^p +\frac{1}{4}
\int_{B(0,\varepsilon^{-1})}(1-|u_2|^2)^2\\
=&\frac{1}{p}\int_B|\nabla u_1|^p+\frac{1}{4}\int_B(1-|u_1|^2)^2
+\frac{1}{p}\int_{B(0,\varepsilon^{-1})\setminus B}|\nabla
\frac{x}{|x|}|^p\\
=&I(1,1)+\frac{(n-1)^{p/2}|S^{n-1}|}{p}
\int_1^{\varepsilon^{-1}}r^{n-p-1}dr\\
=&I(1,1)+\frac{(n-1)^{p/2}|S^{n-1}|}{p(p-n)}(1-\varepsilon^{p-n}) \leq C.
\end{align*}
Similarly, when $p=n$,
$$
I(1,\varepsilon^{-1})\leq I(1,1)+\frac{1}{n}(n-1)^{n/2}
|S^{n-1}||\ln\varepsilon|+C.
$$
Substituting these into (\ref{e2.9}) yields (\ref{e2.7}) and (\ref{e2.8}).


\section{Location of zeros of minimizers}

\begin{proposition} \label{prop3.1}
Let $u_\varepsilon$ be a radial minimizer of $E_\varepsilon(u,B)$.
Then there exists a constant $C$ independent of $\varepsilon \in (0,1]$
such that
\begin{equation}
\frac{1}{\varepsilon^n} \int_B(1-|u_\varepsilon|^2)^2 \leq C.
\label{e3.1}
\end{equation}
\end{proposition}

\paragraph{Proof.}
When $p>n$, (\ref{e3.1}) can be derived by
multiplying (\ref{e2.7}) by $\varepsilon^{p-n}$.
When $p=n$, as in \cite[eqn.(3.6)]{h1},
we derive that
$$
\int_B|\nabla u_{\varepsilon}|^ndx
\geq (n-1)^{n/2}|S^{n-1}||\ln\varepsilon|-C,
$$
where $C$ is independent of $\varepsilon$.
Combining this with (\ref{e2.8}) we obtain (\ref{e3.1}).

\begin{proposition} \label{prop3.2}
Let $u_\varepsilon$ be a radial minimizer of $E_\varepsilon(u,B)$.
Then for any $\eta \in (0,1/2)$, there exist
positive constants $\lambda, \mu$
independent of $\varepsilon \in (0,1)$ such
that if
\begin{equation}
\frac{1}{\varepsilon^n}
\int_{B(0,1-\rho \varepsilon)
\cap B^{2l\varepsilon}}(1-|u_\varepsilon|^2)^2 \leq \mu,
\label{e3.2}
\end{equation}
where $B^{2l\varepsilon}$ is the ball of
radius $2l\varepsilon$ with $l \geq \lambda$, then
\begin{equation}
|u_\varepsilon(x)| \geq 1-\eta,
\quad \forall x \in B(0,1-\rho \varepsilon) \cap B^{l\varepsilon}.
\label{e3.3}
\end{equation}
\end{proposition}

\paragraph{Proof.}
First we observe that there exists a constant $C_2>0$
which is independent of $\varepsilon$ such that
for any $x \in B$ and $0<\rho \leq 1$,
$$
|B(0,1-\rho\varepsilon)\cap B(x,r)|
\geq C_2 r^n.
$$
To prove this proposition, we choose
\begin{equation}
\lambda=\frac{\eta}{2C_1},
\quad \mu=\frac{C_2}{C_1^n}
(\frac{\eta}{2})^{n+2},
\label{e3.4}
\end{equation}
where $C_1$ is the constant in (\ref{e2.3}).
Suppose  that there is a point $x_0 \in B(0,1-\rho\varepsilon)
\cap B^{l\varepsilon}$ such that
$|u_\varepsilon(x_0)| < 1-\eta$.
Then applying (\ref{e2.3})) we have
\begin{eqnarray*}
|u_\varepsilon(x)-u_\varepsilon(x_0)|
&\leq& C_1 \varepsilon^{-1}|x-x_0|
\leq C_1\varepsilon^{-1}(\lambda \varepsilon)\\
&=&C_1\lambda=\frac{\eta}{2},
 \quad \forall x \in B(x_0,\lambda \varepsilon),
\end{eqnarray*}
hence $(1-|u_\varepsilon(x)|^2)^2
 > \frac{\eta^2}{4}, \quad
\forall x \in B(x_0,\lambda \varepsilon)$. Thus
\begin{equation}
\begin{aligned}
\int_{B(x_0,\lambda \varepsilon)
\cap B(0,1-\rho\varepsilon)}(1-|u_\varepsilon|^2)^2
&> \frac{\eta^2}{4}
|B(0,1-\rho\varepsilon) \cap B(x_0,\lambda \varepsilon)| \\
&\geq C_2 \frac{\eta^2}{4}(\lambda \varepsilon)^n
=C_2 \frac{\eta^2}{4}(\frac{\eta}{2C_1})^n
\varepsilon^n=\mu \varepsilon^n.
\end{aligned} \label{e3.5}
\end{equation}
Since $x_0 \in B^{l\varepsilon} \cap B$, and
$(B(x_0,\lambda \varepsilon) \cap B(0,1-\rho\varepsilon))
\subset (B^{2l\varepsilon} \cap
B(0,1-\rho\varepsilon))$, (\ref{e3.5}) implies
$$
\int_{B^{2l\varepsilon} \cap B(0,1-\rho\varepsilon)
}(1-|u_\varepsilon|^2)^2
> \mu \varepsilon^n,
$$
which contradicts (\ref{e3.2}) and thus (\ref{e3.3}) is proved.

Let $u_\varepsilon$ be a radial minimizer of
$E_\varepsilon(u,B)$. Given $\eta \in
(0,1/2)$. Let $\lambda,\mu$ be  constants
in Proposition \ref{prop3.2} corresponding to
$\eta$. If
\begin{equation}
\frac{1}{\varepsilon^n}
\int_{B(x^{\varepsilon},2\lambda \varepsilon)
\cap B(0,1-\rho\varepsilon)}
(1-|u_\varepsilon|^2)^2 \leq \mu,
\label{e3.6}
\end{equation}
then $B(x^{\varepsilon},\lambda \varepsilon)$
is called the ball of type I.
Otherwise it
is called the ball of type II.

Now suppose that
$\{B(x_i^{\varepsilon},\lambda \varepsilon),i \in I\}$
is a family of balls satisfying
\begin{equation}
\begin{gathered}
x_i^{\varepsilon} \in B(0,1-\rho\varepsilon),i \in I; \\
B(0,1-\rho\varepsilon) \subset
\cup_{i \in I}B(x_i^{\varepsilon},\lambda \varepsilon); \\
B(x_i^{\varepsilon},\lambda \varepsilon /4) \cap
B(x_j^{\varepsilon},\lambda \varepsilon /4)=\emptyset,i \neq j.
\end{gathered} \label{e3.7}
\end{equation}
Denote $ J_\varepsilon=\{i \in I;B(x_i^{\varepsilon},
\lambda \varepsilon)\mbox{ is a ball of type II}\}$.

\begin{proposition} \label{prop3.3}
There exists an upper bound for the number of balls of type II.
i.e., there exists a positive integer $N$ such that
$\mathop{\rm Card} J_\varepsilon \leq N$.
\end{proposition}

\paragraph{Proof.}
Since (\ref{e3.7}) implies that every point
in $B$ can be covered by finite,
say $m$ (independent of $\varepsilon$) balls,
from (\ref{e3.6}) and the definition of balls
of type II, we have
\begin{align*}
\mu \varepsilon^n CardJ_\varepsilon
&\leq \sum_{i \in J_\varepsilon}
\int_{B(x_i^{\varepsilon},2\lambda \varepsilon)
 \cap B(0,1-\rho\varepsilon)}(1-|u_\varepsilon|^2)^2\\
&\leq m\int_{\cup_{i \in J_\varepsilon}
B(x_i^{\varepsilon},2\lambda \varepsilon)
\cap B(0,1-\rho\varepsilon)}(1-|u_\varepsilon|^2)^2\\
&\leq m\int_B(1-|u_\varepsilon|^2)^2
\leq mC\varepsilon^n
\end{align*}
and hence for some $n$,
$\mathop{\rm Card} J_\varepsilon \leq \frac{mC}{\mu} \leq N$.
\hfill$\Box$

Proposition \ref{prop3.3} is an important result since the number of balls
of type II is always finite as $\varepsilon$ becomes sufficiently small.

Similar to the argument of \cite[Theorem IV.1]{b1},
we have

\begin{proposition} \label{prop3.4}
There exist a subset $J \subset J_{\varepsilon}$ and a
constant $h\geq \lambda$
such that
\begin{equation}
\begin{gathered}
\cup_{i \in J_{\varepsilon}}B(x_i^{\varepsilon},\lambda \varepsilon)
\subset \cup_{i \in J}B(x_j^{\varepsilon},h \varepsilon), \\
 |x_i^{\varepsilon}-x_j^{\varepsilon}|>
8h\varepsilon,\quad i,j \in J,\quad i \neq j.
\end{gathered} \label{e3.8}
\end{equation}
\end{proposition}

\paragraph{Proof.}
If there are two points $x_1, x_2$ such that (\ref{e3.8}) is not true with
$h=\lambda$, we take $h_1=9\lambda$ and
$J_1=J_{\varepsilon}\setminus\{1\}$. In this case,
if (\ref{e3.8}) holds we are done.
Otherwise we continue to choose a pair points
$x_3, x_4$ which does not satisfy (\ref{e3.8}) and take $h_2=9h_1$ and
$J_2=J_{\varepsilon}\setminus\{1,3\}$.
After at most $N$ steps we may choose $\lambda
\leq h\leq \lambda 9^N$ and conclude this proposition.

Applying Proposition \ref{prop3.4}, we may modify the family of balls
of type II such that the new one, denoted by
$\{B(x_i^{\varepsilon},h\varepsilon);i \in J\}$, satisfies
\begin{gather*}
\cup_{i \in J_\varepsilon}B(x_i^{\varepsilon},\lambda \varepsilon)
\subset \cup_{i \in J}B(x_i^{\varepsilon},h \varepsilon), \\
\mathop{\rm Card}J \leq \mathop{\rm Card}J_\varepsilon, \\
|x_i^{\varepsilon}-x_j^{\varepsilon}|>8h \varepsilon,i,j \in J,i \neq j.
\end{gather*}
The last condition implies that every two balls in the new family are
disjoint.

Now we prove the  main result of this section.

\begin{theorem} \label{thm3.5}
Let $u_\varepsilon
$ be a radial minimizer of $
E_\varepsilon(u,B)$.
Then for any $\eta \in (0,1/2)$, there
exists a constant $h=h(\eta)$ independent of
$\varepsilon \in (0,1)$
such that $Z_\varepsilon=\{x \in B;
|u_\varepsilon(x)|<1-\eta\} \subset
B(0,h \varepsilon)
$. In particular the zeros
of $u_\varepsilon$
are contained in $B(0,h\varepsilon)$.
\end{theorem}

\paragraph{Proof.}
Suppose there exists a point $x_0 \in Z_\varepsilon$ such that
$x_0 \overline{\in}B(0,h \varepsilon)$. Then all points on the circle
$S_0=\{x \in B;~|x|=|x_0|\}$
satisfy $|u_\varepsilon(x)|<1-\eta$ and hence
by virtue of Proposition \ref{prop3.2} and (\ref{e2.4}), all
points on $S_0$ are contained in balls of type II. However, since $|x_0| \geq
h\varepsilon,S_0$ can not be covered by a single
ball of type II. $S_0$ can be covered
by at least two balls of type II. However this is impossible. \hfill$\Box$

This theorem plays the key role in proving the uniqueness of radial
minimizers. Furthermore, it implies that
all the zeros of the radial minimizer
locate near the singularity $0$ of $\frac{x}{|x|}$,
which is not mentioned in [6] when $p=n>2$.

Using Theorem \ref{thm3.5} and (\ref{e2.4}), we can see that
\begin{equation}
|u_{\varepsilon}(x)| \geq \min(\frac{10}{11},1-\eta),
\quad x \in B(0,h(\eta)\varepsilon).
\label{e3.9}
\end{equation}

\section{ Proof of Theorem \ref{thm1.2}}

Fix $\varepsilon \in (0,1)$.
Suppose $u_1(x)=f_1(r)\frac{x}{|x|}$ and $u_2(x)=f_2(r)\frac{x}{|x|}$
are both radial minimizers of
$E_{\varepsilon}(u,B)$ on $W$, then they are
weak radial solutions of (\ref{e2.1}) (\ref{e2.2}). Thus
\begin{align*}
\int_B&(|\nabla u_1|^{p-2}\nabla u_1
-|\nabla u_2|^{p-2}\nabla u_2)\nabla \phi dx\\
&=\frac{1}{\varepsilon^p}\int_B
[(u_1-u_2)-(u_1|u_1|^2-u_2|u_2|^2)] \phi dx.
\end{align*}
Taking $\phi=u_1-u_2=(f_1-f_2)\frac{x}{|x|}$, we have
\begin{align*}
\int_B(&|\nabla u_1|^{p-2}\nabla u_1
-|\nabla u_2|^{p-2}\nabla u_2)\nabla (u_1-u_2)dx\\
=&\frac{1}{\varepsilon^p}\int_B
(f_1-f_2)^2dx-\frac{1}{\varepsilon^p}\int_B
(f_1-f_2)^2(f_1^2+f_2^2+f_1f_2)dx\\
=&\frac{1}{\varepsilon^p}
\int_{B\setminus B(0,h\varepsilon)}
(f_1-f_2)^2[1-(f_1^2+f_2^2+f_1f_2)]dx\\
& +\frac{1}{\varepsilon^p}\int_{B(0,h\varepsilon)}
(f_1-f_2)^2dx-\frac{1}{\varepsilon^p}
\int_{B(0,h\varepsilon)}
(f_1-f_2)^2(f_1^2+f_2^2+f_1f_2)dx.\\
\end{align*}
Letting $\eta<1-\frac{1}{\sqrt{2}}$ in (\ref{e3.9}), we have
$f_1, f_2 \geq 1/\sqrt{2}$, on $B\setminus B(0,h\varepsilon)$
for any given $\varepsilon \in (0,1)$. Hence
$$
\int_B(|\nabla u_1|^{p-2}\nabla u_1
-|\nabla u_2|^{p-2}\nabla u_2)\nabla (u_1-u_2)dx
\leq \frac{1}{\varepsilon^p}
\int_{B(0,h\varepsilon)}(f_1-f_2)^2dx.
$$
Applying \cite[eqn.(2.11)]{t1}, we can see that there exists
a positive constant $\gamma$ independent of $\varepsilon$ and $h$
such that
\begin{equation}
\gamma\int_B|\nabla(u_1-u_2)|^2dx
\leq \frac{1}{\varepsilon^p}\int_{B(0,h\varepsilon)}
(f_1-f_2)^2dx,
\label{e4.1}
\end{equation}
which implies
\begin{equation}
\int_B|\nabla(f_1-f_2)|^2dx
\leq \frac{1}{\gamma\varepsilon^p}\int_{B(0,h\varepsilon)}
(f_1-f_2)^2dx.
\label{e4.2}
\end{equation}
Denote $G=B(0,h\varepsilon)$.
Applying \cite[Theorem 2.1]{l1}, we have
$\|f\|_{\frac{2n}{n-2}} \leq \beta \|\nabla f\|_2$,
where $\beta=2(n-1)/(n-2)$.
Taking $f=f_1-f_2$ and applying (\ref{e4.2}), we obtain $f(|x|)=0$
as $x \in \partial B$ and
$$
[\int_B|f|^{\frac{2n}{n-2}}dx]^{\frac{n-2}{n}}
\leq \beta^2 \int_B|\nabla f|^2dx
\leq \beta^2 \gamma^{-1}
\int_G|f|^2dx \varepsilon^{-p}.
$$
Using Holder's inequality, we derive
$$
 \int_G|f|^2dx \leq |G|^{1-\frac{n-2}{n}}
[\int_G|f|^{\frac{2n}{n-2}}dx]^{\frac{n-2}{n}}
\leq |B|^{1-\frac{n-2}{n}}h^2
\varepsilon^{2-p}\frac{\beta^2}{\gamma}\int_G|f|^2dx.
$$
Hence for any given $\varepsilon \in (0,1)$,
\begin{equation}
\int_G|f|^2dx \leq C(\beta,|B|,\gamma,\varepsilon)
h^2\int_G|f|^2dx.
\label{e4.3}
\end{equation}
Denote $F(\eta)=\int_{B(0,h(\eta)\varepsilon)}|f|^2dx$,
then $F(\eta)\geq 0$ and (\ref{e4.3}) implies that
\begin{equation}
F(\eta)(1-C(\beta,|B|,\gamma,\varepsilon)h^2) \leq 0.
\label{e4.4}
\end{equation}
On the other hand, since $C(\beta,|B|,\gamma,\varepsilon)$
is independent of $\eta$,
we may take $0<\eta<1-\frac{1}{\sqrt{2}}$ so small that
$h=h(\eta)\leq \lambda 9^N=9^N\frac{\eta}{2C_1}$
(which is implied by (\ref{e3.4})) satisfies
$$
0<1-C(\beta,|B|,\gamma,\varepsilon)h^2
$$
for the fixed $\varepsilon \in (0,1)$,
which and (\ref{e4.4})
imply that  $F(\eta)=0$. Namely $f=0$ a.e. on $G$, or
$$
f_1=f_2, \quad a.e. \quad on \quad B(0,h\varepsilon).
$$
Substituting this into (\ref{e4.1}), we know that
$u_1-u_2=C$ a.e. on $B$. Noticing the continuity of $u_1,u_2$
which is implied by Proposition \ref{prop2.1}, and
$u_1=u_2=x$ on $\partial B$, we can see at last that
$$
u_1=u_2,\quad\mbox{on }\overline{B}.
$$

\begin{thebibliography}{00} \frenchspacing

\bibitem{b1} F. Bethuel, H. Brezis, F. Helein:
      {\it  Ginzburg-Landau Vortices,}
      Birkhauser. 1994.

\bibitem{b2} H. Brezis, F. Merle, T. Riviere:
      {\it Quantization effects for $-\Delta u
      =u(1-|u|^2)$ in $\mathbb{R}^2$, }
      Arch. Rational Mech. Anal.,
      {\bf 126} (1994).35-58.

\bibitem{b3} H. Brezis, L. Oswald:
      {\it  Remarks on sublinear elliptic equations, }
      J. Nonlinear Analysis,{\bf 10} (1986).55-64.

\bibitem{c1} Y. Z. Chen, E. DiBenedetto:
      {\it   Boundary estimates for solutions of nonlinear degenerate
      parabolic systems,}
      J. Reine Angew. Math. ,{\bf 395} (1989).102-131.

\bibitem{f1} P. Fife, L. Peletier:
      {\it       On the location of defects in stationary
      solution of the Ginzburg-Landau equation,}
      Quart. Appl. Math. ,{\bf 54} (1996).85-104.

\bibitem{h1} M. C. Hong:
      {\it       Asymptotic behavior for minimizers of a Ginzburg-Landau type functional
      in higher dimensions associated with n-harmonic maps,}
      Adv. in Diff. Equa.,{\bf 1} (1996), 611-634.

\bibitem{h2} R. M. Herve, M. Herve:
      {\it       Etude qualitative des solutions reelles d'une
      equation differentielle liee a l'equation de
      Ginzburg-Landau,}
      Ann. IHP. Analyse nonLineaire,{\bf 11}  (1994),427-440.

\bibitem{l1} O.Ladyzhenskaya, N.Uraltseva:
      {\it  Linear and quasilinear elliptic equations,}
      Acad, Press, NewYork and London, 1968.

\bibitem{l2} Y. T. Lei, Z. Q. Wu, H. J. Yuan:
      {\it       Radial minimizers of a
      Ginzburg-Landau type functional,}
      Electron. J. Diff. Equs. ,{\bf 1999}  (1999), No.30, 1-21.

\bibitem{m1} P.Mironescu:
      {\it     On the stability of radial solution of the
      Ginzburg-Landau equation, }
      J. Functional Analysis. ,{\bf 130}  (1995), 334-344.

\bibitem{s1} M.Struwe:
      {\it  On the asymptotic behavior of minimizers of the
      Ginzburg-Landau model in 2-dimensions, }
      Diff. and Int. Equs. ,{\bf 7}  (1994), 1613-1624.

\bibitem{t1} P.Tolksdorff:
      {\it  Everywhere regularity for some quasilinear systems with a
      lack of ellipticity,}
      Anna. Math. Pura. Appl. ,{\bf 134}  (1983), 241-266.

\end{thebibliography}

\noindent{\sc Yutian Lei} \\
Mathematics  Department\\
SuZhou University\\
Suzhou, Jiangsu, 215006\\
P. R. China \\
e-mail: lythxl@163.com
\end{document}

\end{document}