数理解析研究所講究録 1475 巻 2006 年 107-120

107

Asymptotic profile of a radially symmetric solution withtransition layers for an unbalanced bistable equation
沼津工業高等専門学校 Hiroshi Matsuzawa(Hiroshi Matsuzawa)

Numazu National College of Technology

Introduction and Main Results

1

In this paper, we consider the following boundary value problem:in$(\mathrm{P}_{\epsilon})\{$$\frac{-\partial u}{\theta\nu}=0\epsilon^{2}\Delta u=h(|x|)^{2}(u-a(|x|))(1-u^{2})$on$B_{1}(0)$$\partial B_{1}(0)$centered at thewhere $\epsilon>0$ is a small parameter, $B_{1}(0)$ is a unit ball infunction on $[0, 1]$ satisfying $-1<a(|x|\rangle$ $<1$ andorigin and the function is afunction on $[0, 1]$ satisfying $h’(0)=0$ .
. The function is a positive$\mathbb{R}^{N}$$C^{1}$$a$$C^{1}$$h$$\mathrm{a}’(0)=0$

We set $r=|x|$ .
appears in various models such as population genetics, chemical
Problemreactor theory and phase transition phenomena. See [1] and the references therein.
, thenIf the function satisfies $h(r)\equiv 1$ and the function satisfies
has been studied in [1], [4] and [7], In this case, it is shown
this problem
that there exist radially symmetric solutions with transition layers near the set, then$\{x\in B_{1}(\mathrm{O})|a(|x|)=0\}$ . If the set $\{r\in \mathbb{R}|a(r)=0\}$ contains an intervaldelicate,
the problem to decide the configuration of transition layer on I is moreandOn the other hand, in the case of $N=1$ , if the function satisfies
has been studied in [8]the function satisfies $a(r)\equiv 0$ , then this problem
and [9]. In this case, it is shown that there exist stable solutions with transition
layers near prescribed local minimum points of .
withIn this paper, we consider the case where the function satisfies$a(r)=0$ on some interval $I\subset(0, 1)$ . We show the minimum point of the functionof transition$r^{N-1}h(r)$ on I has very important role to decide the configuration$(\mathrm{P}_{\epsilon})$$a(r)\not\equiv 0$

$a$

$h$

$(\mathrm{P}_{\epsilon})$

$I$

$h$

$h(r)\not\equiv 1$$(\mathrm{P}_{\epsilon})$

$a$

$h$

$a$

$a(r)\not\equiv 0$

layer on I in this case.
to
We note that in [4], Dancer and Shusen Yan considered a problem similar
and the nonlinear term is $u(u-a|x|)(1-u)$
ours. They assume that $N\geq 2$ ,andsatisfying $a(r)=1/2$ on $I=[l_{1}, l_{2}]$ and $a(r)<1/2$ for $l_{1}-r>0$ small
corresponding$a(r)>1/2$ for $-l_{2}>0$ small, then a global minimizer of the$r^{N-1}$functional has a transition layer near the , that is, the minimum point ofTheorem 1.3]). In this sense, we can say that our results are naturalon (seeextention of the results in [4]. We are going to follow throughtout the variationa$h\equiv 1$$r$$l_{1}$$I$$[4_{7}$

108

procedure used in [4] with a few modifications prompted by the presence of thefunction .
:Here we state the energy functional corresponding to$h$$(\mathrm{P}_{\epsilon})$$J_{\epsilon}(u)= \oint_{B_{1}(0)}\frac{\epsilon^{2}}{2}|\nabla u|^{2}-F(|x|, u)dx$,

where $F(|x|, u)= \int_{-1}^{u}f(|x|, s)ds$ and $f(|x|, u)=h(|x|)^{2}(u-a(|x|))(1-u^{2})$ .
It is easy to see that the following minimization problem has a minimizer$\inf\{J_{\epsilon}(u)|u\in H^{1}(B_{1}(0))\}$

.

(1.1)

Let $A_{-}=\{x\in B_{1}(\mathrm{O})|a(|x|)<0\}$ and $A_{+}=\{x\in B_{1}(\mathrm{O})|a(|x|)>0\}$ .
In this paper, we will analyze the profile of the minimizer of (1.1). Our main
theorem is the following:
Theorem 1.1. Letand$u_{\epsilon}$$u_{\epsilon}arrow\{$

be a global minimizer of (1.1). Then$u_{\epsilon}$is radially symmetric

1, uniformly on any compact subset of-1 , uniformly on any compact subset of. In particular$A_{-}$,$A_{+}$,, thatconverges unifor$mly$ near the boundary ofis, if $a(r)<0$ on
for some $r_{0}>0$ ,unifor$mly$ on.
and if $a(r)>0$ on
uniformly on
for some$0<r_{1}\leq
r_{2}<1$$a(r)\neq0$Moreover, for any,
withfor $r_{1}-r>0$, we have:small and for $r-r_{2}>0$ small $a(r)=0$ if

as$\epsilonarrow 0$$B_{1}(0)$$u_{\epsilon}$$[r_{0},1]$$u_{\epsilon}arrow 1$$[r_{0},1]$$r_{0}>0_{2}u_{\epsilon}arrow-1$$\overline{B_{1}(0)}\backslash B_{r_{0}}(0)$$\overline{B_{1}(0)}\backslash B_{r_{0}}(0)$$a(r_{i})=0_{f}\mathrm{i}=1\dot,$ $2$$r\in[r_{1}, r_{2}]$(i) If $a(r)<0$ for $r_{1}-r>0$ small and $a(r)>0$ for $r-r_{2}>0$ , then for any
small $\eta>0$ and for any small $\theta>0$ , there eriste a positive number which, there exist
has the following properties: For any
such
that
$\epsilon_{0}$$\epsilon$$\in(0, \epsilon_{0}]$$t_{\epsilon,1}<t_{\epsilon,2}$(a)$u_{\epsilon}(r)>1-\eta$$u_{\epsilon}(t_{\epsilon,1})=1-\eta$for$r\in[r_{1}-\theta, t_{\epsilon,1})$,

for$r\in(t_{\epsilon,2}, r_{2}+\theta]$

.

,

$\{$

$u_{\epsilon}(t_{\epsilon,2})=-1+\eta$$u_{\epsilon}(r)<-1+\eta$(b) The function$u_{\epsilon}(r)$,,is decreasing in(c) The inequality
constants independent of$(t_{\epsilon,1}, t_{\epsilon,2})$$0<R_{1} \leq\frac{t_{\epsilon.2}-t_{\epsilon,1}}{\epsilon}\leq R_{2}$(d) If$t_{\epsilon_{j},1}$,$jarrow\infty_{f}$$\epsilon$$>0$

Aoids, where$\overline{t}$and$R_{2}$

are two.

for some positive sequencethen satisfies$t_{\epsilon_{\mathrm{j}},2}arrow\overline{t}$$R_{1}$$\{\epsilon_{j}\}$converging to zero as$h(\gamma t$ $\overline{t}^{N-1}=\min_{s\in[r_{1},r_{2}]}h(s)s^{N-1}$

.

109

(ii) ij $a(r)>0$ for $r_{1}-r>0$ small and $a(r)<0$ for $r-r_{2}>0$ , tfien for any
small $\eta>0$ and for any small $\theta>0$ , there exists a positive number whichsuch, there exist
has the folloutirtg properties: For any$\epsilon_{0}$$\epsilon\in(0, \epsilon_{0}]$$t_{\epsilon,1}<t_{\epsilon,2}$

that
(a)
y7

$\mathrm{u}\mathrm{e}(\mathrm{r})<-1+$$u_{\epsilon}(t_{\epsilon,1})=-1+\eta$$\{$$u_{\epsilon}(t_{\epsilon,2})=1-\eta$(b) The function$u_{\epsilon}(r)$,for$r\in(t_{\epsilon,2}, r_{2}+\theta]$.

,

is increasing in(c) The inequality
constants independent of$r\in[r_{1}-\theta, t_{\epsilon,1})$,,$u_{\epsilon}(r)>1$ $-\eta$for$(t_{\epsilon,1}, t_{\epsilon,2})$$0<R_{1} \leq\frac{t_{\epsilon 2}-t_{e,1}}{\epsilon}\leq R_{2}$(d) If$t_{\epsilon_{f},1}$,$jarrow\infty$$\epsilon$$>0$holds, where, then$\mathbb{H}1$$R_{1}$

and$R_{2}$

are two.

for some positive sequencesatisfies$t_{\epsilon_{j},2}arrow\overline{t}$$\overline{t}$.

$\{\epsilon_{j}\}$converging to zero as.

$h( \overline{t})\overline{t}^{N-1}=\min_{s\in[r_{1},r_{2}]}h(s)s^{N-1}$: The profile of the global minimizer$u_{\epsilon}$.

Remarks . (i) We note that results from (a) to (c) both in cases (i) and (ii)are not related to the presence of the function . The effect of presence offunction appears in the result (d) in (i) and (ii).
$h$

$h$

(1i) If $\min_{s\in[r_{1},r_{2}]}s^{N-1}h(s)$ is attained at a unique point , we can show
without taking subsequences.
as$\overline{t}$$\overline{t}$$\epsilon$$t_{\epsilon,1}$,$t_{\epsilon,2}arrow$$arrow 0$(iii) If the function $r^{N-1}h(r)$ is constant on
to know the location of the point$[r_{1},r_{2}]$$\overline{t}\in[r_{1}, r_{2}]$, it is a very difficult problem.

This paper is organized as follows. In section 2, we prepare some preliminary
results. We will prove Theorems 1.1 in section 3.

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2

Preliminary Results

In this section we prepare some preliminary results.
be a function defined on
Let $D$ is a bounded domain in. Let$[-1,1]$ . Suppose is continuous onfor eachwhich is bounded on$D$
R. We also assume
for eachand is measurable in$\mathbb{R}^{N}$$\overline{f}(x, t)$$\overline{D}\mathrm{x}$$t\in \mathbb{R}$$\overline{f}$$\overline{D}\mathrm{x}$$\mathbb{R}$$x\in\overline{D}$$t\in$$\overline{f}(x, t)>0$

for$x\in\overline{D}$, $t>1$ .

(2.1): $u-\eta$ $\in H_{0}^{1}(D)\}$ ,(2.2), $t<-1;\overline{f}(x, t)<0$ , for$x\in\overline{D}$Consider the following minimization problem:$\inf\{\overline{J}_{\epsilon}(u, D):=\int_{D}\frac{\epsilon^{2}}{2}|\nabla u|^{2}-\overline{F}(x, u)dx$

where y7 $\in H^{1}(D)$ with $-1\leq\eta$$\leq 1$

on

$D$

and

$\overline{F}(x, t)=\int_{-1}^{t}\overline{f}(x, s)ds$.

We can prove next two lemmas by methods similar to [4], For readers’s convenience
we prove these lemmas in this section.
Lemma 2,1. Suppose that
Then
on D.

$\overline{f}(x,$t) satisfies (2.1). Let$u_{\epsilon}$

be a minimizer of (2.2).

$-1\leq u_{\epsilon}\leq 1$

Proof We prove$-1\leq u_{\epsilon}$

on

$D$

. Let $M=\{x:u_{\epsilon}(x)<-1\}$ . Define$u_{\epsilon}(x)$et, $(x)=\{$-1$\tilde{u}_{\epsilon}$

as follows:

if
if $x\in M$.

$x\in D\backslash M$, we see $M$ is compactly contained in $D$ . Thus
. If the measure $m(M)$ of $M$ is positive, we have.
Becauseis a minimizer, we see $m(M)$ $=0$ , where $m(A)$ denots the Lebesgue$[]$
measure of the set . Thus
. Similarly we can prove that
.

Since

$u_{\epsilon}(x)=$y7 $\geq-1$ on$\partial D$$\tilde{u}-\eta\in$$H_{0}^{1}(D)$$\overline{J}_{\epsilon}(\tilde{u}_{\epsilon}, D)$$<\overline{J}_{\epsilon}(u_{\epsilon}, D)$$u_{\epsilon}$$A$$u_{\epsilon}\geq-1$$u_{\epsilon}\leq 1$

Lemma 2.2, Suppose that
andboth satisfy (2.1) and the someregularity assumption on . Assume thatsatisfy $-1\leq\eta_{i}\leq 1$ on. Let : be a corresponding minimizer of (2.2), uthere
andfor, $i=1,2$ . Suppose that$[-1,1]$ andfor all. Then.
$\overline{f}_{1}(x, t)$$\mathrm{i}=1,2$$\eta$$\overline{f}=\overline{f}_{i}$$\overline{f}_{1}(x, t)\geq\overline{f}_{2}(x,t)$$1\geq\eta_{1}\geq\eta_{2}\geq-1$

Proof Let$D$$u_{\epsilon},$$=\eta_{i}$we have$\overline{f}_{2}(x, t)$$\eta_{i}\in H^{1}(D)$$\overline{f}$$u_{\epsilon,1}\geq u_{\epsilon,2}$$M=\{x\in D : u_{\epsilon,2}>u_{\epsilon,1}\}$$\varphi_{\mathit{6}}\in H_{0}^{1}(D)$$(x, t)\in\overline{D}\mathrm{x}$. Set. Define$\varphi_{\epsilon}=(u_{\epsilon,2}-u_{\epsilon,1})^{+}$$\overline{F}_{i}(x, u)=\int_{-1}^{u}\overline{f}_{i}(x, s)ds$. Since$u_{\epsilon,i}$$J_{\epsilon,i}(u):= \int_{D}\frac{\epsilon^{2}}{2}|\nabla u|^{2}-\overline{F}_{i}(x, u)dx$. Since$\eta_{1}\geq\eta_{2}$is a minimizer of

,

111

and

$\varphi_{\epsilon}=0$0

for$x\in D\backslash M$, we have$\leq$$J_{\epsilon,1}(u_{\epsilon,1}+\varphi_{\epsilon})-J_{\epsilon,1}(u_{\epsilon,1})$$=$$\oint_{M}\frac{\epsilon^{2}}{2}(|\nabla(u_{\epsilon,1}+\varphi_{\epsilon})|^{2}-|\nabla u_{\epsilon,1}|^{2})dx-\int_{M}\oint_{u_{\epsilon,1}}^{u_{\epsilon,1}+\varphi_{e}}\overline{f}_{1}(x, s)ds$$\int_{h\mathrm{f}}\frac{\epsilon^{2}}{2}(|\nabla(u_{\epsilon,1}+\varphi_{\epsilon})|^{2}-|\nabla u_{\epsilon,1}|^{2})dx-\int_{M}\int_{u_{\epsilon.1}}^{u_{\epsilon,1}+\varphi_{\epsilon}}\overline{f}_{2}(x, s)ds$$\leq$$J_{\epsilon_{1}2}(u_{\epsilon,2})-J_{\epsilon,2}(u_{\epsilon,2}-\varphi_{\epsilon})\leq 0$$=$This implies thatsuch that.

is also a minimizer ofis strictly increasing for$u_{\epsilon,1}+\varphi_{\epsilon}$$\overline{f}_{1}(x, t)+Lt$$J_{\epsilon,1}(u)$$x\in\overline{D}$. Let $L>0$ be large enough, $t\in[-1,1]$ . From$-\epsilon^{2}\Delta(u_{\epsilon,1}+\varphi_{\epsilon})=\overline{f}_{1}(u_{\epsilon,1}+\varphi_{\epsilon})$we obtain,.

$-\epsilon^{2}\Delta\varphi_{\epsilon}=\overline{f}_{1}(u_{\epsilon,1}+\varphi_{\epsilon})-\overline{f}_{1}(u_{\epsilon,1})$Thus$-\epsilon^{2}\triangle\varphi_{\epsilon}+L\varphi_{\epsilon}=\overline{f}_{1}(u_{\epsilon,1}+\varphi_{\epsilon})+L(u_{\epsilon,1}+\varphi_{\epsilon})-(\overline{f}_{1}(u_{\epsilon,1})+Lu_{\epsilon,1})>0$

such thatin $D$ . Fix $z_{0}\in M$ . Let’
Strong maximum principle and Hopf lemma in, where $\nu=(x_{0}-z_{0})/|x_{0}-z_{0}|$ . But. This is a contradiction. Thus we obtain$x_{0}\in\partial M$$|x_{0}$–$z_{0}|=\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}(z_{0}, \partial M)$. Using the, we obtain thatfor $x\not\in M$ . Thus,$B_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}(z_{0},\partial M\rangle}(z_{0})$$\mathrm{s}$$\varphi_{\epsilon}(x)=0$$\frac{\partial\varphi_{\epsilon}}{\partial\nu}(x_{0})<0$$M=\emptyset$$\frac{\partial\varphi_{\zeta}}{\partial\nu}(x_{0})=0$.

$\square$

Proof of Main Theorem

3

In this section we prove Theorem 1.1. The following proposition is the first partof Theorem 1.1.
be a global minimizer of the problem (1.1). Then
Proposition 3.1. Let$u_{\epsilon}$$u_{\epsilon}$satisfies

uniformly on any compact subset of
1-1 uniformly on any compact subset of$u_{\epsilon}arrow\{$as$\epsilon$$arrow 0$$A_{-}$$A_{+}$.

Proof Let$x_{0}\in A_{-}$. Choose. Define$\delta$$>0$ small so that

$B_{\delta}(x\mathrm{o})\subseteq\subset A$$f_{x\mathrm{o}\delta,b},(t)=( \min_{z\in B_{\delta}(x_{9})}h(z)^{2})(t-b)(1-t^{2})$$( \max_{z\in\overline{B_{\delta}(x\mathrm{o})}}a(z), 1/2)$

forminimizer of, $t\in$ $[-1,1]$ , we have$x\in\overline{B_{\delta}(x_{0})}$inf. Take $b\in$. Thenbe the$f(|x|,t)\geq f_{x_{\mathrm{O}},\delta,b}(t)$$\{\int_{B_{\delta}(x\mathrm{o})}\frac{\epsilon^{2}}{2}|\nabla u|^{2}-F_{x_{0},\delta,b}(u)dx$. Let$u_{\epsilon,x\mathrm{o}},\delta,b$: $u+1\in H_{0}^{1}(B_{\delta}(x_{0}))\}$ ,

112

where

$F_{x_{0},\delta,b}(t)= \int_{-1}^{t}f_{x_{0},\delta,b}(s)ds$. It follows from Lemm as 2,1 and 2.2 that$u_{\epsilon,x_{0},\delta,b}(x)\leq u_{\epsilon}(x)\leq 1$

Since

, for

$x\in B_{\delta}(x_{0})$, it follows from $[2, 3]$ thatuniformly inas, thus$u_{\epsilon,x_{0},\delta,b}(x)arrow 1$$\int_{-1}^{1}f_{x_{0},\delta,b}(s)ds>0$

uniformly in$B_{\delta/2}(x_{0})$.

$u_{\epsilon}(x)arrow 1$$\epsilon$$arrow 0$$B_{\delta/2}(x_{0})$.

as$\epsilon$$arrow 0$$\square$

To prove the rest of Theorem 14, we need the following proposition and lemma.

Proposition 3.2. Let u be a local minimizer of the following problem:

inf$\{\int_{B_{1}(0)}\frac{1}{2}|\nabla u|^{2}-G(|x|, u)dx$: $u\in H^{1}(B_{1}(0))\}$ .

for each $r\geq 0$ , $g(r, t)$ and, $g(r, t)<0$ if $t<-1$ or $t>1$
are measurable on
for eachis bounded on $[0, k]$ $[-2,2]$ for any $k>0$ . Then zt is radial,and, $u(x)=u(|x|)$ .

Here $G(r, t)= \int_{-1}^{t}g(r, s)ds$ , $g(r, t)$ is$C^{1}$$t\in \mathbb{R}$$t\in \mathbb{R}$$[0, +\infty)$$g_{t}(r, t)$in$|g(r, t)|+|g_{t}(r, t)|$$\mathrm{x}$$\mathrm{i}.e.$

Proof. See [4, Proposition 2.6],$\square$

Before we prove Theorem 1.1, we prepare a lemma.
Lemma 3.3. Let $0<\eta<1$ be any fixed constant and w satis fies$-w_{zz}=w(1-w^{2})$on$\mathbb{R}$,$w(0)=-1+$ y7 (resp. $u(0)=1-\eta$ ),$w(z)\leq-1+$ y7 (resp. $w(z)\geq 1-\eta$ ) for $z\leq 0$ ,is bounded on R.

$\{$

$w$

Then

$w$

is a unique solution ofon ,$-w_{zz}=w(1-w^{2})$$\{$$\mathbb{R}$$w(0)=-1+$ op (resp. $w(0)=1-\eta$ ),$w’(z)>0$ (resp. $w’(z)<0$ )$w(z)arrow\pm 1$ (resp. $w(z)arrow\mp 1$ )Proof. See for example [6].

$z\in \mathbb{R}$as,$zarrow\pm\infty$.
$\square$Now we prove the rest of Theorem 1.1.

Proof of Theorem 1.1. For the sake of simplicity, we prove for the case where

and $a(r)>0$ on $(r_{2},1]$ for some $0<$$a(r)<0$ on $[0, r_{1})$ , $a(r)=0$ on$r_{1}<r_{2}<1$ (see Figure 1 in Section 1).
,
Part 1. First we show that converges uniformly near the boundary of
for any small $\tau>0$ . We note thatuniformly onthat is,$[\mathrm{r}\mathrm{i},\mathrm{r}2]$$u_{\epsilon}$$u_{\epsilon}arrow-1$$\overline{B_{1}(0)}\backslash B_{r_{2}+\tau}(0)$$B_{1}(0)$113

we have$u_{\epsilon}(r)\leq u_{\epsilon}(1-\mathrm{r})$. Now we claim thatasuniformly on
as follows:
for $r\in[1-\tau, 1]$ . We define the function$\epsilonarrow 0$$\overline{B_{1-\tau}(0)}\backslash B_{\mathrm{r}_{2}+\tau}(0)$$u_{\epsilon}arrow-1$$=:T_{\epsilon}$$\tilde{u}_{\epsilon}$

if $r\in[0, 1-\tau]$ifand $r\in[1-\tau, 1]$ ,and $r\in[1-\tau, 1]$ .
if$u_{\epsilon}(r)$$\tilde{u}_{\epsilon}(r)=\{$$u_{\epsilon}(r)<T_{\epsilon}$$u_{\epsilon}(r)$$u_{\epsilon}(r)\geq T_{\epsilon}$$T_{\epsilon}$

and $-F(r, T_{\epsilon})\leq-F(r, t)$ for $>0$ and $|r-1|$ small
We note thatand we have a contradiction if we. Hence we obtain
andand $r\in[1-\tau,$ } isassume that the measure of the set {.
uniformly on
andpositive. Henceis radially symmetric andPart 2. Next we remark that, by Proposition 3.2,,
is a minimizer of the follow ing problem
we note that for any$\tilde{u}_{\epsilon}\in H^{1}(B_{1}(0))$$\epsilon$$J_{\epsilon}(\tilde{u}_{\epsilon})<J_{\epsilon}(u_{\epsilon})$$t\geq T_{\epsilon}$$1]$$r\in[\mathrm{O},$ $1]|u_{\epsilon}(r)>T_{\epsilon}$$\overline{B_{1}(0)}\backslash B_{\gamma_{2}+\mathcal{T}}(0)$$u_{\epsilon}arrow-1$$-1<u_{\epsilon}(r)\leq T_{\epsilon}$$u_{\epsilon}$$t_{2}>t_{1}$$u_{\epsilon}$$\inf\{J_{\epsilon}(u, B_{t_{2}}(0)\overline{\backslash B_{t_{1}}(0)}) :,u-u_{\epsilon}\in H_{0}^{1}(B_{t_{2}}(0)\overline{\backslash B_{t_{1}}(0)})\}$

where$J_{\epsilon}(u, M)= \oint_{M}\frac{\epsilon^{2}}{2}|\nabla u|^{2}-F(|x|, u)dx$

be the minimum value of this minimization
for any open set $M$ . Let
problem.
.
has exactly one layer near the interval
In this part we show that
Step 2.1. First we estimate the energy of transition layer.
uniformly on
Let $\eta>0$ and $\theta>0$ be small numbers. Sincesuch, $1-\theta$], we can finduniformly on [r2andsmall. Let, $u_{\epsilon}(r)<1-\eta$ for
y7 if
that
small. We may,
y7 forif
be such thatandassume that
. ThenWe employ the so-called blow-up argument. Let$m_{\epsilon,\mathrm{t}_{1},t_{2}}$$[r_{1}, r_{2}]$$u_{\epsilon}$$[0, r_{1}-\theta]$$u_{\epsilon}arrow 1$$\overline{r}_{\epsilon}\in(r_{1}-\theta, r_{2}+\theta)$$+\theta$$u_{\epsilon}arrow-1$$u_{\epsilon}(r)\geq 1-$$r-\overline{r}_{\epsilon}>0$$r\in[0, \overline{r}_{\epsilon}]$$u_{\epsilon}(r)\leq\eta$$r\in[\tilde{r}_{\epsilon}, 1-\theta]$$\overline{r}_{\epsilon}arrow\overline{r}\in[r_{1}, r_{2}]$$\tilde{r}_{\epsilon}>\overline{r}_{\epsilon}$$\tilde{r}_{\epsilon}-r>0$$u_{\epsilon}(r)>$$\tilde{r}_{\epsilon}arrow\tilde{r}\in[r_{1}, r_{2}]$$v_{\epsilon}(t)=u_{\epsilon}(\epsilon t+\overline{r}_{\epsilon})$$-v_{\epsilon}’’- \epsilon\frac{N-1}{\epsilon t+\overline{r}_{\epsilon}}v_{\epsilon}’=f(\epsilon t+\overline{r}_{\epsilon}, v_{\epsilon})$$-1\leq v_{\epsilon}\leq 1$$v_{\epsilon}arrow v$

in

and

$C_{1\mathrm{o}\mathrm{c}}^{1}(\mathbb{R})$$\mathrm{u}\mathrm{e}(0)=1-\eta$. Since,$\overline{r}_{\epsilon}arrow\overline{r}\in[r_{1}, r_{2}]$, it is easy to see that

and$-v’=h(\overline{r})^{2}(v-v^{3})$, $t\in$ R.

and $v(t)\geq 1-\eta$ for $t\leq 0$ . If we set $v(t)=V(h(\overline{r})t)$ , the function$\{\begin{array}{l}-V^{\prime/}=V-V^{3}V(0)=1-\eta V’(\mathrm{t})\geq 1-\eta\end{array}$$\mathrm{o}\mathrm{n}\mathbb{R}t\leq 0’$.

$V(t)$

satisfies
(3.1)

114

Hence by Lemma 3.3, the function$V$

is a unique solution for$-V’=V-V^{3}$on$\mathbb{R}$,$V(0)=1-\eta$ ,$\{$$V’(t)<0$$t\leq 0$$V(t)arrow\pm 1$

as

(3.2)

.

$tarrow\mp\infty$.

Thus, we can find an $R>0$ large, such that $v(R)=\eta$ . Since
we can find an $R_{\epsilon}\in(R-1, R+1)$ , such that $v_{\epsilon}’(r)<0$ if
and$\mathrm{v}(\mathrm{R})=-1+\eta$ . Henceif$v_{\epsilon}arrow v$$u_{\epsilon}^{l}(r)<0$$r\in[\overline{r}_{\epsilon},\overline{r}_{\mathcal{E}}+\epsilon R_{\epsilon}]$in$C_{1\mathrm{o}\mathrm{c}}^{1}(\mathbb{R})$$r\in[0, R_{\epsilon}]$

,

and
.

$u_{\epsilon}(\overline{r}_{\epsilon}+\epsilon R_{\epsilon})=-1+\eta$Then we have
$J_{\epsilon}(u_{\epsilon}, B_{\overline{r}_{\epsilon}+\epsilon R_{\epsilon}}(0)\backslash \overline{B_{\overline{r}_{\epsilon}}(0)})$

(3.3)

$=$

$\omega_{N-1}(\overline{r}_{\epsilon}^{N-1}+o_{\epsilon}(1))\int_{\overline{r}_{\epsilon}}^{\overline{r}_{\epsilon}+\epsilon R_{\epsilon}}(\frac{\epsilon^{2}}{2}|u_{\epsilon}’|^{2}-F(t, u_{\epsilon}))dt$$=$$\omega_{N-1}(\overline{r}_{\epsilon}^{N-1}+o_{\epsilon}(1))\epsilon\int_{0}^{R_{\epsilon}}(\frac{1}{2}|v_{\epsilon}’|^{2}-F(\epsilon t+\overline{r}_{\epsilon}, v_{\epsilon}))dt$$=$$\omega_{N-1}(\overline{r}_{\epsilon}^{N-1}+o_{\epsilon}(1))(\beta_{h(\overline{r})}+O(\eta)+o_{\epsilon}(1))\epsilon$,

is the area of the unit sphere in
wherepositive value defined by$\mathbb{R}^{N}$$\omega_{N-1}$$\beta_{h(s)}$,$o_{\epsilon}(1)arrow 0$

as$\epsilon$$arrow 0$$=$$\int_{-\infty}^{+\infty}(\frac{1}{2}|w_{h(s\}}’(t)|^{2}+h(s)^{2}\frac{(w_{h(s)}^{2}-1)^{2}}{4})dt$$=$$h(s) \int_{-\infty}^{+\infty}\frac{1}{2}|V’(t)|^{2}+\frac{(V(t)^{2}-1)^{2}}{4}dt$$=$$h(s)\beta_{1}$,$\beta_{h(s)}$is theand $w_{h(s)}(t)=V(h(s)t)$ for $s\in[0,1]$ . We note that although the function
depends on , the value$V$$\eta$$\beta_{1}=\int_{-\infty}^{+\infty}\frac{1}{2}|V’(t)|^{2}+\frac{(V(t)^{2}-1)^{2}}{4}dt$

is independent of .
. To show
has exactly one layer near the interval
Step 2,2. We claim, it sufficient to prove the following
has exactly one layer near the interval
claim:.
Claim.
such that
Suppose that the claim is not true. Then we can find a$u_{\epsilon}(r)<-1+\eta$ if. Thus we can use the blow-up
,
if
with
to deduce that there is aargument again at$\eta$$[r_{1}, r_{2}]$$u_{\epsilon}$$[r_{1}, r_{2}]$$u_{\epsilon}$$\tilde{r}_{\epsilon}=\overline{r}_{\epsilon}+\epsilon R_{\epsilon}$$t_{\epsilon}>\overline{r}_{\epsilon}+R_{\epsilon}\epsilon$$r\in(\overline{r}_{\epsilon}+R_{\epsilon}\epsilon, t_{\epsilon})$$u_{\epsilon}(t_{\epsilon})=-1+\eta$$\tilde{t}_{\epsilon}=t_{\epsilon}+\epsilon\tilde{R}_{\epsilon}$$t_{\epsilon}$$u_{\epsilon}’(r)>0$

115$arrow 0$

for some$J_{\epsilon}(u_{\epsilon}, B_{\tilde{t}_{\epsilon}}(0)\backslash \overline{B_{t_{\epsilon}}(0)})=\omega_{N-1}(t_{\epsilon}^{N-1}+o_{\epsilon}(1))(\beta_{h(7t}+O(\eta))\epsilon+o_{\epsilon}(1)$(3.4)$r\in(t_{\epsilon},\tilde{t}_{\epsilon})$$\overline{t}\in[r_{2}, r_{3}]$,$u_{\epsilon}(\tilde{t}_{\epsilon})=1-\eta$. We may assume that$t_{\epsilon},\tilde{t}_{\epsilon}arrow\overline{t}$as$\epsilon$. Moreover.
. Suppose
Now we claim
Let $F_{a}(t)= \int_{-1}^{t}(v-a)(1-v^{2})dv$ . Then for any $t>0$ small and $s\in[-1+t$ , l-tt,$\tilde{t}_{\xi j}<r_{1}$$\overline{t}_{\epsilon}\geq r_{1}$$F_{a}(1-t)-F_{a}(s)$
$=$

$=$

$F_{0}(1-t)-\mathrm{F}\mathrm{O}(\mathrm{s})+\mathrm{F}\mathrm{o}(1-t)-F_{0}(1-t)-F_{a}(s)+F_{0}(s)$(3.5)$\ovalbox{\tt\small REJECT}\frac{(v^{2}-1)^{2}}{4}\ovalbox{\tt\small REJECT}_{s}^{1-\mathrm{t}}-a\oint_{s}^{1-t}(1-v^{2})dv$

Thus it follows from (3.5) that if $a<0$ then
(3.6)

$F_{a}(1-t)-F_{a}(s)>0$

for $s\in[-1+t, 1-t]$ . Define,$1-\eta$$r\in[\overline{r}_{\epsilon},\overline{r}_{\epsilon}+R_{\epsilon}\epsilon]\cup[t_{\epsilon},\overline{t}_{\epsilon}]$$-u_{\epsilon}(r)$$r\in[\overline{r}_{\epsilon}+R_{\epsilon}\epsilon, t_{\epsilon}]$$\overline{u}_{\epsilon}(r):=\{$

By the assumption that
. Hence, we obtain$\tilde{t}_{\epsilon}<r_{1}$.

and using (3.6), we see$F(r, u_{\epsilon})<F(r,\overline{u}_{\epsilon})$if$r\in[\overline{r}_{\epsilon},\tilde{t}_{\epsilon}]$$J_{\epsilon}(\overline{u}_{\epsilon}, B_{\overline{t}_{\epsilon}}(0)\backslash \overline{B_{\overline{r}_{e}}(0)})<J_{\epsilon}(u_{\epsilon}, B_{\tilde{t}_{\mathrm{e}}}(0)\backslash \overline{B_{\overline{\mathrm{r}}_{\epsilon}}(0)})$.

.
Thus we obtain a contradiction. Therefore we have that
Since $a(r)\geq 0$ for $r\in[r_{1},1]$ , we see $F(r, t)\leq F(r, -1)=0$ if $r\in[r_{1},1]$ . Since$u_{\epsilon}(r)\in(-1, -1+\eta)$ for, we have$\tilde{t_{\epsilon},}\geq r_{1}$$r\in[\overline{r}_{\epsilon}+R_{\epsilon}\epsilon, t_{\epsilon}]$$m_{\epsilon,\overline{r}_{\mathrm{g}},\tilde{r}_{\epsilon}}$$=$$J_{\epsilon}(\overline{u}_{\epsilon}, B_{\overline{r}_{\epsilon}+\epsilon R_{\mathrm{e}}}(0)\backslash \overline{B_{\overline{f}_{\text{\’{e}}}}(0)})+J_{\epsilon}(\overline{u}_{\epsilon}, B_{\tilde{t}_{\epsilon}}(0)\backslash \overline{B_{\mathrm{C}_{\epsilon}}(0)})$$+J_{\epsilon}(\overline{u}_{\epsilon}, B_{t_{\epsilon}}(0)\backslash \overline{B_{\overline{\tau}_{e}+\epsilon R_{\epsilon}}(0)})+J_{\epsilon}(\overline{u}_{\epsilon}, B_{\overline{r}_{e}}(0)\backslash \overline{B_{\overline{t}_{\epsilon}}(0)})$$\geq$$\omega_{N-1}$$(\overline{r}_{\epsilon}^{N-1}\beta_{h(\overline{r})}\epsilon +t_{\epsilon}^{N-1}\beta_{h(\overline{t})}\epsilon)+O(\eta\epsilon)+o(\epsilon)$$+ \inf\{-\int_{B_{t_{\xi}}(0)\backslash B_{\overline{\mathrm{r}}_{\mathcal{E}}+\epsilon R_{\xi}}\langle 0)}F(r, w):-1\leq w\leq 1+\eta\}$$+ \inf\{-\int_{B_{\tilde{r}_{\mathrm{S}}}(0)\backslash B_{\tilde{t}_{\zeta}}(0)}F(r, w):-1\leq w\leq 1\}$$\geq$$\omega_{N-1}$$(\overline{r}_{\epsilon}^{N-1}\beta_{h(\overline{r})}\epsilon +t_{\epsilon}^{N-1}\beta_{h\langle\overline{t})}\epsilon)+O(\eta\epsilon)+o(\epsilon)$

(3.7)

118

. Let $R>0$ be such that $V(h(\overline{r})R)=\eta$ ,Now we give an upper bound for
as follows:
where is a unique solution to (3.2). Define$m_{\epsilon,\overline{\tau}_{\epsilon},\overline{r}_{\epsilon}}$$V$$\overline{u}_{\epsilon}$$\overline{u}_{\epsilon}(r)$$:=\{$$V(h(\overline{r})_{\vec{\epsilon}}^{\underline{\mathrm{r}}-\overline{r}})$$r\in[\overline{r}_{\epsilon}, \overline{r}_{e}+\epsilon R]$$-1+\eta^{-q}\epsilon(r-\overline{r}_{\epsilon}-\epsilon R)$$r\in[\overline{r}_{\epsilon}+\epsilon R_{1}\overline{r}_{\epsilon}+\epsilon R+\epsilon]$-1$r\in[\overline{r}_{\epsilon}+\epsilon R+\epsilon,\overline{r}_{\epsilon}-\epsilon]$$-1+1\epsilon$ $(r-\tilde{r}_{\epsilon}+\epsilon)$$r\in[\tilde{r}_{\epsilon}-\epsilon,\tilde{r}_{\epsilon}]$Now we note that $|F(r, t)|=O(\eta)$ for
we have$\leq$$r\in[\overline{r}_{\epsilon},\tilde{r}_{\epsilon}]$(3.8)

and $-1\leq t\leq-1+\eta$ . Then

$J_{\epsilon}(\overline{u}_{\epsilon}, B_{\overline{r}_{\epsilon}}(0)\backslash \overline{B_{\overline{r}_{\mathrm{C}}}(0)})$$m_{\mathcal{E},\overline{T}_{\mathrm{S}},\overline{\Gamma}_{\mathit{9}}}$$\leq$(3.9)$J_{\epsilon}(\overline{u}_{\epsilon}, B_{\overline{r}_{\epsilon}+R\epsilon}(0)\backslash \overline{B_{\overline{r}_{\epsilon}}(0\rangle})+J_{\epsilon}(\overline{u}_{\epsilon}, B_{\tilde{r}_{\xi}}(0)\backslash \overline{B_{\tilde{r}_{\epsilon}-\epsilon}(0)})$$+J_{\epsilon}(\overline{u}_{\epsilon}, B_{\overline{t}_{\epsilon}-\epsilon}(0)\backslash \overline{B_{\overline{\mathrm{r}}_{\epsilon}+\epsilon R}(0)})$$\leq$

Wy $-$ :$=$$\omega_{N-1}\overline{r}_{\epsilon}^{N-1}\beta_{h(\overline{r})}+O(\eta\epsilon)+o(\epsilon)$

By (3.7) and (3.9), we have$\omega_{N-1}(\overline{r}_{\epsilon}^{N-1}\beta_{h(\overline{r})}+t_{\xi j}^{N-1}\beta_{h(\overline{t})})\epsilon\leq\omega_{N-1}\overline{r}_{\epsilon}^{N-1}\beta_{h(\overline{r})}\epsilon+O(\epsilon\eta)+o(\epsilon)$

This is a contradiction. So we can conclude
Part 3. It remains to prove that ifconverging to zero as $jarrow$ oo then satisfies$\tilde{r}_{\epsilon}=\overline{r}_{\xi}+\epsilon R_{\epsilon}$$\overline{r}_{\epsilon_{f}}arrow\overline{r}$.

for some positive sequence$\{\epsilon_{j}\}$$\overline{r}$$\overline{r}^{N-1}h(\overline{r})=\min_{s\in[r_{1},r_{2}]}s^{N-1}h(s)$.

satisfies
Step 3.1. First we note that from Part 1, the function$-1+\eta$ forin this case.
$H(s)=s^{N-\mathrm{I}}h(s)$. Assume that the result is not true. Then
Step 3.2. Set(denoted by ) such that
there exists a subsequence ofsuch thatand $H(r’)> \min_{s\in 1^{r_{1},r_{2}}}{}_{]}H(s)$ . Then we can find a point$-1\leq u_{\epsilon}\leq$$u_{\epsilon}$$r\in[\overline{r}_{\epsilon}+\epsilon R_{\epsilon}, 1]$$\{\overline{r}_{\epsilon}\}$$\overline{r}_{\epsilon}arrow r’\in[r_{1}, r_{2}]$$\overline{r}_{\epsilon}$$\overline{t}\in(r_{1}, r_{2})$$H(r’)>H(]t$ .

Next we give a lower estimate for$J_{\epsilon}(u_{\epsilon})$$=$$J_{\epsilon}(u_{\epsilon})$. We have$J_{\epsilon}(u_{\epsilon}, B_{\overline{r}_{\epsilon}}(\mathrm{O}))+J_{\epsilon}(u_{\epsilon},B_{\overline{r}_{\epsilon}+\epsilon R_{e}}(0)\backslash B_{\overline{r}_{\epsilon}}(0))$$+J_{\epsilon}(u_{\epsilon}, B_{1}(\mathrm{O})\backslash \overline{B_{\overline{r}_{\epsilon}+R_{C}\epsilon}(0)})$First we note that.

(3.10)

forand for sufficiently small y7 $>0$ ,$-F(r, u)\geq-F(r, 1)(u\in[1-\eta, 1])$ . We also remark that since $a(r)<0$ for $r<r_{1}$and $a(r)=0$ for
and $a(r)>0$ for $r>r_{2}$ , we have $-F(r, 1)<0$ for$1-\mathrm{y}7$$\leq u_{\epsilon}(r)\leq 1$$r_{1}\leq r\leq r_{2}$$r\leq\overline{r}_{\epsilon}$

117$r<r_{1}$have

and $-F(r, 1)=0$ for$- \int_{r_{1}}^{\overline{r}_{\epsilon}}r^{N-1}F(r, 1)dr\geq 0$$J_{\epsilon}(u_{\epsilon}, B_{\overline{r}_{\epsilon}}(0))$

and $-F(r, 1)>0$ for $r>r_{2}$ . Hence weand we obtain the following estimate$r_{1}\leq r\leq r_{2}$$\geq$$- \oint_{0}^{\overline{r}_{\epsilon}}r^{N-1}F(r, u_{\epsilon})dr$$\geq$$- \oint_{0}^{\overline{r}_{\epsilon}}r^{N-1}F(r, 1)dr$$=$$- \int_{0}^{r_{1}}r^{N-1}F(r, 1)dr-\int_{r_{1}}^{\overline{r}_{\epsilon}}r^{N-1}F(r, 1)dr$$\geq$$- \oint_{0}^{r_{1}}r^{N-1}F(r, 1)dr=:A$.

We also obtain.

(3.11)$J_{\epsilon}(u_{\epsilon}, B_{\overline{r}_{\epsilon}+R_{e}\epsilon}(0)\backslash B_{\overline{f}_{\xi}}(0))\geq\omega_{N-1}H(r’)\beta_{1}\epsilon+O(\eta\epsilon)+o(\epsilon)$by methods similar to proof of (3.3).
Since $-1\leq u_{\epsilon}(r)\leq-1+$ y7 forand for sufficiently small y7 $>0$ ,$-F(r, u)\geq-F(r, -1)=0(u\in[-1, -1+\eta])$ , we obtain the following estimate:$r\geq\overline{r}_{\epsilon}+\epsilon R_{\epsilon}$$J_{\epsilon}(u_{\epsilon}, B_{1}(0)\backslash B_{\overline{r}_{\epsilon}+R_{c}\epsilon}(0)\}$$\geq$$- \int_{\overline{r}_{e}+\epsilon R_{\epsilon}}^{1}r^{N-1}F(r, u_{\epsilon})dr$$\geq$$- \int_{\overline{r}_{\epsilon}+\epsilon R_{\epsilon}}^{1}r^{N-1}F(r, -1)dr=0$.

(3.12)

Thus we obtain$J(u_{\epsilon})\geq A+\omega_{N-1}H(r’)\beta_{1}\epsilon+O(\eta\epsilon)+o(\epsilon)$

Next we give an upper bound for$\overline{w}_{\epsilon}(r):=\ovalbox{\tt\small REJECT}$$J_{\epsilon}(u_{\epsilon})$. Consider the following function$V(^{\epsilon}1-q(r \overline{t}+\epsilon)1-1_{\epsilon}-\mathrm{i}1(-\overline{t}-\epsilon R’-\epsilon)-1h(\overline{t})\frac{r-\overline{t}-}{r\epsilon})$$\mathrm{W}\mathrm{y}$$\overline{w}_{\epsilon}$:$r\in[0,\overline{t}-\epsilon,]r\in[t-\epsilon,\neg tr\in\ulcorner t,\overline{t}+\epsilon R’]r\in[t+\epsilon R’,\overline{t}+\epsilon R’+\in]r\in\ulcorner t+\epsilon R+\epsilon,1],$

where $R’>0$ is the number satisfying $V(h(\overline{t})R’)=-- 1$$J_{\epsilon}(u_{\epsilon})\leq J_{\epsilon}(\overline{w}_{\epsilon})\leq A+$(3.13).

$+\eta$. Then we can see${}_{-1}H(\overline{t})\beta_{1}\epsilon+O(\eta\epsilon)+o(\epsilon)$.

(3.14)By (3.13) and (3.14) we have a contradiction. The proof of Theorem 1.1 is completed. In the more complicated case, we can show by similar method(see Remark
below),$\square$118$\mathbb{H}2$:

Remark We briefly show in more complicated case, that is, when is theandand I2function as in Figure 2. More precisely we set(r4 , 1] and $a<0$ on $(r3, r4)$ .
we assume $a>0$ on
be small numbers. As in Part 1, we can find pairs of numbers
Let $\eta>0$ and,,satisfying
andoo and,$a$$I_{1}:=[r_{1}, r_{2}]$$:=[r_{3}, r_{4}]$$[0, r_{1})\cup$$\theta>0$$\overline{r}_{1,\epsilon}\in(r_{1}-\theta, r_{2}+\theta)$$(R_{1,\epsilon}, R_{\epsilon,2})$$(\overline{r}_{1,\epsilon}, \overline{r}_{2,\epsilon})$$\overline{r}_{2,\epsilon}\in(r_{3}-\theta, r_{4}+\theta)$$\sup_{\epsilon}|R_{2,\epsilon}|<$$\sup_{\epsilon}|R_{1,\epsilon}|<\infty$$u_{\epsilon}(r)<-1+\eta u_{\epsilon}(_{2,\epsilon}^{\frac{\frac{rr}{}r}{r}}[perp]\epsilon R_{2,\epsilon})=-1+\eta u_{\epsilon}(_{2,\epsilon})=1-\eta u_{\epsilon}()>1-\eta u_{\epsilon}(_{1,\epsilon}^{\frac{r}{}}+\epsilon R_{1,\epsilon})=1-\eta u_{\epsilon}(_{1\epsilon}^{\frac{r}{}})=-1+\eta u\epsilon(),<-1+\eta \mathrm{f}\mathrm{o}\mathrm{r}0<r<\overline{r}_{1,\epsilon}\mathrm{f}\mathrm{o}\mathrm{r}\overline{r}_{1,\epsilon}+\epsilon R_{1,\epsilon}<r<\overline{r}_{2,\epsilon}\mathrm{f}\mathrm{o}\mathrm{r}\overline{r}_{2,\epsilon}+\epsilon R_{2,\epsilon}<r<1$$\ovalbox{\tt\small REJECT}$

which
for some sequenceandWe assume. In this case it is easy to show that the energy of global
converges to 0 as
is estimated as follows:
minimizer$\{\epsilon_{j}\}$$\overline{r}_{1,\epsilon_{\mathrm{j}}}arrow\overline{r}_{1}\in I_{1}$$\overline{r}_{2,\epsilon_{\mathrm{j}}}arrow\overline{r}_{2}\in I_{2}$$jarrow\infty$$J(u_{\epsilon})$$J_{\epsilon_{j}}(u_{\epsilon_{j}})\geq J_{\epsilon_{\mathrm{j}}}(u_{\epsilon_{j}}, B_{r_{2}-\epsilon}(0))+\epsilon_{j}\omega_{N-1}H(\overline{r}_{2})\beta_{1}+B+O(\epsilon_{j}\eta)+o(\epsilon_{j})$where $B=- \int_{r_{2}}^{f}3r^{N-1}F(r, 1)dr$ .
Let us assume the result does not hold. Thenandhold. We assume. We note that if
assumemore easy$H( \overline{r}_{1})>\min_{s\in I_{1}}H(s)$$H( \overline{r}_{1})=\min_{s\in I_{1}}$$\min_{s\in I_{2}}$$r_{1}=\overline{r}_{1}$,(3.15)or. We also$H(\overline{r}_{2})>$$H( \overline{r}_{2})>\min_{s\in I_{2}}H(s)$$H( \overline{r}_{1})>\min_{\epsilon\in I_{1}}H(s)$or$\overline{r}_{1}\in \mathrm{i}\mathrm{n}\mathrm{t}/\mathrm{i}$, the proof isII9

Let we takethe following function:$\tilde{r}_{2}\in \mathrm{i}\mathrm{n}\mathrm{t}I_{2}$$\tilde{u}_{\epsilon}(r):=\ovalbox{\tt\small REJECT}$

such that $H( \overline{r}_{2})>H(\tilde{r}_{2})>\min_{s\in I_{2}}H(s)$ and consider$u_{\epsilon}(r)V(^{\epsilon}11+_{\epsilon}^{q}(r-r_{2})1- \mathrm{i}\mathit{1}(r-2+\epsilon)-1_{\epsilon}-q(r\tilde{r}_{2}-\epsilon R’-\in)-1h(\tilde{r}_{2})f-\frac{-\tilde{\overline{r}r}}{-\epsilon}\mathrm{a})$$\mathrm{o}\mathrm{n}[\tilde{r}_{2}+\epsilon R"+\epsilon, 1]\mathrm{o}\mathrm{n}[\tilde{r}_{2}’+\epsilon R’’,\tilde{r}_{2}+\epsilon,R’+\epsilon]\mathrm{o}\mathrm{n}[\tilde{r}_{2},\tilde{r}_{2}+\epsilon R^{l\prime}]\mathrm{o}\mathrm{n}\mathrm{r}\tilde{r}2-1\epsilon,\tilde{r}_{2}]\mathrm{o}\mathrm{n}[r2\tilde{r}_{2}-\epsilon]\mathrm{o}\mathrm{n}[r_{2}-\epsilon,r_{2}]\mathrm{o}\mathrm{n}[0r_{2}-7\epsilon)$is the unique solution of (3.2) and $R’$ is the unique value such that
$V(h(r_{1})R’’)=-1+\eta$ .
as follows:
Since is global minimizer, we can estimate the energy of

where

$V$

$J_{\epsilon}(\tilde{u}_{\epsilon})$$u_{\epsilon}$$J_{\epsilon}(u_{\epsilon})\leq J_{\epsilon}(\tilde{u}_{\epsilon})\leq J_{\epsilon}(u_{\epsilon}, B_{r_{2}-\epsilon}(0))+\epsilon\omega_{N-1}H(\tilde{r}_{2})\beta_{1}+B+O(\epsilon\eta)+o(\epsilon)$

Then we have a contradiction from (3.15) and (3.16) by taking
ciently large .

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