\documentclass[twoside]{article} \usepackage{amssymb, amsmath} % font used for R in Real numbers \pagestyle{myheadings} \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 $$Z_\varepsilon=\{x \in B; |u_\varepsilon(x)|<1-\eta\} \subset B(0,h \varepsilon) =\{x \in \mathbb{R}^n;|x| n, let h(r)=f(r^{\frac{p-1}{p-n}}). Then \begin{eqnarray*} \int_0^1|h'(r)|^p\,dr &=&(\frac{p-1}{p-n})^p\int_0^1|f'(r^{ \frac{p-1}{p-n}})|^p r^{\frac{p(n-1)}{p-n}}\,dr \\ &=&(\frac{p-1}{p-n})^{p-1} \int_0^1s^{n-1}|f'(s)|^p\,ds<\infty \end{eqnarray*} by noting f_s(s)s^{(n-1)/p} \in L^p(0,1). If p=n, let h(r)=f(r^x) with x>1 to be determined later. Then for any y \in (1,p), \begin{eqnarray*} \int_0^1|h'(r)|^ydr&=&x^y\int_0^1|f'(r^x)|^yr^{(x-1)y}dr\\ &=&x^{y-1}\int_0^1|f'(s)|^ys^{(x-1)(y-1)/x}ds, \end{eqnarray*} where s=r^x. Choose x,y such that (1-\frac{1}{x})(1-\frac{1}{y})=\frac{n-1}{n}. Hence \begin{eqnarray*} \int_0^1|h'(r)|^ydr&=& x^{y-1}\int_0^1|f'(s)|^ys^{y(n-1)/n}ds\\ &\leq& x^{y-1}(\int_0^1|f'(s)|^ns^{n-1}ds)^{y/n}<\infty. \end{eqnarray*} Using an interpolation inequality and Young inequality, \|h\|_{W^{1,y}((0,1),R)}<\infty which implies that h(r) \in C[0,1] and hence f(r)\in C[0,1]. Suppose f(0)>0, then f(r) \geq s>0 for r \in [0,t) with t>0 small enough since f \in C[0,1]. We have$$ \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. $$This implies that |B_+|=0. Thus |u_{\varepsilon}|\leq 1. \begin{proposition} \label{prop2.5} Assume u_{\varepsilon} is a weak radial solution of (\ref{e2.1})-(ref{e2.2}). Then there exist positive constants C_1,\rho which are independent of \varepsilon, such that \begin{gather} \|\nabla u_{\varepsilon}(x)\|_{L(B(x,\rho\varepsilon/8))}\leq C_1\varepsilon^{-1}, \quad\mbox{if}\quad x \in B(0,1-\rho \varepsilon), \label{e2.3} \\ |u_{\varepsilon}(x)|\geq \frac{10}{11}, \quad\mbox{if}\quad x \in \overline{B}\setminus B(0,1-2\rho\varepsilon). \label{e2.4} \end{gather} \end{proposition} \paragraph{Proof.} Let y=x\varepsilon^{-1} in (\ref{e2.1}) and denote v(y)=u(x), B_{\varepsilon}=B(0,\varepsilon^{-1}). Then $$\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}$$ This implies that v(y) is a weak solution of (\ref{e2.5}). By using the standard discuss of the Holder continuity of weak solution of (\ref{e2.5}) on the boundary (for example see Theorem 1.1 and Line 19-21 of Page 104 in \cite{c1}) we can see that for any y_0 \in \partial B_{\varepsilon} and y \in B(y_0,\rho_0) (where \rho_0>0 is a constant independent of \varepsilon), there exist positive constants C=C(\rho_0) and \alpha \in (0,1), both independent of \varepsilon, such that$$ |v(y)-v(y_0)|\leq C(\rho_0)|y-y_0|^{\alpha}. $$Choose \rho>0 sufficiently small such that $$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}$$ then$$ |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{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} 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. $$Substituting these into (\ref{e2.9}) yields (\ref{e2.7}) and (\ref{e2.8}). \section{Location of zeros of minimizers} \begin{proposition} \label{prop3.1} Let u_\varepsilon be a radial minimizer of E_\varepsilon(u,B). Then there exists a constant C independent of \varepsilon \in (0,1] such that $$\frac{1}{\varepsilon^n} \int_B(1-|u_\varepsilon|^2)^2 \leq C. \label{e3.1}$$ \end{proposition} \paragraph{Proof.} When p>n, (\ref{e3.1}) can be derived by multiplying (\ref{e2.7}) by \varepsilon^{p-n}. When p=n, as in \cite[eqn.(3.6)]{h1}, we derive that$$ \int_B|\nabla u_{\varepsilon}|^ndx \geq (n-1)^{n/2}|S^{n-1}||\ln\varepsilon|-C, $$where C is independent of \varepsilon. Combining this with (\ref{e2.8}) we obtain (\ref{e3.1}). \begin{proposition} \label{prop3.2} Let u_\varepsilon be a radial minimizer of E_\varepsilon(u,B). Then for any \eta \in (0,1/2), there exist positive constants \lambda, \mu independent of \varepsilon \in (0,1) such that if $$\frac{1}{\varepsilon^n} \int_{B(0,1-\rho \varepsilon) \cap B^{2l\varepsilon}}(1-|u_\varepsilon|^2)^2 \leq \mu, \label{e3.2}$$ where B^{2l\varepsilon} is the ball of radius 2l\varepsilon with l \geq \lambda, then $$|u_\varepsilon(x)| \geq 1-\eta, \quad \forall x \in B(0,1-\rho \varepsilon) \cap B^{l\varepsilon}. \label{e3.3}$$ \end{proposition} \paragraph{Proof.} First we observe that there exists a constant C_2>0 which is independent of \varepsilon such that for any x \in B and 0<\rho \leq 1,$$ |B(0,1-\rho\varepsilon)\cap B(x,r)| \geq C_2 r^n. To prove this proposition, we choose $$\lambda=\frac{\eta}{2C_1}, \quad \mu=\frac{C_2}{C_1^n} (\frac{\eta}{2})^{n+2}, \label{e3.4}$$ 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{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} 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 $$\frac{1}{\varepsilon^n} \int_{B(x^{\varepsilon},2\lambda \varepsilon) \cap B(0,1-\rho\varepsilon)} (1-|u_\varepsilon|^2)^2 \leq \mu, \label{e3.6}$$ 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 $$\begin{gathered} x_i^{\varepsilon} \in B(0,1-\rho\varepsilon),i \in I; \\ B(0,1-\rho\varepsilon) \subset \cup_{i \in I}B(x_i^{\varepsilon},\lambda \varepsilon); \\ B(x_i^{\varepsilon},\lambda \varepsilon /4) \cap B(x_j^{\varepsilon},\lambda \varepsilon /4)=\emptyset,i \neq j. \end{gathered} \label{e3.7}$$ 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 $$\begin{gathered} \cup_{i \in J_{\varepsilon}}B(x_i^{\varepsilon},\lambda \varepsilon) \subset \cup_{i \in J}B(x_j^{\varepsilon},h \varepsilon), \\ |x_i^{\varepsilon}-x_j^{\varepsilon}|> 8h\varepsilon,\quad i,j \in J,\quad i \neq j. \end{gathered} \label{e3.8}$$ \end{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 $$|u_{\varepsilon}(x)| \geq \min(\frac{10}{11},1-\eta), \quad x \in B(0,h(\eta)\varepsilon). \label{e3.9}$$ \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. $$Applying \cite[eqn.(2.11)]{t1}, we can see that there exists a positive constant \gamma independent of \varepsilon and h such that $$\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}$$ which implies $$\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}$$ Denote G=B(0,h\varepsilon). Applying \cite[Theorem 2.1]{l1}, we have \|f\|_{\frac{2n}{n-2}} \leq \beta \|\nabla f\|_2, where \beta=2(n-1)/(n-2). Taking f=f_1-f_2 and applying (\ref{e4.2}), we obtain f(|x|)=0 as x \in \partial B and$$ [\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. $$Hence for any given \varepsilon \in (0,1), $$\int_G|f|^2dx \leq C(\beta,|B|,\gamma,\varepsilon) h^2\int_G|f|^2dx. \label{e4.3}$$ Denote F(\eta)=\int_{B(0,h(\eta)\varepsilon)}|f|^2dx, then F(\eta)\geq 0 and (\ref{e4.3}) implies that $$F(\eta)(1-C(\beta,|B|,\gamma,\varepsilon)h^2) \leq 0. \label{e4.4}$$ On the other hand, since C(\beta,|B|,\gamma,\varepsilon) is independent of \eta, we may take 0<\eta<1-\frac{1}{\sqrt{2}} so small that h=h(\eta)\leq \lambda 9^N=9^N\frac{\eta}{2C_1} (which is implied by (\ref{e3.4})) satisfies$$ 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. 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