\documentclass[twoside]{article}
\pagestyle{myheadings}
\markboth{\hfil Ginzburg-Landau functional \hfil EJDE--1999/30}
{EJDE--1999/30\hfil Yutian Lei, Zhuoqun Wu, \& Hongjun Yuan \hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc  Electronic Journal of Differential Equations},
Vol. {\bf 1999}(1999), No.~30, pp. 1--21. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp  ejde.math.swt.edu \quad ftp ejde.math.unt.edu (login: ftp)}
 \vspace{\bigskipamount} \\
%
 Radial minimizers of a Ginzburg-Landau functional
\thanks{ {\em 1991 Mathematics Subject Classifications:} 35J70.
\hfil\break\indent
{\em Key words and phrases:} Ginzburg-Landau functional, \hfil\break\indent
radial functional, zeros of a minimizer.
\hfil\break\indent
\copyright 1999 Southwest Texas State University  and University of
North Texas. \hfil\break\indent
Submitted June 8, 1999. Published September 9, 1999.} }
\date{}
%
\author{ Yutian Lei, Zhuoqun Wu, \& Hongjun Yuan}
\maketitle

\begin{abstract}
We consider the functional
$$
E_\varepsilon(u,G) =\frac 1p\int_G|\nabla u|^p
+\frac{1}{4\varepsilon^p}\int_G(1-|u|^2)^2
$$
with $p>2$ and  $d>0$, on the class of functions
$W=\{u(x)=f(r)e^{id\theta} \in W^{1,p}(B,C);
f(1)=1,f(r)\geq 0\}$. The location of the zeroes of the
minimizer and its convergence as $\varepsilon$ approaches zero
are established.
\end{abstract}

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


\section{ Introduction}

Let $G \subset R^2$ be a bounded and simply connected domain
with smooth boundary $\partial G$ and $g$ be a smooth map from $
\partial G$ into $S^1=\{x \in C;|x|=1\}
$. Consider the functional of Ginzburg-Landau type
$$
E_\varepsilon(u,G)
=\frac 1p\int_G|\nabla u|^p
+\frac{1}{4\varepsilon^p}
\int_G(1-|u|^2)^2, \quad (\varepsilon>0)
\eqno{(1.1)}
$$
which has been well-studied in [1] for $p=2$, $d=\deg(g,\partial G)=0$
and in [2] for $p=2$, $\deg(g,\partial G) \neq 0$.
Here $d=\deg(g,\partial G)$
denotes the Brouwer degree of the map $g$. For other related
papers, we refer to [3],[5]--[13].

The first two authors of this paper
studied the general case $p>1$, especially the
case $p>2$ under the restriction
$d=\deg(g,\partial G)=0$. In [9][10] some
results on the asymptotic behaviour of
 the minimizer $u_\varepsilon$ of $E_\varepsilon(u,G)$
are presented, in particular, if
$p>2$, then for some $\alpha \in (0,1)$,
the regularizable minimizer $\tilde{u}_\varepsilon$
of $E_\varepsilon(u,G)$
converges in $C_{\rm loc}^{1,\alpha}(G,C)$ as
$\varepsilon \rightarrow 0$. By the regularizable
minimizer of $E_\varepsilon(u,G)$,
we mean a minimizer of $E_\varepsilon(u,G)$ which
is the limit of a subsequence $u_\varepsilon^{\tau_k}$ of minimizers
$u_\varepsilon^{\tau}$ of the regularized functionals
$$
E_\varepsilon^{\tau}(u,G)=
\frac 1p\int_G(|\nabla u|^2+\tau)^{p/2}
+\frac{1}{4\varepsilon^p}
\int_G(1-|u|^2)^2, \quad (\tau>0)
\eqno{(1.2)}
$$
in $W^{1,p}(G,C)$ as $\tau_k \rightarrow 0$.

In this paper we assume that $d=\deg(g,\partial G) \neq 0$. Under this
condition, if $1<p<2$, then, since $W_g^{1,p}(G,S^1)$ is nonempty, the
existence of the p-harmonic $u_p$ on $G$ with given boundary value $g$
and the convergence to $u_p$ for a subsequence $u_{\varepsilon_k}$
of $u_\varepsilon$ in $W^{1,p}(G,C)$ as
$\varepsilon_k \rightarrow 0$ can be proved similar to [9].

However if $p>2$, then, since $d \neq 0,
W_g^{1,p}(G,S^1)$ must be empty. In this case unlike the case $d=0$ or
$1<p<2$, it is impossible to have some subsequence of $u_\varepsilon$
converging to a p-harmonic map on $G$. Under the condition $d \neq 0,p>2$,
the asymptotic analysis of the minimizers
of $E_\varepsilon(u,G)$ seems to be a very
difficult problem. In this paper, we assume that $G=B=\{x \in
R^2;|x|<1\},
g(x)=e^{id\theta}$, $x=(\cos\theta,\sin\theta)$ on $\partial B=S^1$ and consider
the minimization
of $E_\varepsilon(u,B)$ in the class of radial functions
$$
u(x)=f(r)e^{id\theta} \in W_g^{1,p}(B,C),r=|x|
$$
Such minimizers will be called radial minimizers.

Obviously, $u(x)=f(r)e^{id\theta} \in
W_g^{1,p}(B,C)$ implies $f(1)=1$.
Notice that if $u(x)=f(r)e^{id\theta} \in W_g^{1,p}(B,C)$,
then $|f(r)|e^{id\theta} \in W_g^{1,p}(B,C)$
and \newline $E_\varepsilon(|f(r)|e^{id\theta},B)
=E_\varepsilon(f(r)e^{id\theta},B)$.
So, without loss of generality, we may choose the
class of admissible functions as
$$
W=\{u(x)=f(r)e^{id\theta} \in W^{1,p}(B,C);f(1)=1,f(r)\geq 0\}.
$$

In polar coordinates, for $u(x)=f(r)e^{id\theta}$ we have
\begin{eqnarray*}
&|\nabla u|=(f_r^2+d^2r^{-2}f^2)^{1/2},& \\
&\int_B|u|^p=2\pi \int_0^1r|f|^p \,dr,& \\
&\int_B|\nabla u|^p
=2\pi \int_0^1r(f_r^2+d^2r^{-2}f^2)^{p/2} \,dr.&
\end{eqnarray*}
It is easily seen that $f(r)e^{id\theta} \in W^{1,p}(B,C)$ implies
$f(r)r^{\frac 1p-1},f_r(r)r^{\frac 1p} \in L^p(0,1)$.
Conversely, if $f(r) \in W_{\rm loc}^{1,p}(0,1],
f(r)r^{\frac 1p-1},f_r(r)r^{\frac 1p} \in L^p(0,1)$, then
$f(r)e^{id\theta} \in W^{1,p}(B,C)$. Thus if we denote
$$\begin{array}{ll}
V=\{f \in W_{\rm loc}^{1,p}(0,1];&r^{1/p}f_r\in L^p(0,1),
r^{(1-p)/p}f\in L^p(0,1),\\[2mm]
&f(1)=1,f(r)\geq 0\}\\[2mm]
\end{array}
$$
then $V=\{f(r);u(x)=f(r)e^{id\theta} \in W\}$.

\begin{proposition} \label{prop1.1}
The set $V$ defined  above is a subset of $\{f \in C[0,1];f(0)=0\}$.
\end{proposition}

\paragraph{Proof.}
 Let $f \in V,h(r)=f(r^{1+\frac{1}{p-2}})$.Then
\begin{eqnarray*}
\int_0^1|h'(r)|^p\,dr
&=&(1+\frac{1}{p-2})^p\int_0^1|f'(r^{1
+\frac{1}{p-2}})|^p r^{\frac{p}{p-2}}\,dr  \\
&=&(1+\frac{1}{p-2})^p(1-\frac{1}{p-1})
\int_0^1s|f'(s)|^p\,ds<\infty
\end{eqnarray*}
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 $p>2$, we have
$$
\int_0^1 r^{1-p}f^p \,dr
\geq s^p \int_0^t r^{1-p} \,dr=\infty
$$
which contradicts $r^{1/p-1}f \in L^p(0,1)$. Therefore $f(0)=0$ and the
proof is complete.

Substituting $u(x)=f(r)e^{id\theta} \in W$ into $E_\varepsilon(u,B)
(E_\varepsilon^{\tau}(u,B))$,
we obtain
$$
E_\varepsilon(u,B)=2\pi E_\varepsilon(f)
\eqno{(1.3)}
$$
$$
(E_\varepsilon^{\tau}(u,B)=2\pi E_\varepsilon^{\tau}(f))
$$
where
$$
E_\varepsilon(f)=
\int_0^1[\frac 1p(f_r^2+d^2r^{-2}f^2)^{p/2}
               +\frac{1}{4\varepsilon^p}(1-f^2)^2]r\,dr
\eqno{(1.4)}
$$
$$
(E_\varepsilon^{\tau}(f)=
\int_0^1[\frac 1p
(f_r^2+d^2r^{-2}f^2+\tau)^{p/2}
  +\frac{1}{4\varepsilon^p}(1-f^2)^2]r\,dr)
$$
This shows that $u=f(r)e^{id\theta} \in W$ is the minimizer of
$E_\varepsilon(u,B)
(E_\varepsilon^{\tau}(u,B))$ if and only
if $f(r) \in V$ is the minimizer of
$E_\varepsilon(f)(E_\varepsilon^{\tau}(f))$.

Some basic properties of minimizers are given in $\S2$. The main purpose
of $\S3$ is to prove that for any radial minimizer $u_\varepsilon$ of
$E_\varepsilon(u,B)$ and any given $\eta \in (0,1)$
there exists a constant
$h(\eta)>0$ such that
$$
Z_\varepsilon=\{x \in B;
|u_\varepsilon(x)|<1-\eta\} \subset B(0,h \varepsilon)
=\{x \in R^2;|x| <h\varepsilon\}.
$$
(Theorem~\ref{th3.5}) which implies, in particular,
that the zeroes of $u_\varepsilon$
are contained in $B(0,h\varepsilon)$ and that
$$
\lim_{\varepsilon \rightarrow 0}u_\varepsilon=e^{id\theta},
 \quad in~~C_{\rm loc}(\overline{B} \setminus \{0\},C)
$$
In $\S4$ the convergence
rate for regularizable minimizers
$\tilde{u}_\varepsilon$ is studied (Theorem~\ref{th4.4}). In
$\S5$ we prove the convergence of
radial minimizers $u_\varepsilon$ in
$W_{\rm loc}^{1,p}(\overline{B} \setminus \{0\},C)$
as $\varepsilon \rightarrow 0$ (Theorem~\ref{th5.3}) and the
convergence of regularizable radial minimizers $\tilde{u}_\varepsilon$ in
$C_{\rm loc}^{1,\alpha}({B} \setminus \{0\},C)$
as $\varepsilon \rightarrow 0$ (Theorem~\ref{th5.4}). Finally we
indicate in $\S6$ that our argument can be extended to the higher
dimensional case.


\section{Basic properties of minimizers}

\begin{proposition} \label{prop2.1} The functional
$E_\varepsilon(u,B)$($E_\varepsilon^{\tau}(u,B))$
achieves its minimum on W by a function
$u_\varepsilon(x)=f_\varepsilon(r)e^{id
\theta}$($u_\varepsilon^{\tau}(x)
=f_\varepsilon^{\tau}(r)e^{id\theta}$);
$f_\varepsilon(r)$($f_\varepsilon^{\tau}(r)$)
is the minimizer of $E_\varepsilon(f)$($E_\varepsilon^{\tau}(f)$).
\end{proposition}

\paragraph{Proof.}
$W^{1,p}(B,C)$ is a reflexive Banach space. By a well-known
result of Morrey (see for example [4]) $E_\varepsilon(u,B)$ is weakly
lower-semi-continuous in $W_{\rm loc}^{1,p}(B,C)$. To prove the existence
of the minimizers of $E_\varepsilon(u,B)$
in W, it suffices to verify that W is
a weakly closed subset of $W^{1,p}(B,C)$. Clearly W is a convex subset
of $W^{1,p}(B,C)$. Now we prove that W is a closed subset of
$W^{1,p}(B,C)$.

Let $u_k=f_k(r)e^{id\theta} \in W$ and
$$
\lim_{k \rightarrow \infty}u_k=u, \quad in~~W^{1,p}(B,C)
$$
By the embedding theorem there exists a subsequence of $u_k$, supposed to
be $u_k$ itself, such that

$$
\lim_{k \rightarrow \infty}u_k=u, \quad in~~C(\overline{B},C)
$$
which implies
$$
\lim_{k \rightarrow \infty}f_k=f, \quad in~~C[0,1]
$$
and
$$
u =f(r)e^{id\theta}
$$
Combining this with $f_k(1)=1,f_k(r) \geq 0$,
we see that $f(1)=1,f(r) \geq 0$.
Thus $u \in W$. The existence of minimizers $u_\varepsilon^{\tau}$ of
$E_\varepsilon^{\tau}(u,B)$ can be proved similarly.

\begin{proposition} \label{prop2.2} The minimizer
$f_\varepsilon(r)$($f_\varepsilon^{\tau}(r)$) of the
functional $E_\varepsilon(f)$($E_\varepsilon^{\tau}(f)$) satisfies
$$
-(rAf')'+r^{-1}d^2Af=\frac{r}{\varepsilon^p}f(1-f^2),
~~A=(f_r^2+r^{-2}d^2f^2)^{(p-2)/2}
\eqno{(2.1)}
$$
in the following sense:
$$\begin{array}{ll}
&~~\int_0^1r(f_r^2+r^{-2}d^2f^2)^{(p-2)/2}(f_r\phi_r
     +r^{-2}d^2f\phi)\,dr\\[2mm]
&=\frac{1}{\varepsilon^p}
\int_0^1r(1-f^2)f\phi \,dr,
 \quad \forall \phi \in C_0^{\infty}(0,1)
\end{array}
\eqno{(2.2)}
$$
$$
(-(rAf')'+r^{-1}d^2Af=\frac{r}{\varepsilon^p}f(1-f^2),
~~A=(f_r^2+r^{-2}d^2f^2+\tau)^{(p-2)/2}
\eqno{(2.3)}
$$
in the classical sense).
\end{proposition}

By a limit process we see that the test function $\phi$ in (2.2) can be any
member of
$$
X=\{\phi(r) \in W_{\rm loc}^{1,p}(0,1];\phi(0)=\phi(1)=0,\phi(r) \geq 0,
r^{\frac 1p}\phi ',r^{\frac 1p-1}\phi \in L^p(0,1)\}
$$

\begin{proposition} \label{prop2.3} Let $f_\varepsilon$
($f_\varepsilon^{\tau}$) be a nonnegative solution
of (2.1)((2.3)) satisfying
$f(0)=0,f(1)=1$
Then $f_\varepsilon \leq 1,(f_\varepsilon^{\tau} \leq 1)$ on [0,1].
\end{proposition}

\paragraph{Proof.}
Denote $f=f_\varepsilon$ in (2.2) and set $\phi=f(f^2-1)_+$. Then
\begin{eqnarray*}
\lefteqn{ \int_0^1r(f_r^2+d^2r^{-2}f^2)^{(p-2)/2}[f_r^2(f^2-1)_+ }\\
&&+ff_r[(f^2-1)_+]_r
+d^2r^{-2}f^2(f^2-1)_+] \,dr
+\frac{1}{\varepsilon^p} \int_0^1rf^2(f^2-1)_+^2 \,dr=0
\end{eqnarray*}
from which it follows that
$$
\frac{1}{\varepsilon^p}
\int_0^1rf^2(f^2-1)_+^2 \,dr=0
$$
Thus $f=0$ or $(f^2-1)_+=0$ on $[0,1]$
and hence $f=f_\varepsilon \leq 1$ on
[0,1]. The proof of $f_\varepsilon^{\tau} \leq 1$ is even easier.

\begin{proposition} \label{prop2.4} Let $f_\varepsilon
(f_\varepsilon^{\tau})$ be a minimizer of $E_\varepsilon(f)
(E_\varepsilon^{\tau}(f))$. Then
$$
E_\varepsilon(f_\varepsilon) \leq
C\varepsilon^{2-p},(E_\varepsilon^{\tau}
(f_\varepsilon^{\tau}) \leq C\varepsilon^{2-p})
$$
with a constant $C$ independent of
$\varepsilon \in (0,1)(\varepsilon,\tau \in (0,1))$.
\end{proposition}

\paragraph{Proof.} Denote
$$
I(\varepsilon,R)=Min\{
\int_0^R[\frac 1p
(f_r^2+\frac{d^2}{r^2}f^2)^{
\frac{p}{2}}+\frac{1}
{4\varepsilon^p}(1-f^2)^2]r \,dr;f \in V_R\}
$$
where
$$
V_R=\big\{f(r) \in W_{\rm loc}^{1,p}(0,R]; f(r) \geq0,f(R)=1,
f(r)r^{\frac 1p-1},f'(r)r^{\frac 1p} \in L^p(0,R)\big\}\,.
$$
Then
\begin{eqnarray*}
I(\varepsilon,1)&=&E_\varepsilon(f_\varepsilon)\\
&=&\frac 1p \int_0^1 r((f_\varepsilon)_r^2+d^2r^{-2}f_\varepsilon^2)^{p/2} \,dr
  +\frac{1}{4\varepsilon^p}
  \int_0^1 r(1-f_\varepsilon^2)^2 \,dr\\
&=&\frac 1p \int_0^{1/ \varepsilon}\varepsilon^{2-p}s((f_\varepsilon)_s^2
  +d^2s^{-2}f_\varepsilon^2)^{p/2} \,ds
  +\frac{1}{4\varepsilon^p}\int_0^{\varepsilon
             ^{-1}}\varepsilon^2s(1-f_\varepsilon^2)^2 \,ds\\
&=&\varepsilon^{2-p}I(1,\varepsilon^{-1})
\end{eqnarray*}
Let $f_1$ be the minimizer for $I(1,1)$ and define
$$
f_2=f_1,0<s<1; \quad f_2=1,1 \leq s \leq \varepsilon^{-1}
$$
We have
\begin{eqnarray*}
I(1,\varepsilon^{-1}) &\leq& \frac 1p
\int_0^{\varepsilon^{-1}} s[(f_2')^2+d^2s^{-2}
              f_2^2]^{p/2} \,ds+\frac{1}{4}
              \int_0^{\varepsilon^{-1}}s(1-f_2^2)\,ds\\
&\leq& \frac 1p \int_1^{\varepsilon^{-1}}s^{1-p}d^p \,ds
             +\frac 1p\int_0^1
             s((f'_1)^2+d^2s^{-2}f_1^2)^{p/2} \,ds\\
&&+\frac{1}{4} \int_0^1s(1-f_1^2)^2 \,ds\\
&=&\frac{d^p}{p(p-2)} (1-\varepsilon^{p-2})+I(1,1)\\
&\leq& \frac{d^p}{p(p-2)}+I(1,1)=C
\end{eqnarray*}
Substituting into (2.4) follows the first conclusion of Proposition
~\ref{prop2.4}.

To prove another conclusion, note
\begin{eqnarray*}
E_\varepsilon^{\tau}(f_\varepsilon^{\tau})
&=&\varepsilon^{2-p}[\frac 1p
\int_0^{1/ \varepsilon}s((f_\varepsilon^{\tau})_s^2
  +d^2s^{-2}(f_\varepsilon^{\tau})^2+\varepsilon^2\tau)^{p/2} \,ds\\
&&+\frac{1}{4}\int_0^{\varepsilon
             ^{-1}}s(1-(f_\varepsilon^{\tau})^2)^2 \,ds]
\end{eqnarray*}

Let $f_1$ be the minimizer for $I(1,1)$ and
$f_\varepsilon$ be  the function defined above. Then
$$\begin{array}{ll}
&~~E_\varepsilon^{\tau}(f_\varepsilon^{\tau})
\leq E_\varepsilon^{\tau}(f_\varepsilon)\\[2mm]
& \leq \varepsilon^{2-p}[\frac 1p
\int_0^{\varepsilon^{-1}} s[(f_2')^2+d^2s^{-2}
              f_2^2+\varepsilon^2\tau]^{p/2} \,ds
              +\frac{1}{4}
              \int_0^{\varepsilon^{-1}}s(1-f_2^2)^2
              \,ds]\\[2mm]
             &=\varepsilon^{2-p}[\frac 1p
             \int_1^{\varepsilon^{-1}}s[s^{-2}d^2
             +\varepsilon^2\tau]^{p/2} \,ds
             +\frac 1p\int_0^1
             s((f'_1)^2+d^2s^{-2}f_1^2+\varepsilon^2\tau)^{p/2} \,ds\\[2mm]
& \quad +\frac{1}{4}
\int_0^1s(1-f_1^2)^2 \,ds\\[2mm]
&\leq \varepsilon^{2-p}[\frac{C}{p}
\int_1^{\varepsilon^{-1}}s[s^{-p}d^p+\varepsilon^p]^{p/2} \,ds
              +\frac{C}{p}\int_0^1
             s[((f'_1)^2+d^2s^{-2}f_1^2)^{p/2}+\varepsilon^p] \,ds\\[2mm]
& \quad +\frac{1}{4}\int_0^1s(1-f_1^2)^2 \,ds]\\[2mm]
&\leq \varepsilon^{2-p}[CI(1,1)+C\varepsilon^p+C
+C\varepsilon^{p-2}] \leq C\varepsilon^{2-p}
\end{array}
$$
The proof of Proposition~\ref{prop2.4} is complete.



\section{Location of zeroes and $C_{\rm loc}$ convergence for minimizers}

By the embedding theorem we first derive from Proposition~\ref{prop2.3}
and Proposition~\ref{prop2.4}
the following

\begin{proposition} \label{prop3.1} Let $u_\varepsilon
(u_\varepsilon^{\tau})$ be a radial minimizer of
$E_\varepsilon(u,B)(E_\varepsilon^{\tau}(u,B))$.
Then there exists a constant $C$
independent of $\varepsilon \in (0,1)
(\varepsilon,\tau \in (0,1))$ such that
$$
|u_\varepsilon(x)-u_\varepsilon(x_0)|
\leq C{\varepsilon^{(2-p)/p}}|x-x_0|^{1-2/p},
 \quad \forall x,x_0 \in B
$$
$$
(|u_\varepsilon^{\tau}(x)-u_\varepsilon^{\tau}(x_0)|
\leq C{\varepsilon^{(2-p)/p}}|x-x_0|^{1-2/p})
 \quad \forall x,x_0 \in {B}
$$
\end{proposition}

As a corollary of Proposition~\ref{prop2.4} we have


\begin{proposition} \label{prop3.2}
Let $u_\varepsilon(u_\varepsilon^{\tau})$
be a radial minimizer of $E_\varepsilon(u,B)
(E_\varepsilon^{\tau}(u,B))$. Then for
some constant $C$ independent of $\varepsilon(\varepsilon,\tau) \in (0,1]$
$$
\frac{1}{\varepsilon^2}
\int_B(1-|u_\varepsilon|^2)^2 \leq C
\eqno{(3.1)}
$$
$$
(\frac{1}{\varepsilon^2}
\int_B(1-|u_\varepsilon^{\tau}|^2)^2 \leq C)
$$
\end{proposition}

Based on Proposition~\ref{prop3.1}, we have the following interesting result:

\begin{proposition} \label{prop3.3} Let $u_\varepsilon
(u_\varepsilon^{\tau})$ be a radial minimizer of
$E_\varepsilon(u,B)(E_\varepsilon^{\tau}(u,B))$.
Then for any $\eta \in (0,1)$, there exist
positive constants $\lambda, \mu$
independent of $\varepsilon(\varepsilon,\tau) \in (0,1)$ such
that if
$$
\frac{1}{\varepsilon^2}\int_{B
\cap B^{2l\varepsilon}}(1-|u_\varepsilon|^2)^2 \leq \mu
\eqno{(3.2)}
$$
$$
(\frac{1}{\varepsilon^2}\int_{B
\cap B^{2l\varepsilon}}(1-|u_\varepsilon^{\tau}|^2)^2 \leq \mu)
$$
where $B^{2l\varepsilon}$ is some disc of
radius $2l\varepsilon$ with $l \geq \lambda$, then
$$
|u_\varepsilon(x)| \geq 1-\eta,
 \quad \forall x \in B \cap B^{l\varepsilon}
\eqno{(3.3)}
$$
$$
(|u_\varepsilon^{\tau}(x)| \geq 1-\eta,
 \quad \forall x \in B \cap B^{l\varepsilon})
$$
\end{proposition}

\paragraph{Proof.}
First we observe that there exists a constant $\beta>0$ such that
for any $x \in B$ and $0<\rho \leq 1$,
$$
mes(B \cap B(x,\rho)) \geq \beta \rho^2
$$
To prove the proposition, we choose
$$
\lambda=(\frac{\eta}{2C})^{\frac{p}{p-2}}, \quad
\mu=\frac{\beta}{4}
(\frac{1}{2C})^{\frac{2p}{p-2}}\eta^{2+\frac{2p}{p-2}}
$$
where $C$ is the constant in Proposition~\ref{prop3.1}.

Suppose  that there is a point $x_0 \in B \cap B^{l\varepsilon}$ such that
$|u_\varepsilon(x_0)| < 1-\eta$.
Then applying Proposition~\ref{prop3.1} we have
\begin{eqnarray*}
|u_\varepsilon(x)-u_\varepsilon(x_0)|
&\leq& C \varepsilon^{(2-p)/p}|x-x_0|^{1-2/p}
\leq C\varepsilon^{(2-p)/p}(\lambda \varepsilon)^{1-2/p} \\
&=&C\lambda^{1-2/p}=\frac{\eta}{2},
 \quad \forall x \in B(x_0,\lambda \varepsilon)
\end{eqnarray*}
Hence
$$
(1-|u_\varepsilon(x)|^2)^2 > \frac{\eta^2}{4}, \quad
\forall x \in B(x_0,\lambda \varepsilon)
$$
$$\begin{array}{ll}
&~~\int_{B(x_0,\lambda \varepsilon)
\cap B}(1-|u_\varepsilon|^2)^2
> \frac{\eta^2}{4}
mes(B \cap B(x_0,\lambda \varepsilon)) \\[2mm]
&\geq \beta \frac{\eta^2}{4}(\lambda \varepsilon)2
=\beta \frac{\eta^2}{4}(\frac{\eta}{2C})^{
\frac{2p}{p-2}}\varepsilon^2=\mu \varepsilon^2
\end{array}
\eqno{(3.4)}
$$
Since $x_0 \in B^{l\varepsilon} \cap B$, and
$(B(x_0,\lambda \varepsilon) \cap B)
\subset (B^{2l\varepsilon} \cap B)$, (3.4) implies
$$
\int_{B^{2l\varepsilon} \cap B}(1-|u_\varepsilon|^2)^2
> \mu \varepsilon^2
$$
which contradicts (3.2) and thus the proposition is proved.

Let $u_\varepsilon$ be a radial minimizer of
$E_\varepsilon(u,B)$. Given $\eta \in
(0,1)$. Let $\lambda,\mu$ be  constants
in Proposition~\ref{prop3.3} corresponding to
$\eta$. If
$$
\frac{1}{\varepsilon^2}
\int_{B(x^{\varepsilon},2\lambda \varepsilon)
\cap B}(1-|u_\varepsilon|^2)^2 \leq \mu
\eqno{(3.5)}
$$
then $B(x^{\varepsilon},\lambda \varepsilon)$
is called $\eta-$ good disc, or simply good disc.
Otherwise $B(x^{\varepsilon},\lambda\varepsilon)$
is called $\eta-$ bad disc or simply bad disc.

Now suppose that $\{B(x_i^{\varepsilon},\lambda \varepsilon),
i \in I\}$ is a family of discs
satisfying
$$
(i):x_i^{\varepsilon} \in B,i \in I;
 \quad (ii):B \subset
\cup_{i \in I}B(x_i^{\varepsilon},\lambda \varepsilon)
$$
$$
(iii):B(x_i^{\varepsilon},\lambda \varepsilon /4) \cap
B(x_j^{\varepsilon},\lambda \varepsilon /4)=\emptyset,i \neq j
\eqno{(3.6)}
$$
Denote
$$
J_\varepsilon=\{i \in I;B(x_i^{\varepsilon},
\lambda \varepsilon)~~is~~a~~bad~~disc\}
$$

\begin{proposition} \label{prop3.4}
There exists a positive integer $N$ such that the
number of bad discs
$\mathop{\rm card}J_\varepsilon \leq N$
\end{proposition}

\paragraph{Proof.}
Since (3.6) implies that every point in $B$ can be covered by finite,
say m (independent of $\varepsilon$) discs,
from (3.2) and the definition of bad
discs,we have
\begin{eqnarray*}
\mu \varepsilon^2 \mathop{\rm card}  J_\varepsilon
&\leq& \sum_{i \in J_\varepsilon}
\int_{B(x_i^{\varepsilon},2\lambda \varepsilon)
                        \cap B}(1-|u_\varepsilon|^2)^2\\
&\leq& m\int_{\cup_{i \in J_\varepsilon}
B(x_i^{\varepsilon},2\lambda \varepsilon)
\cap B}(1-|u_\varepsilon|^2)^2\\
&\leq& m\int_B(1-|u_\varepsilon|^2)^2
\leq mC\varepsilon^2
\end{eqnarray*}
and hence card$\,J_\varepsilon\leq \frac{mC}{\mu} \leq N$.

Applying Theorem IV.1 in [2], we may modify the family of bad discs such
that the new one, denoted by
$\{B(x_i^{\varepsilon},h\varepsilon);i \in J\}$, satisfies
$$
\cup_{i \in J_\varepsilon}B(x_i^{\varepsilon},\lambda \varepsilon)
\subset \cup_{i \in J}B(x_i^{\varepsilon},h \varepsilon),
$$
$$
\lambda \leq h; \quad \mathop{\rm card}  J \leq \mathop{\rm card}  J_\varepsilon
\eqno{(3.7)}
$$
$$
|x_i^{\varepsilon}-x_j^{\varepsilon}|>8h \varepsilon,i,j \in J,i \neq j
$$
The last condition implies that every two discs in the new family are
Dis-intersected.

The argument on the good and bad discs can be applied to the radial
minimizer $u_\varepsilon^{\tau}$ of
$E_\varepsilon^{\tau}(u,B)$. In particular, we may
obtain a family of discs $\{B(x_i^{\varepsilon,\tau},
\lambda \varepsilon),i \in I\}$  such that
the number of bad discs is bounded by a positive integer N independent
of both $\varepsilon \in (0,1)$ and
$\tau \in (0,1)$. The family of bad discs can
be modified such that the new one satisfies the conditions corresponding
to (3.7).

Now we prove our main result of this section.

\begin{theorem} \label{th3.5} Let $u_\varepsilon
(u_\varepsilon^{\tau})$ be a radial minimizer of $
E_\varepsilon(u,B)(E_\varepsilon^{\tau}(u,B))$.
Then for any $\eta \in (0,1)$, there
exists a constant $h=h(\eta)$ independent of
$\varepsilon(\varepsilon, \tau) \in (0,1)$
such that $Z_\varepsilon=\{x \in B;
|u_\varepsilon(x)|<1-\eta\} \subset
B(0,h \varepsilon)(Z_\varepsilon^{\tau}=\{x \in B;
|u_\varepsilon^{\tau}(x)|<1-\eta\} \subset
B(0,h \varepsilon))$.In particular the zeroes
of $u_\varepsilon(u_\varepsilon^{\tau})$
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{prop3.3} all
points on $S_0$ are contained in bad discs. However, since $|x_0| \geq
h\varepsilon,S_0$ can not be covered by a single
bad disc. $S_0$ can be covered
by at least two bad discs. However this is impossible. The same is true
for $u_\varepsilon^{\tau}$.

\begin{theorem} \label{th3.6} Let $u_\varepsilon
=f_\varepsilon(r)e^{id\theta}$ be a radial
minimizer of $E_\varepsilon(u,B)$. Then
$$
\lim_{\varepsilon \rightarrow 0}
f_\varepsilon=1, \quad in~~C_{\rm loc}((0,1],R)
$$
$$
\lim_{\varepsilon \rightarrow 0}
u_\varepsilon=e^{id\theta},
 \quad in~~C_{\rm loc}(\overline{B} \setminus \{0\},C)
$$
\end{theorem}


\section{Convergence rate for minimizers}

\begin{proposition} \label{prop4.1} Let $u_\varepsilon^{\tau}$
be a radial minimizer of
$E_\varepsilon^{\tau}(u,B)$. Then there exists a subsequence
$u_\varepsilon^{\tau_k}$ of
$u_\varepsilon^{\tau}$ with $\tau_k \rightarrow 0$ such that
$$
\lim_{\tau_k \rightarrow 0}
u_\varepsilon^{\tau_k}=\tilde{u}_\varepsilon,
 \quad in~~W^{1,p}(B,C)
\eqno{(4.1)}
$$
and $\tilde{u}_\varepsilon$ is a
radial minimizer of $E_\varepsilon(u,B)$.
\end{proposition}

\paragraph{Proof.} Since $u_\varepsilon \in W$ and
$u_\varepsilon^{\tau}$ is a radial minimizer of
$E_\varepsilon^{\tau}(u,B)$ in W, we have
$$
E_\varepsilon^{\tau}(u_\varepsilon^{\tau},B)
\leq E_\varepsilon^{\tau}(u_\varepsilon,B)
\leq C
$$
with a constant $C$ independent of $\tau \in (0,1)$. This and
$|u_\varepsilon^{\tau}| \leq 1$ on $\overline{B}$
imply the existence of a subsequence
$u_\varepsilon^{\tau_k}$ of
$u_\varepsilon^{\tau}$ with $\tau_k \rightarrow 0$
and a function $\tilde{u}_\varepsilon \in W^{1,p}(B,C)$ such that
$$
\lim_{\tau_k \rightarrow 0}
u_\varepsilon^{\tau_k}=\tilde{u}_\varepsilon,
 \quad weakly~~in~~W^{1,p}(B,C)
\eqno{(4.2)}
$$
$$
\lim_{\tau_k \rightarrow 0}u_\varepsilon^{\tau_k}
=\tilde{u}_\varepsilon, \quad in~~C(\overline{B},C)
\eqno{(4.3)}
$$
Thus, $\tilde{u}_\varepsilon \in W$ and we have
$$
\liminf_{\tau_k
\rightarrow 0}E_\varepsilon^{\tau_k}(u_\varepsilon^{\tau_k},B)
\leq \limsup_{\tau_k
\rightarrow 0}E_\varepsilon^{\tau_k}(u_\varepsilon^{\tau_k},B)
\leq \lim_{\tau_k \rightarrow 0}
E_\varepsilon^{\tau_k}(\tilde{u}_\varepsilon,B)
$$
$$
\lim_{\tau_k \rightarrow 0}\int_B
(1-|u_\varepsilon^{\tau_k}|^2)^2 =
\int_B(1-|\tilde{u}_\varepsilon|^2)^2
$$
Hence
$$\begin{array}{ll}
&~~\liminf_{\tau_k \rightarrow 0}
\int_B(|\nabla u_\varepsilon^{\tau_k}|^2+\tau_k)^{p/2}
\leq \limsup_{\tau_k \rightarrow 0}
\int_B(|\nabla u_\varepsilon^{\tau_k}|^2
+\tau_k)^{p/2}\\[2mm]
&\leq \lim_{\tau_k \rightarrow 0}
\int_B(|\nabla \tilde{u}_\varepsilon|^2
+\tau_k)^{p/2}
=\int_B|\nabla \tilde{u}_\varepsilon|^p
\end{array}
\eqno{(4.4)}
$$
On the other hand, (4.2) and the lower
semicontinuity of $\int
_B|\nabla v|^p$ imply
$$
\int_B|\nabla \tilde{u}_\varepsilon|^p
 \leq \liminf_{\tau_k \rightarrow 0}
\int_B|\nabla u_\varepsilon^{\tau_k}|^p
$$
>From this and (4.4) we obtain
$$
\lim_{\tau_k \rightarrow 0}
\int_B|\nabla u_\varepsilon^{\tau_k}|^p
=\int_B|\nabla \tilde{u}_\varepsilon|^p
$$
which combined with (4.2) gives
$$
\lim_{\tau_k \rightarrow 0}
\int_B|\nabla (u_\varepsilon^{\tau_k}
-\tilde{u}_\varepsilon)|^p=0
\eqno{(4.5)}
$$
(4.1) follows from (4.3) and (4.5).

For any $v \in W$, we have
$$
E_\varepsilon^{\tau_k}(u_\varepsilon^{\tau_k},B)
\leq E_\varepsilon^{\tau_k}(v,B)
$$
Letting $\tau_k \rightarrow 0$ and noticing that
$$
\lim_{\tau_k \rightarrow 0}
E_\varepsilon^{\tau_k}(u_\varepsilon^{\tau_k},B)
=E_\varepsilon(\tilde{u}_\varepsilon,B)
$$
we are led to $
E_\varepsilon(\tilde{u}_\varepsilon,B) \leq E_\varepsilon(v,B)$
Thus $\tilde{u}_\varepsilon$ is a
radial minimizer of $E_\varepsilon(u,B)$

\begin{proposition} \label{prop4.2}
Let $f_\varepsilon^{\tau}$ be a minimizer of the
regularized functional $E_\varepsilon^{\tau}(f)$
in $V$. Then there exist a
subsequence $f_\varepsilon^{\tau_k}$
of $f_\varepsilon^{\tau}$ with $\tau_k \rightarrow 0$
and a function $\tilde{f}_\varepsilon \in V$, such that
$$
\lim_{\tau_k \rightarrow 0}
\int_0^1r(f_\varepsilon^{\tau_k}
-\tilde{f}_\varepsilon)_r^p\,dr=0;
$$
$\tilde{f}_\varepsilon$ is a minimizer of $E_\varepsilon(f)$ in $V$.
\end{proposition}

Now we prove the main result of this section.

\begin{theorem} \label{th4.3} Suppose $p>4$. Let
$\tilde{f}_\varepsilon$ be a regularizable
minimizer of $E_\varepsilon(f)$.
Then there exists a constant $C$ independent of
$\varepsilon \in (0,1)$ such that
$$
\|(\tilde{f}_\varepsilon)'\|_{L^2(r_0,r_1)}
\leq C(r_0,r_1)\varepsilon
\eqno{(4.6)}
$$
where $[r_0,r_1]$ is an arbitrary closed interval of (0,1).
\end{theorem}

\paragraph{Proof.} Substitute $f=f_\varepsilon^{\tau}$
into (2.3) and let $w=1-f$. Then $w$
satisfies
$$
w-\varepsilon^p(2-w)^{-1}(1-w)^{-1}[(Aw')'
+Ar^{-1}w'+d^2r^{-2}A(1-w)]=0
$$
Differentiate with respect to r, multiply by $rw'\zeta^2$ with
$\zeta \in C_0^{\infty}(0,1)$, such that
$0 \leq \zeta \leq 1$ on $[0,1],
\zeta =1$ on $[t_1,t_2],
\zeta =0$ on $[0,1]-[t,
t_3]$, where $0<t<t_1<t_2<t_3<1,
|\zeta '| \leq C$,
and integrate over $(0,1)$. Then we have
$$
\int_{t}^{1}r(w')^2\zeta^2\,dr
+\varepsilon^p \int_{t}^{1}
(rw'\zeta^2)'(2-w)^{-1}(1-w)^{-1}
$$
$$
\cdot [(Aw')'+Ar^{-1}w'+d^2r^{-2}A(1-w)]=0
\eqno{(4.7)}
$$
>From Theorem~\ref{th3.5}, $f$ has a positive
uniform lower bound on $[t,1]$ for
$\varepsilon >0$ small enough. Hence
$$
C^{-1} \leq (2-w)^{-1}(1-w)^{-1} \leq C
$$
for some constant $C>0$ independent of
$\varepsilon \in (0,\eta),\tau \in (0,1)$.
Substituting
$$
A'=(p-2)A^{\frac{p-4}{p-2}}
      \cdot (w'w''-d^2r^{-2}(1-w)w'-2(1-w)^2d^2r^{-3})
$$
into (4.7), we obtain
$$
\int_{t}^{1}r(w')^2\zeta^2\,dr
+\frac{\varepsilon^p}{C}\int_{t}^{1}
rA(w'')^2\zeta^2\,dr
+\frac{p-2}{C}\varepsilon^p\int_{t}
^{1}r(w'w'')^2A^{\frac{p-4}{p-2}}\zeta^2\,dr
$$
$$\begin{array}{ll}
&\leq C\varepsilon^p\int_{t}^{1}
[Aw'w''\zeta^2+d^2r^{-1}A(1-w)w''\zeta^2\\[2mm]

&~~+(w'\zeta^2+2\zeta \zeta'rw')
(A'w'+Aw''+r^{-1}Aw'+d^2r^{-2}A(1-w))\\[2mm]

&~~-(p-2)A^{\frac{p-4}{p-2}}rw'w''\zeta^2
(d^2r^{-2}(1-w)w'-2(1-w)^2d^2r^{-3})]\,dr
\end{array}
$$
and after putting in order
$$
\int_{t}^{1}r(w')^2\zeta^2\,dr
+\frac{\varepsilon^p}{Ct}\int_{t}^{1}
A(w'')^2\zeta^2\,dr
+\frac{p-2}{Ct}\varepsilon^p\int_{t}
^{1}(w'w'')^2A^{\frac{p-4}{p-2}}\zeta^2\,dr
$$
$$\begin{array}{ll}
&\leq C(t,d)\varepsilon^p\int_{t}^{1}
[Aw'w''(\zeta^2+\zeta \zeta')+Aw''\zeta^2\\[2mm]

&~~+A(w')^2(\zeta^2+\zeta \zeta')+Aw'(\zeta^2+\zeta \zeta')] \,dr\\[2mm]

&~~+C(t,d,p)\varepsilon^p
\int_t^1A^{\frac{p-4}{p-2}}[(w')^3w''(\zeta^2
+\zeta \zeta')+(w')^3(\zeta^2+\zeta \zeta')\\[2mm]
&~~+(w')^2(\zeta^2+\zeta \zeta')+(w')^2w''\zeta^2+w'w''\zeta^2]\,dr\\[2mm]
&=C(t,d)\varepsilon^p J_1+C(t,d,p)\varepsilon^p J_2
\end{array}
\eqno{(4.8)}
$$
Using the Young inequality we see that for any $\delta \in (0,1)$
$$
J_1 \leq \delta\int_t^1A(w'')^2\zeta^2 \,dr
+C(\delta)\int_t^1A[(w')^2+1] \,dr
\eqno{(4.9)}
$$
Noticing that $p>4$ and using the Young inequality again we have for any
$\delta \in (0,1)$
$$\begin{array}{ll}
J_2 &\leq \delta\int_t^1
A^{\frac{p-4}{p-2}}(w'w'')^2\zeta^2 \,dr
+C(\delta)\int_t^1A^{\frac{p-4}{p-2}}[(w')^4+1] \,dr\\[2mm]
&\leq \delta\int_t^1A^{\frac{p-4}{p-2}}(w'w'')^2\zeta^2 \,dr
+C(\delta)\int_t^1(A^{\frac{p}{p-2}}+1) \,dr
\end{array}
\eqno{(4.10)}
$$
Combining (4.8) with (4.9)(4.10) and
choosing $\delta$ small enough we are
led to
$$
\int_{t}^{1}r(w')^2\zeta^2\,dr
+\varepsilon^p\int_{t}^{1}
A(w'')^2\zeta^2\,dr
$$
$$
+\varepsilon^p\int_{t}
^{1}(w'w'')^2A^{\frac{p-4}{p-2}}\zeta^2\,dr
\leq C\varepsilon^p(1+\int_{t}^{1}
A^{\frac{p}{p-2}}\,dr)
$$
In particular
$$\begin{array}{ll}
&~~\int_{t}^{1}r(w')^2\zeta^2\,dr
\leq C\varepsilon^p (\int_{t}
^{1}A^{\frac{p}{p-2}}\,dr+1)\\[2mm]

&\leq C\varepsilon^p(1+t^{-1} \int_{t}
^{1}rA^{\frac{p}{p-2}}\,dr) \leq C(t)\varepsilon^{2-p}
\end{array}
$$
Here Proposition~\ref{prop2.4} is applied. Thus we have
$$
\int_{t_1}^{t_2}(w')^2r\,dr \leq C\varepsilon^2
$$
namely
$$
\int_{t_1}^{t_2}(f_\varepsilon^{\tau})_r^2r\,dr
\leq C\varepsilon^2
\eqno{(4.11)}
$$

As a regularizable minimizer of $E_\varepsilon(f)$,
$\tilde{f}_\varepsilon$ is the limit of a
subsequence $f_\varepsilon^{\tau_k}$ of
$f_\varepsilon^{\tau}$ in the sense of Proposition~\ref{prop4.2}.
Therefore, taking
$\tau=\tau_k$ in (4.11) and letting $\tau_k \rightarrow 0$, we finally obtain
$$
\int_{t_1}^{t_2}(\tilde{f}_\varepsilon)_r^2 \,dr
\leq Ct_1^{-1}\varepsilon^2
$$
which is just (4.6).

It follows from Theorem~\ref{th4.3} immediately

\begin{theorem} \label{th4.4} Suppose $p>4$.
Let $\tilde{u}_\varepsilon=\tilde{f}_\varepsilon
e^{id\theta}$ be a regularizable radial
minimizer of $E_\varepsilon(u,B)$.
Then there exists a constant C independent of $\varepsilon$, such that
$$
\|1-\tilde{f}_\varepsilon\|_{H^1(r_0,r_1)} \leq C(r_0,r_1)\varepsilon
$$
$$
\|\tilde{u}_\varepsilon-e^{id \theta}\|_{H^1(K,C)} \leq C(K)\varepsilon
$$
where $[r_0,r_1]$ is an arbitrary closed interval of (0,1) and K is an
arbitrary compact subset of $B \setminus \{0\}$.
\end{theorem}


\section {$W_{\rm loc}^{1,p}$ convergence and $C_{\rm loc}^{1,\alpha}$
convergence for minimizers}

Let $u_\varepsilon(x)=f_\varepsilon(r)
e^{id\theta}$ be a radial minimizer of
$E_\varepsilon(u,B)$, namely $f_\varepsilon$ be a minimizer of
$$
E_\varepsilon(f)=\frac 1p\int_0^1(f_r^2+
d^2r^{-2}f^2)^{p/2}r\,dr
+\frac{1}{4\varepsilon^p}\int_0^1(1-f^2)^2r\,dr
$$
in $V$. From Proposition~\ref{prop2.4}, we have
$$
E_\varepsilon(f_\varepsilon) \leq
C\varepsilon^{2-p}
\eqno{(5.1)}
$$
for some constant $C$ independent of $\varepsilon \in (0,1)$.

In this section we further prove that for any $\eta \in (0,1)$,
there exists a constant $C(\eta)$ such that
$$
E_\varepsilon(f_\varepsilon;\eta) \leq C(\eta)
\eqno{(5.2)}
$$
for $\varepsilon \in (0,\varepsilon_0)$ with
$\varepsilon_0>0$  small be enough, where
$$
E_\varepsilon(f_\varepsilon;\eta)
=\frac 1p\int_{\eta}^1
(f_r^2+d^2r^{-2}f^2)^{p/2}r\,dr
+\frac{1}{4\varepsilon^p}
\int_{\eta}^1(1-f^2)^2r\,dr
$$
In fact we can prove a more accurate estimate
on $E_\varepsilon(f_\varepsilon;
\eta)$ (see Proposition 5.2).
Based on this estimate  and Theorem~\ref{th3.5}, we may obtain better
convergence for minimizers, namely the $W_{\rm loc}^{1,p}$ convergence and
$C_{\rm loc}^{1,\alpha}$ convergence.

We first prove

\begin{proposition} \label{prop5.1}
Given $\eta \in (0,1)$. There exist constants
$$
\eta_j \in [\frac{(j-1)\eta}{N+1},
\frac{j\eta}{N+1}],
(N=[p])
$$
and $C_j$, such that
$$
E_\varepsilon(f_\varepsilon,\eta_j)
\leq C_j\varepsilon^{j-p}
\eqno{(5.3)}
$$
for $j=2,...,N$, where $\varepsilon \in (0,\varepsilon_0)$.
\end{proposition}

\paragraph{Proof.}
For $j=2$, the inequality (5.3) is just the one in Proposition
~\ref{prop2.4}.

Suppose that (5.3) holds for all $j \leq n$. Then we have, in particular
$$
E_\varepsilon(f_\varepsilon;\eta_n) \leq C_n\varepsilon^{n-p}
\eqno{(5.4)}
$$
If $n=N$ then we are done. Suppose $n<N$. We want to prove (5.3)
for $j=n+1$.

Obviously (5.4) implies
$$\begin{array}{ll}
\frac 1p\int_{\frac{n\eta}{N+1}}
^{\frac{(n+1)\eta}{N+1}}
[(f_\varepsilon)_r^2+d^2r^{-2}f_\varepsilon^2]^{p/2}r\,dr
&+\frac{1}{4\varepsilon^p}
\int_{\frac{n\eta}{N+1}}
^{\frac{(n+1)\eta}{N+1}}
(1-f_\varepsilon^2)^2r\,dr\\[2mm]
&\leq C_n\varepsilon^{n-p}
\end{array}
$$
from which we see by integral mean value theorem that there exists
$$
\eta_{n+1} \in [\frac{n\eta}{N+1},
\frac{(n+1)\eta}{N+1}]
$$
such that
$$
[(f_\varepsilon)_r^2+d^2r^{-2}f_\varepsilon^2]_{r=\eta_{n+1}}
\leq C_n\varepsilon^{n-p}
\eqno{(5.5)}
$$
$$
[\frac{1}{\varepsilon^p}
(1-f_\varepsilon^2)^2]_{r=\eta_{n+1}} \leq C_n\varepsilon^{n-p}
\eqno{(5.6)}
$$

Consider the functional
$$
E(\rho,\eta_{n+1})=\frac 1p
\int_{\eta_{n+1}}^1
(\rho_r^2+1)^{p/2}\,dr
+\frac{1}{2\varepsilon^p}\int_{\eta_{n+1}}^1
(1-\rho)^2\,dr
$$
It is easy to prove that the minimizer $\rho_1$ of $E(\rho,\eta_{n+1})$
on $W_{f_\varepsilon}^{1,p}((\eta_{n+1},1),
R^+)$ exists and satisfies
$$
-\varepsilon^p(v^{(p-2)/2}\rho_r)_r=1-\rho, \quad in~~(\eta_{n+1},1)
\eqno{(5.7)}
$$
$$
\rho|_{r=\eta_{n+1}}=f_\varepsilon,~~
\rho|_{r=1}=f_\varepsilon(1)=1
\eqno{(5.8)}
$$
where $v=\rho_r^2+1$.

Applying Theorem~\ref{th3.5} and (5.4) we see easily that
$$
E(\rho_1;\eta_{n+1}) \leq E(f_\varepsilon;\eta_{n+1})
\leq C_nE_\varepsilon(f_\varepsilon;\eta_{n+1})
\leq C_n\varepsilon^{n-p}
\eqno{(5.9)}
$$
for $\varepsilon \in (0,\varepsilon_0)$ with
$\varepsilon_0 >0$ small enough.

Since $f_\varepsilon \leq 1$, it follows from the maximum principle
$$
\rho_1 \leq 1
\eqno{(5.10)}
$$

Now choosing a smooth function $\zeta(r)$ such that $\zeta=1$ on
$(0,\eta),\zeta=0$ near $r=1$, multiplying (5.7) by $\zeta \rho_r
(\rho=\rho_1)$ and integrating over $(\eta_{n+1},1)$ we obtain
$$
\begin{array}{ll}
v^{(p-2)/2}\rho_r^2|_{r=\eta_{n+1}}
&+\int_{\eta_{n+1}}^1
v^{(p-2)/2}\rho_r(\zeta_r\rho_r+\zeta\rho_{rr})\,dr\\[2mm]
&=\frac{1}{\varepsilon^p}\int_{\eta_{n+1}}^1
(1-\rho)\zeta\rho_r\,dr
\end{array}
\eqno{(5.11)}
$$
Using (5.9) we have
$$\begin{array}{ll}
&~~|\int_{\eta_{n+1}}^1
v^{(p-2)/2}\rho_r(\zeta_r\rho_r+\zeta\rho_{rr})\,dr|\\[2mm]
&\leq \int_{\eta_{n+1}}^1v^{(p-2)/2}|\zeta_r|\rho_r^2\,dr
+\frac 1p|
\int_{\eta_{n+1}}^1(v^{p/2}\zeta)_r\,dr
-\int_{\eta_{n+1}}^1v^{p/2}\zeta_r\,dr|\\[2mm]
&\leq C\int_{\eta_{n+1}}^1v^{p/2}
+\frac 1pv^{p/2}
|_{r=\eta_{n+1}}+\frac{C}{p}
\int_{\eta_{n+1}}^1v^{p/2}\\[2mm]
&\leq C\int_{\eta_{n+1}}^1v^{p/2}
+\frac 1pv^{p/2}|_{r=\eta_{n+1}}
\leq C_n\varepsilon^{n-p}+\frac 1pv^{p/2}|_{r=\eta_{n+1}}
\end{array}
\eqno{(5.12)}
$$
and using (5.6)(5.9) we have
$$\begin{array}{ll}
&~~|\frac{1}{\varepsilon^p}\int_{\eta_{n+1}}^1
(1-\rho)\zeta\rho_r\,dr|
=\frac{1}{2\varepsilon^p}|
\int_{\eta_{n+1}}^1((1-\rho)^2\zeta)_r \,dr
-\int_{\eta_{n+1}}^1(1-\rho)^2\zeta_r \,dr|\\[2mm]
&\leq \frac{1}{2\varepsilon^p}(1-\rho)^2|_{r=\eta_{n+1}}
+\frac{C}{2\varepsilon^p}
\int_{\eta_{n+1}}^1(1-\rho)^2 \,dr|
\leq C_n\varepsilon^{n-p}
\end{array}
\eqno{(5.13)}
$$
Combining (5.11) with (5.12)(5.13) yields
$$
v^{(p-2)/2}\rho_r^2|_{r=\eta_{n+1}}
\leq C_n\varepsilon^{n-p}
+\frac 1pv^{p/2}|_{r=\eta_{n+1}}
$$
Hence
\begin{eqnarray*}
v^{p/2}|_{r=\eta_{n+1}}
&=&v^{(p-2)/2}(\rho_r^2+1)|_{r=\eta_{n+1}}
=v^{(p-2)/2}\rho_r^2|_{r=\eta_{n+1}}
+v^{(p-2)/2}|_{r=\eta_{n+1}}\\
&\leq& C_n\varepsilon^{n-p}
+\frac 1pv^{p/2}|_{r=\eta_{n+1}}
+v^{(p-2)/2}|_{r=\eta_{n+1}}\\
&\leq& C_n\varepsilon^{n-p}
+\frac 1pv^{p/2}|_{r=\eta_{n+1}}
+\delta v^{p/2}|_{r=\eta_{n+1}}+C(\delta)\\
&=&C_n\varepsilon^{n-p}+(\frac 1p
+\delta)v^{p/2}|_{r=\eta_{n+1}}
+C(\delta)
\end{eqnarray*}
from which it follows by choosing $\delta>0$ small enough that
$$
v^{p/2}|_{r=\eta_{n+1}} \leq C_n\varepsilon^{n-p}
\eqno{(5.14)}
$$

Now we multiply both sides of (5.7) by $\rho-1$ and integrate. Then
$$
-\varepsilon^p\int_{\eta_{n+1}}^1
[v^{(p-2)/2}\rho_r(\rho-1)]_r\,dr
+\varepsilon^p\int_{\eta_{n+1}}^1
v^{(p-2)/2}\rho_r^2\,dr
+\int_{\eta_{n+1}}^1
(\rho-1)^2\,dr=0
$$
>From this, using(5.8)(5.14)(5.6) and noticing that $n<p$, we obtain
$$\begin{array}{ll}
&~~E(\rho_1;\eta_{n+1})\leq
|\int_{\eta_{n+1}}^1
[v^{(p-2)/2}\rho_r(\rho-1)]_r\,dr|\\[2mm]
&=v^{(p-2)/2}\rho_r|\rho-1|_{r=\eta_{n+1}}
\leq v^{(p-1)/2}|\rho-1|_{r=\eta_{n+1}}\\[2mm]
&\leq (C_n\varepsilon^{n-p})^{(p-1)/p}(C_n\varepsilon^n)^{1/2}
\leq C_{n+1}\varepsilon^{n+1-p+(n/2-n/p)}
\end{array}
$$
which implies
$$
E(\rho_1;\eta_{n+1}) \leq C_{n+1}\varepsilon^{n+1-p}
\eqno{(5.15)}
$$

Define
$$
w_\varepsilon=f_\varepsilon,~for~ r \in (0,\eta_{n+1}); \quad
w_\varepsilon=\rho_1,~for~r \in [\eta_{n+1},1]
$$
Since $f_\varepsilon$ is a minimizer of $E_\varepsilon(f)$, we have
$$
E_\varepsilon(f_\varepsilon) \leq E_\varepsilon(w_\varepsilon)
$$
namely
\begin{eqnarray*}
E_\varepsilon(f_\varepsilon;\eta_{n+1})
&\leq& \frac 1p\int_{\eta_{n+1}}^1
(\rho_r^2+d^2r^{-2}\rho^2)^{p/2}r\,dr
+\frac{1}{4e^p}\int_{\eta_{n+1}}^1
(1-\rho_r^2)^2r\,dr\\
&\leq& \frac{C}{p}\int_{\eta_{n+1}}^1
(\rho_r^2+1)^{p/2}\,dr
+ \frac{C}{2\varepsilon^p}\int_{\eta_{n+1}}^1
(1-\rho_r)^2\,dr+C\\
&=&CE(\rho_1;\eta_{n+1})+C
\end{eqnarray*}
Thus, using (5.15) yields
$$
E_\varepsilon(f_\varepsilon;\eta_{n+1})
\leq C_{n+1}\varepsilon^{n-p+1}
$$
for $\varepsilon \in (0,\varepsilon_0)$. This is just (5.3) for $j=n+1$.


\begin{proposition} \label{prop5.2}
Given $\eta \in (0,1)$. There exist constants
$\eta_{N+1} \in [\frac{N\eta}{N+1},\eta]$
and $C_{N+1}$ such that
$$
E_\varepsilon(f_\varepsilon;\eta_{N+1})
\leq C_{N+1}\varepsilon^{2(N-p+1)/p}+\frac
{1}{p}\int_{\eta_{N+1}}^1\frac{d^p}{r^{p-1}}\,dr
\eqno{(5.16)}
$$
where $N=[p]$.
\end{proposition}

\paragraph{Proof.}
Similar to the derivation of (5.6) we may obtain from Proposition
~\ref{prop5.1} for $j=N$ that there exists
$\eta_{N+1} \in [\frac{N\eta}{N+1},
\frac{(N+1)\eta}{N+1}]$, such that
$$
\frac{1}{\varepsilon^p}
(1-f_\varepsilon^2)^2|_{r=\eta_{N+1}} \leq C_N\varepsilon^{N-p}
\eqno{(5.17)}
$$
Also similarly, consider the functional
$$
E(\rho,\eta_{N+1})=\frac 1p
\int_{\eta_{N+1}}^1
(\rho_r^2+1)^{p/2}\,dr
+\frac{1}{2\varepsilon^p}\int_{\eta_{N+1}}^1
(1-\rho)^2\,dr
$$
whose minimizer $\rho_2$ on $W_{f_\varepsilon}^{1,p}((\eta_{N+1},1),
R^+)$ exists and satisfies
$$
-\varepsilon^p(v^{(p-2)/2}\rho_r)_r=1-\rho, \quad in~~(\eta_{N+1},1)
\eqno{(5.18)}
$$
$$
\rho|_{r=\eta_{N+1}}=f_\varepsilon,~~
\rho|_{r=1}=f_\varepsilon(1)=1
$$
where $v=\rho_r^2+1$. By the maximum principle we have
$$
\rho_2 \leq 1
\eqno{(5.19)}
$$
>From (5.4) for $n=N$ it follows immediately that
$$
E(\rho_2;\eta_{N+1}) \leq E(f_\varepsilon;\eta_{N+1})
\leq C_NE_\varepsilon(f_\varepsilon;\eta_{N+1})
\leq C_NE_\varepsilon(f_\varepsilon;\eta_{N})
\leq C_N\varepsilon^{N-p}
\eqno{(5.20)}
$$

Similar to the proof of (5.14) and (5.15), we get from (5.17) that
$$
v^{p/2}|_{r=\eta_{N+1}} \leq C_N\varepsilon^{N-p}
$$
$$
E(\rho_2;\eta_{N+1}) \leq C_{N+1}\varepsilon^{N+1-p}
\eqno{(5.21)}
$$

Now we define
$$
w_\varepsilon=f_\varepsilon,~for~r \in (0,\eta_{N+1}); \quad
w_\varepsilon=\rho_2,~for~r \in [\eta_{N+1},1]
$$
and then we have
$$
E_\varepsilon(f_\varepsilon) \leq E_\varepsilon(w_\varepsilon)
$$
Notice  that
$$\begin{array}{ll}
&~~\int_{\eta_{N+1}}^1
(\rho_r^2+d^2r^{-2}\rho^2)^{p/2}r\,dr
-\int_{\eta_{N+1}}^1
(d^2r^{-2})^{p/2}\,dr\\[2mm]
&= \frac{p}{2}\int_{\eta_{N+1}}^1
\int_0^1
[(\rho_r^2+d^2r^{-2}\rho^2)s
+(d^2r^{-2}\rho^2)(1-s)]^{(p-2)/2}]
\,ds\rho_r^2r\,dr\\[2mm]
&\leq C\int_{\eta_{N+1}}^1\int_0^1
[(\rho_r^2+d^2r^{-2}\rho^2)^{(p-2)/2}s^{(p-2)/2} \\
&~~+(d^2r^{-2}\rho^2)^{(p-2)/2}(1-s)^{(p-2)/2}]
\,ds\rho_r^2r\,dr\\[2mm]
&=C\int_{\eta_{N+1}}^1
(\rho_r^2+d^2r^{-2}\rho^2)^{(p-2)/2}\rho_r^2r\,dr
\int_0^1s^{(p-2)/2}\,ds\\[2mm]
&~~+C\int_{\eta_{N+1}}^1
(d^2r^{-2}\rho^2)^{(p-2)/2}\rho_r^2r\,dr\int_0^1
(1-s)^{(p-2)/2}\,ds\\[2mm]
&\leq C(\int_{\eta_{N+1}}^1\rho_r^p\,dr
+\int_{\eta_{N+1}}^1\rho_r^2\,dr)
\end{array}
$$
Hence
\begin{eqnarray*}
\lefteqn{ E_\varepsilon(f_\varepsilon;\eta_{N+1}) }\\
&\leq& \frac 1p\int_{\eta_{N+1}}^1
((\rho_2)_r^2+d^2r^{-2}(\rho_2)^2)^{p/2}r\,dr
+\frac{1}{4e^p}\int_{\eta_{N+1}}^1
(1-(\rho_2)^2)^2r\,dr\\
&\leq& \frac 1p\int_{\eta_{N+1}}^1
(d^2r^{-2})^{p/2}\,dr
+\frac{1}{4\varepsilon^p}\int_{\eta_{N+1}}^1
(1-(\rho_2)^2)^2\,dr \\
&&+C(\int_{\eta_{N+1}}^1(\rho_2)_r^p\,dr
+\int_{\eta_{N+1}}^1(\rho_2)_r^2\,dr)
\end{eqnarray*}
Using (5.21) we have
$$
E_\varepsilon(f_\varepsilon;\eta_{N+1})
\leq \frac 1p\int_{\eta_{N+1}}^1
(d^2r^{-2})^{p/2}\,dr
+C_{N+1}\varepsilon^{2(N-p+1)/p}\,.
$$


\begin{theorem} \label{th5.3}
Let $u_\varepsilon=f_\varepsilon(r)e^{id\theta}$
be a radial minimizer of $E_\varepsilon(u,B)$.
Then
$$
\lim_{\varepsilon \rightarrow 0}f_\varepsilon=1
, \quad ~in~~W^{1,p}((\eta,1],R)
\eqno{(5.22)}
$$
$$
\lim_{\varepsilon \rightarrow 0}u_\varepsilon=
e^{id\theta}, \quad in~~W^{1,p}(K,C)
\eqno{(5.23)}
$$
for any $\eta \in (0,1)$ and compact subset
$K \subset \overline{B} \setminus \{0\}$.
\end{theorem}

\paragraph{Proof.}
 It suffices to prove (5.23), since (5.23) implies (5.22).
Without loss of generality, we may
assume $K=B \setminus B(0,\eta_{N+1})$.
>From Proposition~\ref{prop5.2}, We have
$$
E_\varepsilon(u_\varepsilon,K)
=2\pi E_\varepsilon(f_\varepsilon,\eta_{N+1}) \leq C
$$
where $C$ is independent of $\varepsilon$, namely
$$
\int_K|\nabla u_\varepsilon|^p \leq C
\eqno{(5.24)}
$$
$$
\int_K(1-|u_\varepsilon|^2)^2 \leq C\varepsilon^p
\eqno{(5.25)}
$$
(5.24) and $|u_\varepsilon| \leq 1$
imply the existence of  a subsequence
$u_{\varepsilon_k}$ of $u_\varepsilon$
and a function $u_* \in W^{1,p}(K,C)$,
such that
$$
\lim_{\varepsilon_k \rightarrow 0}
u_{\varepsilon_k}=u_*, \quad\mbox{weakly in }W^{1,p}(K,C)
\eqno{(5.26)}
$$
$$
\lim_{\varepsilon_k \rightarrow 0}
u_{\varepsilon_k}=u_*,
 \quad\mbox{in }C^{\alpha}({K},C),\alpha \in (0,1- \frac{2}{p})
\eqno{(5.27)}
$$
(5.27) implies $u_*=e^{id\theta}$. Noticing that any subsequence of
$u_\varepsilon$ has a convergence subsequence and the limit is always
$e^{id\theta}$, we can assert
$$
\lim_{\varepsilon \rightarrow 0}u_\varepsilon=e^{id\theta},
 \quad\mbox{weakly in }W^{1,p}(K,C)
\eqno{(5.28)}
$$
>From this and the weakly lower
semicontinuity of $\int_K|\nabla u|^p$,
using Proposition~\ref{prop5.2}, we have
\begin{eqnarray*}
\int_K|\nabla e^{id\theta}|^p
&\leq& \liminf_{\varepsilon_k \rightarrow 0}
\int_K|\nabla u_\varepsilon|^p
\leq \limsup_{\varepsilon_k \rightarrow 0}
\int_K|\nabla u_\varepsilon|^p\\
&\leq& C\lim_{\varepsilon \rightarrow 0}\varepsilon^{2(N+1-p)/p}
+2\pi \int_{\eta_{N+1}}^1
(d^2r^{-2})^{p/2}r\,dr
\end{eqnarray*}
and hence
$$
\lim_{\varepsilon \rightarrow 0}
\int_K|\nabla u_\varepsilon|^p
=\int_K|\nabla e^{id\theta}|^p
$$
since
$$
\int_K|\nabla e^{id\theta}|^p
=2\pi \int_{\eta_{N+1}}^1
(d^2r^{-2})^{p/2}r\,dr
$$
Combining this with (5.28)(5.27) completes the proof of (5.23).

For the regularizable radial minimizer
$\tilde{u}_\varepsilon=\tilde{f}_\varepsilon(r)e^{id\theta}$,
we may prove
\begin{eqnarray*}
E_\varepsilon^{\tau}(f_\varepsilon^{\tau};\eta)
&=&\frac 1p\int_{\eta}^1
[(f_\varepsilon^{\tau})_r^2+d^2r^{-2}
(f_\varepsilon^{\tau})^2+\tau]^{p/2}r\,dr
+\frac{1}{4\varepsilon^p}\int_{\eta}^1
(1-(f_\varepsilon^{\tau})^2)^2r\,dr\\
&\leq& C(\eta),
\end{eqnarray*}
where $f_\varepsilon^{\tau}$ is the regularized
minimizer of $E_\varepsilon(f)$. On
the basis of this fact and the conclusion for
$f_\varepsilon^{\tau}$ similar to
Theorem~\ref{th3.5}, we may obtain better convergence for the regularizable

minimizer $\tilde{f}_\varepsilon$ by means of the argument applied in [10].
Precisely we have

\begin{theorem} \label{th5.4} Let $\tilde{u}_\varepsilon
=\tilde{f}_\varepsilon(r)e^{id\theta}$ be a
regularizable radial minimizer of
$E_\varepsilon(u,B)$. Then for some $\alpha \in
(0,1)$
$$
\lim_{\varepsilon \rightarrow 0}
\tilde{f}_\varepsilon=1 \mbox{ in }C_{\rm loc}^{1,\alpha}((0,1),R),
\qquad
\lim_{\varepsilon \rightarrow 0}
\tilde{u}_\varepsilon=e^{id\theta} \mbox{ in }
C_{\rm loc}^{1,\alpha}(B \setminus \{0\},C)\,.
$$
\end{theorem}


\section{Generalization}

Let $G \subset R^n$ be a bounded and simply connected domain with smooth
boundary $\partial G,n>2,g:\partial G \rightarrow S^{n-1}
=\{x \in R^n;|x|=1\}$ be a smooth map with $d
=\deg(g,\partial G) \neq 0$. Consider the minimization of the functional
$$
E_\varepsilon(u,G)=\frac 1p
\int_G |\nabla u|^p
+\frac{1}{4\varepsilon^p}\int_G
(1-|u|^2)^2
$$
on $W=\{v \in W^{1,p}(G,R^n);v|_{\partial G}=g\}$.
When $1<p<n$, we have $W_g^{1,p}(G,S^{n-1})
\neq \emptyset$ and hence it is easy to prove that $\int_G
|\nabla u|^p$ achieves its minimum on $W_g^{1,p}(G,S^{n-1})$
by a p-harmonic map with boundary value $g$.
One can also prove that the
minimizer $u_\varepsilon$ of
$E_\varepsilon(u,G)$ on $W$ exists and for a subsequence
$u_{\varepsilon_k}$ of $u_\varepsilon$ there holds,
$$
\lim_{\varepsilon_k \rightarrow 0}
u_{\varepsilon_k}=u_p \quad\mbox{in }W^{1,p}(G,R^n)
$$
where $u_p$ is a p-harmonic map with boundary value $g$.

In case $p=n$, M.C.Hong studied in [6] the asymptotic behavior of
the regularizable minimizer of $E_\varepsilon(u,G)$. He proved that the
minimizer $u_\varepsilon^{\tau}$ of
$$
E_\varepsilon^{\tau}(u,G)
=\frac{1}{n}\int_G(|\nabla u|^2+\tau)^{n/2}
+\frac{1}
{4\varepsilon^n}\int_G(1-|u|^2)^2
$$
on $W_g^{1,n}(G,R^n)$ converges to a minimizer
$\tilde{u_\varepsilon}$ (called
regularizable minimizer) of
$E_\varepsilon(u,G)$ on $W^{1,p}(G,R^n)$ as $\tau \rightarrow
0$ and  that $\tilde{u_\varepsilon}$ contains
a subsequence $\tilde{u}_{\varepsilon_k}$
such that
$$
\lim_{\varepsilon_k \rightarrow 0}\tilde{u_{\varepsilon_k}}=u_n,
 \quad\mbox{ weakly in }W_{\rm loc}^{1,n}(G \setminus \cup _{j=1}^J\{a_j\},R^n)
$$
where $a_j(j=1,2,...,J) \in G$ and $u_n$ is an n-harmonic map on
$G \setminus \cup_{j=1}^J\{a_j\}$. In case $G=B
=\{x \in R^n;|x|<1\},g=x$, he proved that for a
subsequence $\tilde{u}_{\varepsilon_k}$ of the regularizable radial
minimizer $\tilde{u}_\varepsilon$
$$
\lim_{\varepsilon_k \rightarrow 0}
\tilde{u}_{\varepsilon_k}=\frac{x}{|x|},
quad\mbox{ weakly in }W_{\rm loc}^{1,n}(B \setminus \{0\},R^n).
$$

In this section we are concerned with the case $p>n$. Assume that
$G=B$,  and $g=x$
where $B$ is the unit ball centered at the origin, and consider the
minimizers of $E_\varepsilon(u,B)$ on the class of radial functions
$$
W=\{u \in W_g^{1,p}(B,R^n);u(x)=f(r)x|x|^{-1},f(r) \geq 0,r=|x|\}
$$
we call them radial minimizers.

Denote as in $\S1$
$$
V=\{f(r) \in W_{\rm loc}^{1,p}(0,1];r^{(1-p)/p}f,r^{1/p}f_r
\in L^p(0,1),f(1)=1,f(r)\geq 0\}
$$
Substituting $u=f(r)x|x|
^{-1}$ into $E_\varepsilon(u,B)$ we obtain
$$
E_\varepsilon(u,B)=\mathop{\rm meas}(S^{n-1})E_\varepsilon(f)
$$
where
$$
E_\varepsilon(f)=\int_0^1
r^{n-1}[\frac 1p(f_r^2+(n-1)r^{-2}f^2)^{p/2}
  +\frac{1}{4\varepsilon^p}(1-f^2)^2] \,dr
$$
This means that $u_\varepsilon(x)
=f_\varepsilon(r)x|x|^{-1}$ is the minimizer of
$E_\varepsilon(u,B)$ on $W$ if and only
if $f_\varepsilon(r)$ is the minimizer of
$E_\varepsilon(f)$ on $V$.

Parallel to the discussions in the previous sections we can obtain the
corresponding results. In particular, we have the results on the location
of zeroes of minimizers and on the convergence rate for minimizers. Also
it can be proved that if $u_\varepsilon(x)
=f_\varepsilon(r)x|x|^{-1}$ is a radial
minimizer of $E_\varepsilon(u,B)$, then
$$
\lim_{\varepsilon \rightarrow 0}
f_\varepsilon=1 \quad\mbox{in }W^{1,p}((\eta,1],R), \qquad
\lim_{\varepsilon \rightarrow 0}
u_\varepsilon=\frac{x}{|x|} \quad\mbox{in }W^{1,p}(K,R^n)
$$
for any $\eta \in (0,1)$ and any compact subset $K \subset
\overline{B} \setminus \{0\}$.
If $p>2n-2$, then for the regularizable minimizer
$\tilde{u}_\varepsilon(x)=\tilde{f}_\varepsilon(r)x|x|^{-1}$, we have
$$
\lim_{\varepsilon \rightarrow 0}\tilde{f}_\varepsilon=1 \quad\mbox{in }
C_{\rm loc}^{1,\alpha}((0,1),R)\, \qquad
\lim_{\varepsilon \rightarrow
0}\tilde{u}_\varepsilon=\frac{x}{|x|} \quad\mbox{in }
C_{\rm loc}^{1,\alpha}(B \setminus \{0\},R^n)
$$
with some constant $\alpha \in (0,1)$.

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\end{thebibliography} \bigskip

\noindent{\sc  Yutian Lei,  Zhuoqun Wu, \&  Hongjun Yuan} \\
Institute of Mathematics,   Jilin University\\
130023 Changchun China \\
E-mail:wzq@mail.jlu.edu.cn

\end{document}