## Asymptotic behavior of regularizable minimizers of a Ginzburg-Landau functional in higher

## dimensions ^{∗}

### Yutian Lei

Abstract

We study the asymptotic behavior of the regularizable minimizers of a Ginzburg-Landau type functional. We also dicuss the location of the zeroes of the minimizers.

### 1 Introduction

LetG⊂R^{n} (n≥2) be a bounded and simply connected domain with smooth
boundary ∂G. Let g be a smooth map from ∂G into S^{n}^{−}^{1} satisfying d =
deg(g, ∂G)6= 0. Consider the Ginzburg-Landau-type functional

Eε(u, G) = 1 p Z

G

|∇u|^{p}+ 1
4ε^{p}

Z

G

(1− |u|^{2})^{2}, (p >1)

with a small parameter ε > 0. It is known that this functional achieves its minimum on

Wp={v∈W^{1,p}(G,R^{n}) :v|∂G=g}

at a functionu_{ε}. We are concerned with the asymptotic behavior ofu_{ε}and the
location of the zeroes ofu_{ε}as ε→0.

The functionalEε(u, G) was introduced in the study of the Ginzburg-Landau
vortices by F. Bethue, H. Brezis and F. Helein [1] in the casep=n= 2. Similar
models are also used in many other theories of phase transition. The minimizer
uεofEε(u, G) represents a complex order parameter. The zeroes ofuεand the
module |uε| both have physics senses, for example, in superconductivity |uε|^{2}
is proportional to the density of supercoducting electrons, and the zeroes ofuε

are the vortices, which were introduced in the type-II superconductors.

In the case 1< p < n, it is easily seen that W_{g}^{1,p}(G, S^{n}^{−}^{1})6=∅. It is not
difficult to prove that the existence of solutionupfor the minimization problem

min{ Z

G

|∇u|^{p} :u∈W_{g}^{1,p}(G, S^{n}^{−}^{1})}

∗Mathematics Subject Classifications: 35J70.

Key words: Ginzburg-Landau functional, module and zeroes of regularizable minimizers.

2001 Southwest Texas State University.c

Submitted December 13, 2000. Published February 23, 2001.

1

by taking the minimizing sequence. This solution is called a map of the least p-energy with boundary valueg. Using the variational methods, we can proved that the solutionup is also p-harmonic map onG with the boundary data g, namely, it is a weak solution of the following equation

−div(|∇u|^{p}^{−}^{2}∇u) =u|∇u|^{p}.

Asε→0, there exists a subsequenceuε_{k} ofuε, the minimizer ofEε(u, G), such
that

uε_{k}→up, in W^{1,p}(G,R^{n}).

In the case p > n, W_{g}^{1,p}(G, S^{n}^{−}^{1}) = ∅. Thus there is no map of least
p-energy on G with the boundary value g. It seems to be very difficult to
study the convergence for minimizers of E_{ε}(u, G) in W_{p}. Some results on the
asymptotic behavior of the radial minimizers ofEε(u, G) were presented in [7].

Whenp=n, this problem was introduced in [1] (the open problem 17). M.

C. Hong studied the asymptotic behavior for the regularizable minimizers of
Eε(u, G) inWn [6]. He proved that there exist{a1, a2, . . . , aJ} ⊂G,J ∈N and
a subsequenceuε_{k} of the regularizable minimizersuε such that

uε_{k}

→w un, inW_{loc}^{1,n}(G\ {a1, a2, . . . , aJ},R^{n}) (1.1)
asεk →0, whereun is an n-harmonic map.

In this paper we shall discuss the asymptotic behavior for the regularizable minimizers of Eε(u, G) on Wn in the case p = n. Without loss of generality, we may assumed >0. Recalling a minimizer ofEε(u, G) on Wn be called the regularizable minimizer, if it is the limit of the minimizer of the regularized functional

E_{ε}^{τ}(u, G) = 1
p
Z

G

(|∇u|^{2}+τ)^{p/2}+ 1
4ε^{p}

Z

G

(1− |u|^{2})^{2}, (τ∈(0,1))
onW_{n} in W^{1,p}. It is not difficult to prove that the regularizable minimizer is
also a minimizer ofE_{ε}(u, G). In order to find the zeroes of the minimizers, we
should first locate the singularities of the n-harmonic mapu_{n}.

Theorem 1.1 If aj ∈ G, j = 1,2, . . . , J are the singularities of n-harmonic
map u_{n}, then J =d, the degree deg(u_{n}, a_{j}) = 1, and {a_{j}}^{d}j=1 ⊂G. Moreover,
for everyj, there exists at least one zero of the regularizable minimizeru_{ε}near
toa_{j}.

Because the module of the minimizer has the physics sense, we have also studied its asymptotic behavior.

Theorem 1.2 Let uε be a regularizable minimizer of Eε(u, G),ρ=|uε|, then there exists a constant C independent ofε such that

Z

G

|∇ρ|^{n}≤C, and 1
ε^{n}

Z

G

(1−ρ^{2})≤C(1 +|lnε|).

For any given η >0, denote Gη=G\ ∪^{d}j=1B(aj, η), then asε→0,
1

ε^{n}
Z

G_{η}

(1−ρ^{2})^{2}→0,
ρ→1, inC_{loc}(G_{η}, R).

At last, we develop the conclusion of (1.1) into following

Theorem 1.3 There exists a subsequenceuε_{k} of uε such that as ε→0,
uε_{k}→un, in W_{loc}^{1,n}(G\ ∪^{d}j=1{aj},R^{n}).

We shall prove Theorems 1.2 and 1.3 in§5 and§7 respectively, and the proof of Theorem 1.1 will be given in §6.

### 2 Basic properties of the regularizable minimiz- ers

First we recall the minimizer of the regularized functional
E_{ε}^{τ}(u, G) = 1

n Z

G

(|∇u|^{2}+τ)^{n/2}+ 1
4ε^{n}

Z

G

(1− |u|^{2})^{2}, τ ∈(0,1)
on Wn, denoted by u^{τ}_{ε}. As τ → 0, there exists a subsequence u^{τ}_{ε}^{k} of u^{τ}_{ε} such
that

τlim_{k}→0u^{τ}_{ε}^{k} =uε, inW^{1,n}(G,R^{n}), (2.1)
and the limitu_{ε}is one minimizer ofE_{ε}(u, G) onW_{n}, which is named the regu-
larizable minimizer. It is not difficult to prove thatu^{τ}_{ε} solves the problem

−div[(|∇u|^{2}+τ)^{(n}^{−}^{2)/2}∇u] = 1

ε^{n}u(1− |u|^{2}), onG, (2.2)
u|∂G=g

and satisfies the maximum principle: |u^{τ}_{ε}| ≤1 on G. Moreover

Proposition 2.1 (Theorem 2.2 in [6]) For any δ > 0, there exists a con- stant C independent ofε such that

limτ→0|∇u^{τ}_{ε}| ≤Cε^{−}^{1}, onG^{δε}, (2.3)
where G^{δε} ={x∈G: dist(x, ∂G)≥δε}.

In this section we shall present some basic properties of the regularizable minimizeruε. Clearly it is a weak solution of the equation

−div(|∇u|^{n}^{−}^{2}∇u) = 1

ε^{n}u(1− |u|^{2}), onG, (2.4)
and it is known that|uε| ≤1 a.e. onG[6]. We also have

Proposition 2.2 For any δ > 0, there exists a constant C independent of ε such that

k∇u_{ε}kL^{∞}(B(x,δε/8,R^{n})≤Cε^{−}^{1}, ifx∈G^{δε}.

Proof. Let y=xε^{−}^{1} in (2.4) and denote v(y) =u(x),Gε={y=xε^{−}^{1} :x∈
G}, G^{δ} = {y ∈ Gε : dist(y, ∂Gε) > δ}. Since that u is a weak solution of
(2.4), we have

Z

Gε

|∇v|^{n}^{−}^{2}∇v∇φ=
Z

Gε

v(1− |v|^{2})φ, φ∈W_{0}^{1,n}(Gε,R^{n}).

Takingφ=vζ^{n}, ζ ∈C_{0}^{∞}(Gε, R), we obtain
Z

Gε

|∇v|^{n}ζ^{n}≤n
Z

Gε

|∇v|^{n}^{−}^{1}ζ^{n}^{−}^{1}|∇ζ||v|+
Z

Gε

|v|^{2}(1− |v|^{2})ζ^{n}.

Setting y ∈ G^{δ}, B(y, δ/2) ⊂ Gε, and ζ = 1 in B(y, δ/4), ζ = 0 in Gε\
B(y, δ/2),|∇ζ| ≤C(δ), we have

Z

B(y,δ/2)

|∇v|^{n}ζ^{n} ≤C(δ)
Z

B(y,δ/2)

|∇v|^{n}^{−}^{1}ζ^{n}^{−}^{1}+C(δ).

Using Holder inequality we can derive R

B(y,δ/4)|∇v|^{n} ≤C(δ). Combining this
with the theorem of [9] yields

k∇vk^{n}L^{∞}(B(y,δ/8))≤C(δ)
Z

B(y,δ/4)

(1 +|∇v|)^{n} ≤C(δ)
which implies

k∇ukL^{∞}(B(x,εδ/8))≤C(δ)ε^{−}^{1}.

Proposition 2.3 (Lemma 2.1 in [6]) There exists a constantCindependent of εsuch that forε∈(0,1),

E_{ε}(u_{ε}, G)≤d(n−1)^{n/2}

n |S^{n}^{−}^{1}||lnε|+C. (2.5)
Proposition 2.4 There exists a constant C independent ofε such that

1
ε^{n}

Z

G

(1− |ue|^{2})^{2}≤C. (2.6)

Proof. By (3.6) in [6], Z

G

|∇uε|^{n}≥d(n−1)^{n/2}|S^{n}^{−}^{1}||lnε| −C.

Applying Proposition 2.3 we may obtain (2.6).

### 3 A class of bad balls

Fixρ >0. For the regularizable minimizeruε, from Theorem 2.2 in [6] we know

|uε| ≥ 1

2, onG\G^{ρε}, (3.1)

where G^{ρε} ={x ∈G : dist(x, ∂G) ≥ρε}. Thus there exists no zero of u_{ε} on
G\G^{ρε}.

Proposition 3.1 Let u_{ε} be a regularizable minimizer of E_{ε}(u, G), There exist
positive constants λ, µ which are independent ofε∈(0,1)such that if

1
ε^{n}

Z

G^{ρε}∩B^{2lε}

(1− |uε|^{2})^{2}≤µ, (3.2)
where B^{2lε} is some ball of radius2lεwithl≥λ, then

|uε| ≥ 1

2, ∀x∈G^{ρε}∩B^{lε}. (3.3)

Proof. First it is known that there exists a constantβ >0 such that for any
x∈G^{ρε}and 0< r≤1,

|G^{ρε}∩B(x, r)| ≥βr^{n}.
Next we take

λ= min( 1 4C,1

8ρ), µ= βλ^{n}
16
where Cis the constant in Proposition 2.2.

Suppose that there is a pointx0∈G^{ρε}∩B^{lε} such that|uε(x0)|<1/2, then
applying Proposition 2.2 we have

|uε(x)−uε(x0)| ≤Cε^{−}^{1}|x−x0|=1

4, x∈B(x0, λε)∩G^{ρε}.
Hence

(1− |u_{ε}(x)|^{2})^{2}> 1

16, ∀x∈B(x_{0}, λε)∩G^{ρε},
Z

B(x_{0},λε)∩G^{ρε}

(1− |u_{ε}|^{2})^{2}> 1

16|G^{ρε}∩B(x_{0}, λε)| ≥β 1

16(λε)^{n} =µε^{n}. (3.4)
Since x0 ∈ B^{lε}∩G^{ρε}, we have (B(x0, λε)∩G^{ρε}) ⊂ (B^{2lε}∩G^{ρε}), thus (3.4)
implies

Z

B^{2lε}∩G^{ρε}

(1− |uε|^{2})^{2}> µε^{n}
which contradicts (3.2) and thus the proposition is proved.

To find the zeroes of the regularizable minimizer uε based on Proposition 3.1, we may take (3.2) as the ruler to distinguish the ball of radius λε which contain the zeroes.

Letλ, µ be constants in Proposition 3.1. If 1

ε^{n}
Z

G^{ρε}∩B(x^{ε},2λε)

(1− |u_{ε}|^{2})^{2}≤µ,

thenB(x^{ε}, λε) is called good ball. OtherwiseB(x^{ε}, λε) is called bad ball. From
Proposition 3.1 we are led to

|u_{ε}| ≥ 1

2, onG^{ρε}\ ∪x^{ε}∈ΛB(x^{ε}, λε), (3.5)
where Λ is the set of the centres of all bad balls. (3.5) and (3.1) imply that the
zeroes ofuεare contained in these bad balls.

Now suppose that{B(x^{ε}_{i}, λε), i∈I} is a family of balls satisfying
(i)x^{ε}_{i} ∈G^{ρε}, i∈I

(ii)G^{ρε}⊂ ∪i∈IB(x^{ε}_{i}, λε)
(iii)

B(x^{ε}_{i}, λε/4)∩B(x^{ε}_{j}, λε/4) =∅, i6=j . (3.6)
LetJ_{ε}={i∈I:B(x^{ε}_{i}, λε) is a bad ball}.

Proposition 3.2 There exists a positive integer N which is independent of ε
such that the number of bad balls cardJ_{ε}≤N.

Proof. Since (3.6) implies that every point in G^{ρε} can be covered by finite,
say m (independent of ε) balls, from (2.6) and the definition of bad balls,we
have

µε^{n}cardJε ≤ X

i∈Jε

Z

B(x^{ε}_{i},2λε)∩G^{ρε}

(1− |uε|^{2})^{2}

≤ m Z

∪i∈JεB(x^{ε}_{i},2λε)∩G^{ρε}

(1− |uε|^{2})^{2}

≤ m Z

G

(1− |u_{ε}|^{2})^{2}≤mCε^{n}
and hence cardJ_{ε}≤ ^{mC}_{µ} ≤N.

Similar to the argument of Theorem IV.1 in [1], we have

Proposition 3.3 There exist a subsetJ ⊂Jε and a constant h≥λsuch that

∪i∈J_{ε}B(x^{ε}_{i}, λε)⊂ ∪i∈JB(x^{ε}_{j}, hε),

|x^{ε}_{i}−x^{ε}_{j}|>8hε, i, j∈J, i6=j. (3.7)

Proof. If there are two pointsx1, x2 such that (3.7) is not true withh=λ, we take h1 = 9λ and J1 =Jε\ {1}. In this case, if (3.7) holds we are done.

Otherwise we continue to choose a pair pointsx3, x4which does not satisfy (3.7) and takeh2= 9h1andJ2=Jε\ {1,3}. After at mostN steps we may conclude this proposition.

Applying Proposition 3.3 we may modify the family of bad balls such that
the new one, denoted by {B(x^{ε}_{i}, hε) :i∈J}, satisfies

∪i∈J_{ε}B(x^{ε}_{i}, λε)⊂ ∪i∈JB(x^{ε}_{i}, hε),

λ≤h; cardJ ≤cardJ_{ε}, (3.8)

|x^{ε}_{i} −x^{ε}_{j}|>8hε, i, j∈J, i6=j.

The last condition implies that every two balls in the new family do not intersect.

Asε→0, there exist a subsequencex^{ε}_{i}^{k} ofx^{ε}_{i} and a_{i} ∈Gsuch that
x^{ε}_{i}^{k}→ai, i= 1,2, . . . , N1= cardJ.

Perhaps there may be at least two subsequences converge to the same point, we denote by

a1, a2, . . . , aN2, N2≤N1

the collection of distinct points in{ai}^{N}1^{1}.

To proveaj∈∂G, it is convenient to enlarge a littleG. AssumeG^{0}⊂R^{n} is a
bounded, simply connected domain with smooth boundary such that G⊂G^{0},
and take a smooth map ¯g : (G^{0} \G) → S^{n}^{−}^{1} such that ¯g = g on ∂G. We
extend the definition domain of every element in {u:G→R^{n} :u|∂G =g} to
G^{0} such thatu=gonG^{0}\G. In particular, the regularizable minimizeruεcan
be defined onG^{0}.

Fix a small constantσ >0 such that

B(a_{j}, σ)⊂G^{0}, j= 1,2, . . . , N_{2};
4σ <|aj−ai|, i6=j; 4σ <dist(G, ∂G^{0}).

Writing Λ_{j}={i∈J:x^{ε}_{i}^{k}→a_{j}}, j= 1,2, . . . , N_{2}, we have

∪i∈Λ_{j}B(x^{ε}_{i}^{k}, hεk)⊂B(aj, σ), j= 1,2, . . . , N2

∪j∈JB(x^{ε}_{j}^{k}, hεk)⊂ ∪^{N}j=1^{2} B(aj, σ/4)
B(x^{ε}_{i}^{k}, hε_{k})∩B(x^{ε}_{j}^{k}, hε_{k}) =∅, i, j∈J, i6=j

as long asεk is small enough. Letuεis the regularizable minimizer ofEε(u, G)
and denote d^{k}_{i} =deg(uε_{k}, ∂B(x^{ε}_{i}^{k}, hεk)), l_{j}^{k}=deg(uε_{k}, ∂B(aj, σ)), thus

l^{k}_{j} = X

i∈Λj

d^{k}_{i}, d=

N2

X

j=1

l^{k}_{j}. (3.9)

To prove that the degrees d^{k}_{i} and l^{k}_{j} are independent of εk, we recall a
proposition stated in [6] (Lemma 3.3) or [2] (Theorem 8.2).

Proposition 3.4 Let φ:S^{n}^{−}^{1}→S^{n}^{−}^{1} be aC^{0}-map withdegφ=d. Then
Z

S^{n−1}

|∇τφ|^{n}^{−}^{1}dx≥ |d|(n−1)^{(n}^{−}^{1)/2}|S^{n}^{−}^{1}|.

Proposition 3.5 There exists a constant C which is independent of ε_{k} such
that

|d^{k}_{i}| ≤C, i∈J; |l^{k}_{j}| ≤C, j= 1,2, . . . , N_{2}.

Proof. Sinceu=u_{ε}is a weak solution of (2.4), applying the theory of the local
regularity in [9], we know u∈C(∂B(x^{ε}_{i}^{k}, hεk)). Since (3.5) implies |u| ≥ 1/2
on∂B(x^{ε}_{i}^{k}, hεk), thusφ= ^{u}

|u| ∈C(∂B(x^{ε}_{i}^{k}, hεk), S^{n}^{−}^{1}). From Proposition 3.4,
we have

|d^{k}_{i}| ≤ |S^{n}^{−}^{1}|^{−}^{1}(n−1)^{(1}^{−}^{n)/2}
Z

∂B(x^{εk}_{i} ,hε_{k})

|( u

|u|)_{τ}|^{n}^{−}^{1}.
Since|u| ≥ ^{1}_{2} onG^{0}\G^{ρε}, there is no zero ofu_{ε}in it. Thus

deg(uε_{k}, ∂B(x^{ε}_{i}^{k}, hεk)) = deg(uε_{k}, ∂(B(x^{ε}_{i}^{k}, hεk)∩G^{ρε}^{k}))
and

|d^{k}_{i}| ≤ |S^{n}^{−}^{1}|^{−}^{1}(n−1)^{(1}^{−}^{n)/2}
Z

∂[B(x^{εk}_{i} ,hε_{k})∩G^{ρε}]

|( u

|u|)_{τ}|^{n}^{−}^{1}. (3.10)
Substituting (2.3) and the fact|uε_{k}| ≥^{1}_{2} on∂[B(x^{ε}_{i}^{k}, hεk)∩G^{ρε}] into (3.10), we
obtain

|d^{k}_{i}| ≤Cε^{1}_{k}^{−}^{n}|S^{n}^{−}^{1}|^{−}^{1}(n−1)^{(1}^{−}^{n)/2}(hεk)^{n}^{−}^{1}≤C,

where C is a constant which is independent ofεk. Combining this with (3.9) we can complete the proof of the proposition.

Proposition 3.5 implies that there exist a number k_{j} which is independent
ofε_{k} and a subsequence ofl^{k}_{j} denoted itself such that

l^{k}_{j} →kj, as k→ ∞.

Since l^{k}_{j}, k_{j} ∈N,{l_{j}^{k}} must be constant sequence for any fixed j, namely l^{k}_{j} =
kj. The same reason shows d^{k}_{i} can be writen as di which is also a number
independent ofεk later.

### 4 An estimate for the lower bound

Write Ω^{0} =G^{0}\ ∪^{N}j=1^{2} B(aj, σ). Fixingj∈ {1,2, . . . , N2} and takingi0∈Λj, we
havexi_{0} →aj as ε→0. Thus

∪i∈ΛjB(x^{ε}_{i}, hε)⊂B(x_{i}_{0}, σ/4)⊂B(a_{j}, σ) (4.1)
holds withεsmall enough.

Denote Ωj =B(aj, σ)\ ∪i∈Λ_{j}B(x^{ε}_{i}, hε),Ωjσ =B(xi_{0}, σ/4)\ ∪i∈Λ_{j}B(x^{ε}_{i}, hε).

To estimate the lower bound of k∇uεkL^{n}(Ωj), the following proposition is nec-
essary that was given by Theorem 3.9 in [6].

Proposition 4.1 Let As,t(xi) = (B(xi, s)\B(xi, t))∩G with ε≤t < s≤R.

Assume thatu∈W_{g}^{1,n}(G,R^{n})and^{1}_{2} ≤ |u| ≤1onAs,t(xi). If there is a constant
C such that

1
ε^{n}

Z

A_{s,t}(x_{i})

(1− |u|^{2})^{2}≤C.

Then for ε < ε0 there holds Z

A_{s,t}(x_{i})

|∇u|^{n}≥ |di|^{n/(n}^{−}^{1)}(n−1)^{n/2}|S^{n}^{−}^{1}|lns
t −C,

where C is a constant which is independent of ε and di is the degree of u on
each ∂(B(x_{i}, r)∩G), t≤r≤s.

Proposition 4.2 AssumeCardΛj =N. Then Z

Ω_{j}

|∇u_{ε}|^{n}≥
Z

Ω_{j,σ}

|∇u_{ε}|^{n}≥(n−1)^{n/2}|S^{n}^{−}^{1}||k_{j}|lnσ

ε −C (4.2) where C is a constant which is independent ofε.

Proof. We give the proof following that in [6] (see Theorem 3.10), and the idea
comes from [8]. Supposex1, x2, . . . , xN converge toaj, anddi,R(i= 1,2, . . . , N)
is the degree of uε around ∂B(xi, R). Let R^{σ}_{ε} denote the set of all numbers
R ∈ [ε, σ] such that∂B(xi, R)∩B(xj, ε) = ∅ for all i 6= j and such that for
some collection JR ⊂ {1,2, . . . , N}, satisfying JR ⊂JR^{0} ifR^{0} ≤R, the family
{B(xi, R)}i∈JR is disjoint and

∪^{N}i=1B(xi, ε)⊂ ∪i∈J_{R}0B(xi, R^{0})⊂ ∪i∈J_{R}B(xi, R), R^{0}≤R.

Note that R^{σ}_{ε} is the union of closed intervals [R^{l}_{0},R^{l}],1 ≤l ≤L, whose right
endpoints correspond to a number R=R^{l} such that∂B(x_{i}, R)∩B(x_{j}, R)6=∅
for some pairi6=j∈JR and whose left endpoints correspond to a number R^{l}_{0}
such thatB(xi,R^{l}−1)\ ∪j∈J_{0}B(xj, R^{l}_{0})6=∅fori∈J_{R}l

0. JR=J^{l}is a constant for
R ∈[R^{l}_{0},R^{l}] andJ^{l+1} ⊂J^{l}, J^{l+1}6=J^{l}. ThusL≤N. Moreover, there exists a
constant M =M(h)>0 such that

R^{l}_{0}≤M ε, R^{L} ≥σ/M, R^{l+1}_{0} ≤M R^{l} (4.3)
for alll= 1,2, . . . , L−1. Finally, observe that for allR∈R^{σ}_{ε} andJ ∈JR,

|kj|=| X

i∈JR

di,R| ≤ X

i∈JR

|di,R|^{n/(n}^{−}^{1)}. (4.4)
Applying (4.3)(4.4) and proposition 4.1 we have

Z

Ω_{j,σ}

|∇uε|^{n} ≥

L

X

l=1

X

i∈J^{l}

| Z

A_{Rl ,Rl}

0

(x_{i})

∇uε|^{n}

≥

L

X

l=1

X

i∈J^{l}

|S^{n}^{−}^{1}|(n−1)^{n/2}|d_{i,R}l|ln(R^{l}/R^{l}_{0})−C

≥ |S^{n}^{−}^{1}|(n−1)^{n/2}|kj|X

l

(lnR^{l}−lnR^{l}_{0})−C

≥ (n−1)^{n/2}|S^{n}^{−}^{1}||kj|lnσ
ε −C.

This and (4.1) imply that (4.2) holds.

Remark In fact the following results Z

Ω_{j}

|∇ uε

|u_{ε}||^{n}≥(n−1)^{n/2}|S^{n}^{−}^{1}||k_{j}|^{n/(n}^{−}^{1)}lnσ
ε,

and Z

Ω_{j}

(1− |uε|^{n})|∇ u_{ε}

|uε||^{n}≤C

had been presented in the proof of Theorem 3.9 in [6], where C which is inde- pendent ofε. Noticing

Z

Ω_{j}

|u_{ε}|^{n}|∇u_{ε}

|uε||^{n} =
Z

Ω_{j}

|∇ u_{ε}

|uε||^{n}−
Z

Ω_{j}

(1− |u_{ε}|^{n})|∇ u_{ε}

|uε||^{n},
we have

Z

Ω_{j}

|uε|^{n}|∇ u_{ε}

|uε||^{n}≥(n−1)^{n/2}|kj|^{n/(n}^{−}^{1)}|S^{n}^{−}^{1}|lnσ
ε −C.

Theorem 4.3 There exists a constant C which is independent ofε, σ ∈(0,1) such that

Z

∪^{N}j=1^{2}Ω_{j}

|∇uε|^{n}≥(n−1)^{n/2}|S^{n}^{−}^{1}|dlnσ

ε −C, (4.5)

1 n

Z

G_{σ}

|∇uε|^{n}+ 1
4ε^{n}

Z

G

(1− |uε|^{2})^{2}≤ 1

n(n−1)^{n/2}|S^{n}^{−}^{1}|dln 1

σ+C (4.6)
whereGσ=G\ ∪^{N}j=1^{2} B(aj, σ).

Proof. From (4.2) and Proposition 2.3 we have
(n−1)^{n/2}|S^{n}^{−}^{1}|(

N_{2}

X

j=1

|k_{j}|) lnσ

ε ≤(n−1)^{n/2}|S^{n}^{−}^{1}|dln1
ε +C
or (PN_{2}

j=1|kj| −d) ln^{1}_{ε} ≤C. It is seen asεsmall enough

N_{2}

X

j=1

|kj| ≤d=

N_{2}

X

j=1

kj

which implies

kj ≥0. (4.7)

This and (3.9) imply

N_{2}

X

j=1

|kj|=

N_{2}

X

j=1

kj =d. (4.8)

Substituting (4.8) into (4.2) yields (4.5), and (4.6) may be concluded from (4.5) and Proposition 2.3.

From (4.6) and the fact |u_{ε}| ≤ 1 a.e. on G, we may conclude that there
exists a subsequenceu_{ε}_{k} ofu_{ε} such that

uε_{k}

→w u_{∗}, W^{1,n}(Gσ,R^{n}) (4.9)
as εk→0. Compare (4.9) with (1.1) we knownu_{∗}=un onGσ, and

{a_{j}}^{N}_{j=1}^{2} ={a_{j}}^{J}j=1. (4.10)
These points were called the singularities ofun.

To show these singularitiesaj∈∂G, the following conclussion is necessary.

Proposition 4.4 Assumea∈∂Gandσ∈(0, R)with a small constant R. If
u∈W^{1,n}(A_{R,σ}(a), S^{n}^{−}^{1})∩C^{0}, u=g

on (G^{0}\G)∩B(a, R)anddeg(u, ∂B(a, R)) = 1, then there exists a constantC
which is independent of σsuch that

Z

A_{R,σ}(a)

|∇u|^{n} ≥2^{1}^{n}(n−1)^{n/2}|S^{n}^{−}^{1}|ln1

σ−C . (4.11)

Proof. Similar to the proof of Lemma VI.1 in [1], we may writeGas the half space

{(x1, x2, . . . , xn) :xn>0} locally and aas 0 by a conformal change.

Denote St =∂B(0, t), t∈(σ, R). Noticing that g is smooth on G^{0}\G, we
have

sup

G^{0}\G

|g_{τ}| ≤C_{1}.
Takingtsufficiently small such that

t≤(n−1)^{1/2}(2^{n}^{−}^{1}−1)^{1/(n}^{−}^{1)}
2C1

, then

Z

S_{t}^{−}

|g¯_{τ}|^{n}^{−}^{1}≤ |S_{t}^{−}|C_{1}^{n}^{−}^{1}≤ |S^{n}^{−}^{1}|t^{n}^{−}^{1}C_{1}^{n}^{−}^{1}≤(n−1)^{(n}^{−}^{1)/2}|S^{n}^{−}^{1}|(1−2^{1}^{−}^{n})
(4.12)

with R <1 small enough, where S_{t}^{−} =St∩ {xn <0}. On the other hand we
can be led to

(n−1)^{(n}^{−}^{1)/2}|S^{n}^{−}^{1}| ≤
Z

St

|u_{τ}|^{n}^{−}^{1}=
Z

S_{t}^{+}

|u_{τ}|^{n}^{−}^{1}+
Z

S_{t}^{−}

|g¯_{τ}|^{n}^{−}^{1}
from Proposition 3.4. HereS_{t}^{+}=St\S^{−}_{t} . Combining this with (4.12) yields

Z

S^{+}_{t}

|uτ|^{n} ≥ |S_{t}^{+}|^{−}^{1/(n}^{−}^{1)}(
Z

S^{+}_{t}

|uτ|^{n}^{−}^{1})^{n/(n}^{−}^{1)} (4.1)

≥ 2^{n}^{1}|S^{n}^{−}^{1}|(n−1)^{n/2}t^{−}^{1}. (4.2)
Integrating this over (σ, R), we obtain

Z

A_{R,σ}

|∇u|^{n}≥2^{1}^{n}|S^{n}^{−}^{1}|(n−1)^{n/2}lnR
σ

which implies (4.11). To provek_{j} = 1 for anyj, we supposeR >2σ is a small
constant such that

B(aj, R)⊂G^{0}; B(aj, R)∩B(ai, R) =∅, i6=j. (4.13)
Denote Π ={v∈W^{1,n}(Ω^{0}, S^{n}^{−}^{1})∩C^{0}: deg(v, ∂B(a_{j}, r)) =k_{j}, r ∈(σ, R), j =
1,2, . . . , N2}.

Proposition 4.5 For anyv∈Π, ifkj ≥0, j= 1,2, . . . , N2, then there exists a constant C=C(R)which is independent of σsuch that

Z

Ω^{0}

|∇v|^{n}≥(n−1)^{n/2}|S^{n}^{−}^{1}|(

N_{2}

X

j=1

k

n n−1

j ) ln1

σ−C. (4.14)

Proof. WriteA_{R,σ}(a_{j}) =B(a_{j}, R)\B(a_{j}, σ), thus∪^{N}_{j=1}^{2} A_{R,σ}(a_{j})⊂Ω^{0}. From
Proposition 3.4 we have

kj=|kj| ≤ (n−1)^{(1}^{−}^{n)/2}|S^{n}^{−}^{1}|^{−}^{1}
Z

S^{n−1}

|vτ|^{n}^{−}^{1}

≤ (n−1)^{(1}^{−}^{n)/2}|S^{n}^{−}^{1}|^{(n}^{−}^{1)/n}(
Z

S^{n−1}

|v_{τ}|^{n})^{(n}^{−}^{1)/n}
namely

Z

S^{n−1}

|v_{τ}|^{n}≥(n−1)^{n/2}|S^{n}^{−}^{1}|k^{n/(n}_{j} ^{−}^{1)}.
On the other hand, we may obtain

Z

Ω^{0}

|∇v|^{n} ≥

N_{2}

X

j=1

Z

AR,σ(aj)

|∇v|^{n}

≥

N2

X

j=1

Z R

σ

Z

S^{n−1}

r^{−}^{n}|∇τv|^{n}r^{n}^{−}^{1}dζdr

≥ (n−1)^{n/2}|S^{n}^{−}^{1}|

N_{2}

X

j=1

k_{j}^{n/(n}^{−}^{1)}
Z R

σ

r^{−}^{1}dr

= (n−1)^{n/2}|S^{n}^{−}^{1}|(

N2

X

j=1

k^{n/(n}_{j} ^{−}^{1)}) lnR
σ
which implies (4.14).

### 5 The proof of Theorem 1.2

Let uεbe a regularizable minimizer ofEε(u, G). Proposition 2.4 has given one estimate of convergence rate of |uε|. Moreover, we also have

Theorem 5.1 There exists a constantCwhich is independent ofε∈(0,1)such

that 1

ε^{n}
Z

G

(1− |u_{ε}|^{2})≤C(1 + ln1

ε). (5.1)

Proof. The minimizer u =u^{τ}_{ε} of the regularized functional E_{ε}^{τ}(u, G) solves
(2.2). Taking the inner product of the both sides of (2.2) withuand integrating
overGwe have

1
ε^{n}

Z

G

|u|^{2}(1− |u|^{2}) = −
Z

G

div(v^{(n}^{−}^{2)/2}∇u)u

= Z

G

v^{(n}^{−}^{2)/2}|∇u|^{2}−
Z

∂G

v^{(n}^{−}^{2)/2}uun (5.2)

≤ Z

G

v^{(n}^{−}^{2)/2}|∇u|^{2}+C
Z

∂G

v^{n/2}+C

where n denotes the unit outward normal to ∂G and u_{n} the derivative with
respect to n.

To estimateR

∂Gv^{n/2}, we choose a smooth vector fieldν such thatν|∂G=n.

Multiplying (2.2) by (ν· ∇u) and integrating overG, we obtain 1

ε^{n}
Z

G

u(1− |u|^{2})(ν· ∇u) = −
Z

G

div(v^{(n}^{−}^{2)/2}∇u)(ν· ∇u)

= Z

G

v^{(n}^{−}^{2)/2}∇u·(ν· ∇u)−
Z

∂G

v^{(n}^{−}^{2)/2}|un|^{2}.
Combining this with

1
ε^{n}

Z

G

u(1− |u|^{2})(ν· ∇u) = 1
2ε^{n}

Z

G

(1− |u|^{2})(ν· ∇(|u|^{2}))

= − 1

4ε^{n}
Z

G

(1− |u|^{2})^{2}divν

and Z

G

v^{(n}^{−}^{2)/2}∇u· ∇(ν· ∇u)

= Z

G

v^{(n}^{−}^{2)/2}|∇u|^{2}divν+1
n

Z

G

ν· ∇(v^{n/2})

= Z

G

v^{(n}^{−}^{2)/2}|∇u|^{2}divν+1
n

Z

∂G

v^{n/2}− 1
n

Z

G

v^{n/2}divν
we obtain

Z

∂G

v^{(n}^{−}^{2)/2}|un|^{2}≤ C
4ε^{n}

Z

G

(1− |u|^{2})^{2}+C
Z

G

v^{n/2}+ 1
n

Z

∂G

v^{n/2}.
Thus

Z

∂G

v^{n/2} =
Z

∂G

v^{(n}^{−}^{2)/2}(|u_{n}|^{2}+|g_{t}|^{2}+τ)

≤ C Z

∂G

v^{(n}^{−}^{2)/2}+1
n

Z

∂G

v^{n/2}+CE_{ε}^{τ}(u^{τ}_{ε}, G).

Substituting this into (5.2) yields 1

ε^{n}
Z

G

|u|^{2}(1− |u|^{2})≤CE_{ε}^{τ}(u^{τ}_{ε}, G).

Letτ→0, applying (2.1) and Proposition 2.3 we have 1

ε^{n}
Z

G

|uε|^{2}(1− |uε|^{2})≤CEε(uε, G)≤C(1 +|lnε|)
which and (2.6) imply (5.1).

Theorem 5.2 Denoteρ=|u_{ε}|. There exists a constantCwhich is independent
of ε∈(0,1)such that

k∇ρkL^{n}(G)≤C. (5.3)

Proof. Denote u=uε. From the Remark in§4 we know Z

Ωj

|u|^{n}|∇ u

|u||^{n}dx≥(n−1)^{n/2}|kj|^{n−1}^{n} |S^{n}^{−}^{1}|lnσ
ε −C.

Thus we may modify (4.5) as Z

∪^{N}_{j=1}^{2}Ω_{j}

ρ^{n}|∇ u

|u||^{n}≥(n−1)^{n/2}|S^{n}^{−}^{1}|dlnσ
ε −C.

Combining this with Z

∪^{N}j=1^{2} Ω_{j}

|∇u|^{n} ≥
Z

∪^{N}j=1^{2}Ω_{j}

ρ^{n}|∇ u

|u||^{n}+
Z

∪^{N}j=1^{2} Ω_{j}

|∇ρ|^{n}−C

and Proposition 2.3, we derive Z

∪^{N}j=1^{2}Ω_{j}

|∇ρ|^{n}≤C. (5.4)

On the other hand, from (2.1) and Proposition 2.1 we are led to Z

G^{ρε}∩B(xi,hε)

|∇uε|^{n}= lim

τ_{k}→0

Z

G^{ρε}∩B(xi,hε)

|∇uετ_{k}

|^{n} ≤C(λε)^{n}(C/ε)^{n}≤C,
fori∈Λj. Summarizing fori and using (5.4) we can obtain (5.3).

Theorem 5.3 For theσ >0 in Theorem 4.4, then as ε→0, 1

ε^{n}
Z

G_{3σ}

(1−ρ^{2})^{2}→0, (5.5)

where G3σ=G\ ∪^{N}j=1^{2} B(aj,3σ).

Proof. The regularizable minimizeru_{ε}satisfies
Z

Gσ

|∇u|^{n}^{−}^{2}∇u∇φ= 1
ε^{n}

Z

Gσ

uφ(1− |u|^{2}), (5.6)
where φ ∈ W_{0}^{1,n}(G_{σ},R^{n}) since u_{ε} is a weak solution of (2.4). Denoting u =
u^{τ}_{ε} = ρw, ρ =|u|, w = ^{u}

|u| in G_{σ} and taking φ =ρwζ, ζ ∈ W_{0}^{1,n}(G_{σ},R^{n}), we
have

Z

G_{σ}

|∇u|^{n}^{−}^{2}(w∇ρ+ρ∇w)(ρζ∇w+ρw∇ζ+wζ∇ρ) = 1
ε^{n}

Z

G_{σ}

ρ^{2}ζ(1−ρ^{2}). (5.7)
Substituting 2w∇w=∇(|w|^{2}) = 0 into (5.7), we obtain

Z

Gσ

|∇u|^{n}^{−}^{2}(ρ∇ρ∇ζ+|∇u|^{2}ζ) = 1
ε^{n}

Z

Gσ

ρ^{2}ζ(1−ρ^{2}). (5.8)
SetS={x∈Gσ:ρ(x)>1−ε^{β}}for some fixedβ∈(0, n/2) andρ= max(ρ,1−
ε^{β}), thusρ=ρonS. In (5.8) takingζ= (1−ρ)ψ, whereψ∈C^{∞}(Gσ, R), ψ= 0
onGσ\G2σ,0< ψ <1 onG2σ\G3σ, ψ= 1 onG3σ, we have

Z

G_{σ}

|∇u|^{n}^{−}^{2}ρ∇ρ· ∇ρψ¯ + 1
ε^{n}

Z

G_{σ}

l^{2}(1−ρ^{2})(1−ρ)ψ¯ (5.9)

= Z

Gσ

|∇u|^{n}^{−}^{2}ρ∇ρ∇ψ(1−ρ) +¯
Z

Gσ

|∇u|^{n}ψ(1−ρ)
Noticing 1/2≤l≤1 inGσ and applying (4.6) we obtain

1
ε^{n}

Z

G3σ

(1−ρ)(1−ρ^{2}) +
Z

S∩G3σ

|∇u|^{n}^{−}^{2}|∇ρ|^{2}≤Cε^{β}. (5.10)

On the other hand, (2.6) implies
ε^{2β}|G_{σ}\S| ≤

Z

Gσ\S

(1−l^{2})^{2}≤Cε^{n},

namely|G_{σ}\S| ≤Cε^{n}^{−}^{2β}. Then there exists a small constantε_{0}>0 such that
G_{3σ} ⊂S∪E

asε∈(0, ε0) whereE is a set, the measure of which converges to zero. Thus lim

ε→0

Z

G_{3σ}

(1−ρ^{2})(1−ρ) = lim

ε→0

Z

G_{3σ}

(1 +ρ)(1−ρ)^{2}.
By (5.10),

εlim→0

1
ε^{n}

Z

G_{3σ}

(1 +ρ)^{2}(1−ρ)^{2}

≤ lim

ε→0

2
ε^{n}

Z

G3σ

(1−ρ)(1−ρ^{2}) = 0
This is our conclusion.

Theorem 5.4 Assume B(x,2σ)⊂G_{σ} satisfies
1

ε^{n}
Z

B(x,σ)

(1− |uε|^{2})^{2}→0, asε→0, (5.11)
then|uε| →1 inC(B(x, σ), R).

Proof. SinceB(x,2σ)⊂Gσ, there existsε0sufficiently small so thatB(x, σ)⊂
G^{2δε}^{0}. We always assumeε < ε0. Forx0∈B(x, σ), setα=|uε(x0)|. Proposi-
tion 2.2 implies

|u_{ε}(x)−u_{ε}(x_{0})|< Cε^{−}^{1}τ ε, ifx∈B(x_{0}, τ ε),

whereτ= (1−α)(N C)^{−}^{1}, Cis the constant in Proposition 2.2 andN is a large
number such thatτ < δ. ThusB(x0, τ ε)⊂B(x, σ) and

|uε(x)| ≤α+Cτ, ifx∈B(x0, τ ε), Z

B(x0,τ ε)

(1− |uε(x)|^{2})^{2}≥(1−1/N)^{2}(1−α)^{n+2}πε^{n}(N C)^{−}^{n}.

Combining this with (5.11) we obtain (1−α)^{n+2} =o(1) as ε→0. Thus it is
not difficult to complete the proof of Theorem.

### 6 The proof of Theorem 1.1

It is known that the singularities ofun are inGfrom the discussion in§3. Since
deg(g, ∂G) > 0, we can see that the zeroes of u_{ε} are also in G . Moreover,
the zeroes are contained in finite bad balls, i.e. B(x^{ε}_{i}, hε), i ∈ J. As ε →
0, B(x^{ε}_{i}, hε)→a_{j}, i∈Λ_{j}. This implies that the zeroes ofu_{ε}distribute near these
singularities of u_{n} as ε→0. Thus it is necessary to describe these singularities
{a_{j}}, j= 1,2, . . . , N_{2}.

Proposition 6.1 kj = deg(un, aj).

Proof. Denote Ω^{0}=G^{0}\ ∪^{N}j=1^{2} B(a_{j}, σ). Combining (4.6) and
Z

G^{0}\G

|∇u_{ε}|^{n}=
Z

G^{0}\G

|∇g¯|^{n}≤C,
we have

Z

Ω^{0}

|∇uε|^{n} ≤C+ (n−1)^{n/2}|S^{n}^{−}^{1}|d|lnσ|, (6.1)
whereC is a constant which is independent ofε. ForRin (4.13), from (6.1) we

have Z

AR,σ(aj)

|∇uε|^{n} ≤C.

Then we know that there exists a constantr∈(σ, R) such that Z

∂B(a_{j},r)

|∇uε|^{n}≤C(r)

by using integral mean value theorem. Thus there exists a subsequence uε_{k} of
uεsuch that

uε_{k} →un, inC(∂B(aj, r))
as ε_{k}→0, which implies

k_{j}= deg(u_{ε}, ∂B(a_{j}, σ)) = deg(u_{n}, a_{j}).

Proposition 6.2 k_{j} = 0or k_{j}= 1.

Proof. From the regularity results on n-harmonic maps (see [3][5] or [9]), we
knowun∈C^{0}(Gσ,R^{n}). Set

w=

g¯ onG^{0}\G,
u_{n} onG_{σ},
thenw∈Π. Using Proposition 4.5 and (4.7) we have

Z

Ω^{0}

|∇w|^{n}≥(n−1)^{n/2}|S^{n}^{−}^{1}|(

N_{2}

X

j=1

k

n n−1

j ) ln1

σ−C(R). (6.2)

On the other hand, (6.1) and (4.9) imply
uε_{k}

→w w, inW^{1,n}(Ω^{0},R^{n}).

Noting this and the weak lower semicontinuity of R

Ω^{0}|∇u|^{n}, applying (6.1) we
have

Z

Ω^{0}

|∇w|^{n} ≤lim_{ε}_{k}_{→}_{0}
Z

Ω^{0}

|∇uε_{k}|^{n}≤(n−1)^{n/2}|S^{n}^{−}^{1}|dln 1

σ+C. (6.3) Combining this with (6.2), we obtain

(

N2

X

j=1

k

n n−1

j −d) ln1

σ ≤C or

N2

X

j=1

k

n n−1

j ≤d=

N2

X

j=1

k_{j}

forσsmall enough. Thus (k_{j}^{1/(n}^{−}^{1)}−1)kj ≤0 which implies that the Proposition
holds.

Proposition 6.3 kj >0,j= 1,2, . . . , N2.

Proof. Supposek1= 0 andk2, k3, . . . , kN_{2}>0. Similar to the proof of Theo-
rem 4.3 we have

Z

∪^{N}j=2^{2}Ωj

|∇uε|^{n}≥(n−1)^{n/2}|S^{n}^{−}^{1}|dlnσ
ε −C.

By this we can rewrite (4.6) as Z

G\∪^{N}j=2^{2} B(aj,σ)

|∇u_{ε}|^{n}+ 1
4ε^{n}

Z

G

(1− |u_{ε}|^{2})^{2}≤C(σ).

Thus similar to the proof of Theorem 5.3 we may modify (5.5) as 1

ε^{n}
Z

G\∪^{N}j=2^{2}B(a_{j},3σ)

(1− |u_{ε}|^{2})^{2}→0 (6.4)
asε→0. Noticing

G∩B(a1, σ)⊂G∩B(a1, R)⊂G\ ∪^{N}j=2^{2} B(aj, R)⊂G\ ∪^{N}j=2^{2} B(aj,3σ)
we have

1
ε^{n}

Z

G∩B(a1,σ)

(1− |uε|^{2})^{2}→0. (6.5)
On the other hand, the definition ofa1implies that there exists at least one bad
ballB(x^{ε}_{0}, hε) such that

G∩B(x^{ε}_{0}, hε)⊂G∩B(a1, σ).

Applying the definition of bad ball we obtain 1

ε^{n}
Z

G∩B(a1,σ)

(1− |uε|^{2})^{2}≥ 1
ε^{n}

Z

G∩B(x^{ε}_{0},hε)

(1− |uε|^{2})^{2}≥µ >0
which is contrary to (6.5). This contradiction showsk1>0.