\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 $12$, then, since $d \neq 0, W_g^{1,p}(G,S^1)$ must be empty. In this case unlike the case $d=0$ or $12$, 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 $$Z_\varepsilon=\{x \in B; |u_\varepsilon(x)|<1-\eta\} \subset B(0,h \varepsilon) =\{x \in R^2;|x| 0 such that for any x \in B and 0<\rho \leq 1,$$ 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 $$Differentiate with respect to r, multiply by rw'\zeta^2 with \zeta \in C_0^{\infty}(0,1), such that 0 \leq \zeta \leq 1 on [0,1], \zeta =1 on [t_1,t_2], \zeta =0 on [0,1]-[t, t_3], where 0From Theorem~\ref{th3.5}, f has a positive uniform lower bound on [t,1] for \varepsilon >0 small enough. Hence$$ 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)} $$If n=N then we are done. Suppose n0 small enough. Since f_\varepsilon \leq 1, it follows from the maximum principle$$ \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 $$>From this, using(5.8)(5.14)(5.6) and noticing that nFrom (5.4) for n=N it follows immediately that$$ 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 $$on W=\{v \in W^{1,p}(G,R^n);v|_{\partial G}=g\}. When 1n. Assume that G=B, and g=x where B is the unit ball centered at the origin, and consider the minimizers of E_\varepsilon(u,B) on the class of radial functions$$ 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 $\alpha \in (0,1)$. \begin{thebibliography}{20} \bibitem{BBH1;93} F.Bethuel, H.Brezis, F.Helein: {\it Asymptotics for the minimization of a Ginzburg-Landau functional, } Calc. Var.PDE.,{\bf 1} (1993).123-148. \bibitem{BBH2} F.Bethuel, H.Brezis, F.Helein: {\it Ginzburg-Landau Vortices, } Birkhauser. 1994. \bibitem{DL;97} S.J.Ding, Z.H.Liu: {\it On the zeroes and asymptotic behaviour of minimizers to the Ginzburg-Landau functional with variable coefficient } J. Partial Diff. Eqs.,{\bf 10} No.1. (1997).45-64. \bibitem{Gi;89} M.Giaquinta: {\it Multiple integrals in the calculus of variations and nonlinear elliptic systems, } Ann. Math. Stud.,{\bf 105} 1989. \bibitem{HL2;95} R.Hardt, F.H.Lin: {\it Singularities for p-energy minimizing unit vector fields on planner domains, } Cal. Var. PDE.,{\bf 3} (1995), 311-341. \bibitem{Ho2;96} 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{Ho1;95} M.C.Hong: {\it On a problem of Bethuel, Brezis and Helein concerning the Ginzburg-Landau functional, } C. R. Acad. Sic. Paris ,{\bf 320} (1995), 679-684. \bibitem{Ho3;97} M.C.Hong: {\it Two estimates concerning asymptotics of the minimizations of a Ginzburg-Landau functional, } J. Austral. Math. Soc. Ser. A ,{\bf 62} No.1, (1997), 128-140. \bibitem{Ho3;96} Y.T.Lei: {\it Asymptotic behavior of minimizers for functional } Acta. Scien. Natur. Jilin Univ. ,{\bf 1} (1996), 1-6. \bibitem{Ho4;96} Y.T.Lei, Z.Q.Wu: {\it $C^{1,\alpha}$ convergence for minimizers of a Ginzburg-Landau- type functional } to appear. \bibitem{Lin2;96} F.H.Lin: {\it Solutions of Ginzburg-Landau equations and critical points of the renormalized energy, } Ann. Inst. P. Poincare Anal. Nonlineare, {\bf 12} No.5, (1995), 599-622. \bibitem{St1;94} M.Struwe: {\it On the asymptotic behavior of minimizers of the Ginzburg-Landau model in 2-dimensions, } Diff. and Int. Equa.,{\bf 7} (1994), 1613-1624. \bibitem{St2;93} M.Struwe: {\it An asymptotic estimate for the Ginzburg-Landau model, } C.R.Acad. Sci. Paris.,{\bf 317} (1993), 677-680. \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}