Electronic Journal of Qualitative Theory of Differential Equations 2003, No. 22, 1-21; http://www.math.u-szeged.hu/ejqtde/

Symmetric solutions to minimization of

a p-energy functional with ellipsoid value

Yutian Lei

Depart. of Math., Nanjing Normal University, Nanjing, 210097, P.R.China Email: leiyutian@pine.njnu.edu.cn

Abstract The author proves the W^{1,p} convergence of the symmetric minimizers u_{ε} = (u_{ε1}, u_{ε2}, u_{ε3}) of a p-energy functional as ε → 0, and the zeros of u_{ε1}^{2} + u_{ε2}^{2} are located roughly. In addition, the estimates of the convergent rate of u_{ε3}^{2} (to 0) are presented. At last, based on researching the Euler-Lagrange equation of symmetric solutions and establishing its C^{1,α} estimate, the author obtains the C^{1,α} convergence of some symmetric minimizer. Keywords: symmetric minimizer, p-energy functional, convergent rate MSC 35B25, 35J70

1 Introduction

Denote B = \{x ∈ R^{2}; x_{1}^{2}+x_{2}^{2} < 1\}. For b > 0, let E(b) = \{x ∈ R^{3}; x_{1}^{2}+x_{2}^{2}+ \frac{x_{3}^{2}}{b^{2}} = 1} be a surface of an ellipsoid. Assume g(x) = (e^{idθ}, 0) where x = (cos θ, sin θ) on ∂B, d ∈ N. We concern with the minimizer of the energy functional

E_{ε}(u, B) = \frac{1}{p} ∫_{B} |∇u|^{p}dx + \frac{1}{2ε^{p}} ∫_{B} u_{3}^{2}dx (p > 2)

in the function class

W = {u(x) = (sin f(r)e^{idθ}, b cos f(r)) ∈ W^{1,p}(B, E(b)); u|_{∂B} = g},

which is named the symmetric minimizer of E_{ε}(u, B).

When p = 2, the functional E_{ε}(u, B) was introduced in the study of some simplified model of high-energy physics, which controls the statics of planar ferromagnets and antiferromagnets (see [5][8]). The asymptotic behavior of minimizers of E_{ε}(u, B) has been considered in [3]. In particular, they discussed the asymptotic behavior of the symmetric minimizer with E(1)-value of E_{ε}(u, B) in §5. When the term \frac{u_{3}^{2}}{ε^{2}} is replaced by \frac{(1−|u|^{2})^{2}}{2ε^{2}} , the functional is the Ginzburg-Landau functional, which was well studied in [1], [4] and [7]. The works in [1] and [3] enunciated that the study of minimizers of the functional E_{ε}(u, B) is connected tightly with the study of harmonic map with E(1)-value. Due to this we may also research the asymptotic behavior of minimizers of E_{ε}(u, B) by referring to the p-harmonic map with ellipsoid value (which was discussed in [2]).

In this paper, we always assume p > 2. As in [1] and [3], we are interested in the behavior of minimizers of E_{ε}(u, B) as ε → 0. We will prove the W_{loc}^{1,p} convergence of the symmetric minimizers. In addition, some estimates of the convergent rate of the symmetric minimizer will be presented and we will discuss the location of the points where u_{3}^{2} = b^{2}.

In polar coordinates, for u(x) = (sin f(r)e^{idθ}, b cos f(r)), we have

|∇u|^{2} = (1 + (b^{2} − 1) sin^{2} f)f_{r}^{2} + d^{2}r^{−2} sin^{2} f,

∫_{B} |∇u|^{p}dx = 2π ∫_{0}^{1} r((1 + (b^{2} − 1) sin^{2} f)f_{r}^{2} + d^{2}r^{−2} sin^{2} f)^{p/2}dr.

If we denote

V = \{f ∈ W_{loc}^{1,p}(0, 1]; r^{1/p}f_{r}, r^{(1−p)/p} sin f ∈ L^{p}(0, 1), f(r) ≥ 0, f(1) = \frac{π}{2} \},

then V = {f(r); u(x) = (sin f(r)e^{idθ}, b cos f(r)) ∈ W }. It is not difficult to see V ⊂ {f ∈ C[0, 1]; f(0) = 0}. Substituting u(x) = (sin f(r)e^{idθ}, b cos f(r)) ∈ W into E_{ε}(u, B) we obtain

E_{ε}(u, B) = 2πE_{ε}(f, (0, 1)),

where

E_{ε}(f, (0, 1)) = ∫_{0}^{1}[\frac{1}{p}(f_{r}^{2}(1+(b^{2} −1) sin^{2} f)+d^{2}r^{−2} sin^{2} f)^{p/2} + \frac{1}{2ε^{p}} b^{2} cos^{2} f]rdr.

This shows that u = (sin f(r)e^{idθ}, b cos f(r)) ∈ W is the minimizer of E_{ε}(u, B) if and only if f(r) ∈ V is the minimizer of E_{ε}(f, (0, 1)). Applying the direct method in the calculus of variations we can see that the functional E_{ε}(u, B) achieves its minimum on W by a function u_{ε}(x) = (sin f_{ε}(r)e^{idθ}, b cos f_{ε}(r)), hence f_{ε}(r) is the minimizer of E_{ε}(f, (0, 1)) in V. Observing the expression of the functional E_{ε}(f, (0, 1)), we may assume that, without loss of generality, the function f satisfies 0 ≤ f ≤ \frac{π}{2} .

We will prove the following

Theorem 1.1 Let u_{ε} be a symmetric minimizer of E_{ε}(u, B) on W. Then for any small positive constant γ ≤ b, there exists a constant h = h(γ) which is independent of ε ∈ (0, 1) such that Z_{ε} = {x ∈ B; |u_{ε3}| > γ} ⊂ B(0, hε).

This theorem shows that all the points where u_{ε3}^{2} = b^{2} are contained in B(0, hε). Hence as ε → 0, these points converge to 0.

Theorem 1.2 Let u_{ε}(x) = (sin f_{ε}(r)e^{idθ}, b cos f_{ε}(r)) be a symmetric minimizer of E_{ε}(u, B) on W. Then

lim_{ε→0} u_{ε} = (e^{idθ}, 0), in W^{1,p}(K, R^{3})  (1.1)

for any compact subset K ⊂ \overline{B} \ \{0\}.

Theorem 1.3 (convergent rate) Let u_{ε}(x) = (sin f_{ε}(r)e^{idθ}, b cos f_{ε}(r)) be a symmetric minimizer of E_{ε}(u, B) on W. Then for any η ∈ (0, 1) and K = \overline{B} \ B(0, η), there exist C, ε_{0} > 0 such that as ε ∈ (0, ε_{0}),

∫_{η}^{1} r[(f′_{ε})^{p} + \frac{1}{ε^{p}} cos^{2} f_{ε}]dr ≤ Cε^{p}.  (1.2)
sup_{x∈K} |u_{ε3}(x)| ≤ Cε^{\frac{p−2}{2}} .  (1.3)

(1.2) gives the estimate of the convergent rate of f_{ε} to π/2 in W^{1,p}(η, 1] sense, and that of convergence of |u_{ε3}(x)| to 0 in C(K) sense is showed by (1.3).

However, there may be several symmetric minimizers of the functional in W. We will prove that one of the symmetric minimizer ũ_{ε} can be obtained as the limit of a subsequence u_{ε}^{τ_{k}} of the symmetric minimizer u_{ε}^{τ} of the regularized functionals

E_{ε}^{τ} (u, B) = \frac{1}{p} ∫_{B}(|∇u|^{2} + τ)^{p/2}dx + \frac{1}{2ε^{p}} ∫_{B} u_{3}^{2}dx, (τ ∈ (0, 1))

on W as τ_{k} → 0. In fact, there exist a subsequence u_{ε}^{τ_{k}} of u_{ε}^{τ} and ũ_{ε} ∈ W such that

lim_{τ_{k}→0} u_{ε}^{τ_{k}} = ũ_{ε}, in W^{1,p}(B, E(b)).  (1.4)

Here ũ_{ε} is a symmetric minimizer of E_{ε}(u, B) in W. The symmetric minimizer ũ_{ε} is called the regularized minimizer. Recall that the paper [3] studied the asymptotic behavior of minimizers u_{ε} ∈ H_{g}^{1}(B, E(1)) of the energy functional E_{ε}(u, B) as ε → 0. It turns out that

lim_{ε→0} u_{ε} = (u_{∗}, 0), in C_{loc}^{1,α}(\overline{B} \ A)  (1.5)

for some α ∈ (0, 1), where u_{∗} is a harmonic map, A is the set of singularities of u_{∗}. Theorem 1.2 has shown the W_{loc}^{1,p}(\overline{B} \ \{0\}) convergence (weaker than (1.5)) of the symmetric minimizer. We will prove that the convergence of (1.5) is still true for the regularized minimizer. The result holds only for the regularized minimizer, since the Euler-Lagrange equation for the symmetric minimizer u_{ε} is degenerate. To derive the C^{1,α} convergence of the regularized minimizer ũ_{ε}, we try to set up the uniform estimate of u_{ε}^{τ} by researching the classical Euler-Lagrange equation which u_{ε}^{τ} satisfies. By this and applying (1.4), one can see the C^{1,α} convergence of ũ_{ε}. So, the following theorem holds only for the regularized minimizer.

Theorem 1.4 Let ũ_{ε} be a regularized minimizer of E_{ε}(u, B). Then for any compact subset K ⊂ \overline{B} \ \{0\}, we have

lim_{ε→0} ũ_{ε} = (e^{idθ}, 0), in C^{1,α}(K, E(b)), α ∈ (0, 1/2).

At the same time, the estimates of the convergent rate of the regularized minimizer, which is better than (1.3), will be presented as following

Theorem 1.5 Let ũ_{ε}(x) be the regularized minimizer of E_{ε}(u, B). Then for any compact subset K of (0, 1] there exist positive constants ε_{0} and C (independent of ε), such that as ε ∈ (0, ε_{0}),

sup_{K} |ũ_{ε3}| ≤ Cε^{λp},  (1.6)

where λ = \frac{1}{2}. Furthermore, if K is any compact subset of (0, 1), then (1.6) holds with λ = 1.

The proof of Theorem 1.1 will be given in §2. In §3, we will set up the uniform estimate of E_{ε}(u_{ε}, K) which implies the conclusion of Theorem 1.2. By virtue of the uniform estimate we can also derive the proof of Theorem 1.3 in §4. For the regularized minimizer, we will give the proofs of Theorems 1.4 and 1.5 in §5 and §6, respectively.

2 Proof of Theorem 1.1

Proposition 2.1 Let f_{ε} be a minimizer of E_{ε}(f, (0, 1)). Then

E_{ε}(f_{ε}, (0, 1)) ≤ Cε^{2−p}

with a constant C independent of ε ∈ (0, 1).

Proof. Denote

I(ε, R) = Min\{∫_{0}^{R}[\frac{1}{p}(f_{r}^{2}(1 + (b^{2} − 1) sin^{2} f) + \frac{d^{2}}{r^{2}} sin^{2} f)^{\frac{p}{2}}
+\frac{1}{2ε^{p}} b^{2} cos^{2} f]rdr; f ∈ V_{R}\},

where V_{R} = \{f(r) ∈ W_{loc}^{1,p}(0, R]; f(R) = \frac{π}{2} , sin f(r)r^{\frac{1}{p} −1}, f′(r)r^{\frac{1}{p}} ∈ L^{p}(0, R)\}. Then

I(ε, 1) = E_{ε}(f_{ε}, (0, 1)) = \frac{1}{p} ∫_{0}^{1} r((f_{ε})_{r}^{2}(1 + (b^{2} − 1) sin^{2} f)
+d^{2}r^{−2}(sin f_{ε})^{2})^{p/2}dr + \frac{1}{2ε^{p}} ∫_{0}^{1} rb^{2} cos^{2} f_{ε}dr
= \frac{1}{p} ∫_{0}^{1/ε} ε^{2−p}s((f_{ε})_{s}^{2}(1 + (b^{2} − 1) sin^{2} f) + d^{2}s^{−2} sin^{2} f_{ε})^{p/2}ds
+\frac{1}{2ε^{p}} ∫_{0}^{ε^{−1}} ε^{2}sb^{2} cos^{2} f_{ε}ds = ε^{2−p}I(1, ε^{−1}).

(2.1)

Let f_{1} be the minimizer for I(1, 1) and define

f_{2} = f_{1}, as 0 < s < 1; f_{2} = \frac{π}{2} , as 1 ≤ s ≤ ε^{−1}.

We have

I(1, ε^{−1})
≤ \frac{1}{p} ∫_{0}^{ε^{−1}} s[(f′_{2})^{2}(1 + (b^{2} − 1) sin^{2} f) + d^{2}s^{−2} sin^{2} f_{2}]^{p/2}ds
+\frac{1}{2} ∫_{0}^{ε^{−1}} sb^{2} cos^{2} f_{2}ds

≤ \frac{1}{p} ∫_{1}^{ε^{−1}} s^{1−p}d^{p}ds + \frac{1}{p} ∫_{0}^{1} s((f′_{1})^{2}(1 + (b^{2} − 1) sin^{2} f) + d^{2}s^{−2} sin^{2} f_{1})^{p/2}ds

+ \frac{1}{2} ∫_{0}^{1} sb^{2} cos^{2} f_{1}ds
= \frac{d^{p}}{p(p−2)}(1 − ε^{p−2}) + I(1, 1) ≤ \frac{d^{p}}{p(p−2)} + I(1, 1) = C.

Substituting into (2.1) follows the conclusion of Proposition 2.1.

By the embedding theorem we derive, from |u_{ε}| = max{1, b} and proposition 2.1, the following

Proposition 2.2 Let u_{ε} be a symmetric minimizer of E_{ε}(u, B). Then there exists a constant C independent of ε ∈ (0, 1) such that

|u_{ε}(x) − u_{ε}(x_{0})| ≤ Cε^{(2−p)/p}|x − x_{0}|^{1−2/p}, ∀x, x_{0} ∈ B.

As a corollary of Proposition 2.1 we have Proposition 2.3 Let u_{ε} be a symmetric minimizer of E_{ε}(u, B). Then

\frac{1}{ε^{2}} ∫_{B} u_{ε3}^{2}dx ≤ C

with some constant C > 0 independent of ε ∈ (0, 1).

Proposition 2.4 Let u_{ε} be a symmetric minimizer of E_{ε}(u, B). Then for any γ ∈ (0, γ_{0}) with γ_{0} < b sufficiently small, there exist positive constants λ, μ independent of ε ∈ (0, 1) such that if

\frac{1}{ε^{2}} ∫_{B∩B^{2lε}} u_{ε3}^{2}dx ≤ μ  (2.2)

where B^{2lε} is some disc of radius 2lε with l ≥ λ, then

|u_{ε3}(x)| ≤ γ,  ∀x ∈ B ∩ B^{lε}.  (2.3)

Proof. First we observe that there exists a constant β > 0 such that for any x ∈ B and 0 < ρ ≤ 1, mes(B ∩ B(x, ρ)) ≥ βρ^{2}. To prove the proposition, we choose λ = (\frac{γ}{2C})^{\frac{p}{p−2}} , μ = \frac{β}{4} (\frac{1}{2C})^{\frac{2p}{p−2}} γ^{2+ \frac{2p}{p−2}} where C is the constant in Proposition 2.2.

Suppose that there is a point x_{0} ∈ B ∩ B^{lε} such that (2.3) is not true, i.e.

|u_{ε3}(x_{0})| > γ.  (2.4)

Then applying Proposition 2.2 we have

\begin{aligned}|u_{ε}(x) − u_{ε}(x_{0})| & ≤ Cε^{(2−p)/p}|x − x_{0}|^{1−2/p} ≤ Cε^{(2−p)/p}(λε)^{1−2/p} \\ & = Cλ^{1−2/p} = \frac{γ}{2} , ∀x ∈ B(x_{0}, λε) \end{aligned}

which implies |u_{ε3}(x) − u_{ε3}(x_{0})| ≤ \frac{γ}{2} . Noticing (2.4), we obtain |u_{ε3}(x)|^{2} ≥ [|u_{ε3}(x_{0})| − \frac{γ}{2} ]^{2} > \frac{γ^{2}}{4} , ∀x ∈ B(x_{0}, λε). Hence

∫_{B(x_{0},λε)∩B} u_{ε3}^{2}dx > \frac{γ^{2}}{4} mes(B ∩ B(x_{0}, λε)) ≥ β \frac{γ^{2}}{4} (λε)^{2} = με^{2}.  (2.5)

Since x_{0} ∈ B^{lε} ∩ B, and (B(x_{0}, λε) ∩ B) ⊂ (B^{2lε} ∩ B), (2.5) implies

∫_{B^{2lε}∩B} u_{ε3}^{2}dx > με^{2},

which contradicts (2.2) and thus the proposition is proved.

To find the points where u_{ε3}^{2} = b^{2} based on Proposition 2.4, we may take (2.2) as the ruler to distinguish the discs of radius λε which contain these points.

Let u_{ε} be a symmetric minimizer of E_{ε}(u, B). Given γ ∈ (0, 1). Let λ, μ be constants in Proposition 2.4 corresponding to γ. If

\frac{1}{ε^{2}} ∫_{B(x^{ε},2λε)∩B} u_{ε3}^{2}dx ≤ μ,

then B(x^{ε}, λε) is called γ− good disc, or simply good disc. Otherwise B(x^{ε}, λε) is called γ− bad disc or simply bad disc.

Now suppose that {B(x_{i}^{ε}, λε), i ∈ I} is a family of discs satisfying

(i) : x_{i}^{ε} ∈ B, i ∈ I; (ii) : B ⊂ ∪_{i∈I}B(x_{i}^{ε}, λε);
(iii) : B(x_{i}^{ε}, λε/4) ∩ B(x_{j}^{ε}, λε/4) = ∅, i ≠ j.  (2.6)

Denote J_{ε} = {i ∈ I; B(x_{i}^{ε}, λε) is a bad disc}. Then, one has

Proposition 2.5 There exists a positive integer N (independent of ε) such that the number of bad discs Card J_{ε} ≤ N.

Proof. Since (2.6) implies that every point in B can be covered by finite, say m (independent of ε) discs, from Proposition 2.3 and the definition of bad discs,we have

\begin{matrix}με^{2}CardJ_{ε} ≤ ∑_{i∈J_{ε}} ∫_{B(x_{i}^{ε},2λε)∩B} u_{ε3}^{2}dx \\ ≤ m ∫_{∪_{i∈Jε} B(x_{i}^{ε},2λε)∩B} u_{ε3}^{2}dx ≤ m ∫_{B} u_{ε3}^{2}dx ≤ mCε^{2} \end{matrix}

and hence Card J_{ε} ≤ \frac{mC}{μ} ≤ N.

Applying TheoremIV.1 in [1], we may modify the family of bad discs such that the new one, denoted by {B(x_{i}^{ε}, hε); i ∈ J}, satisfies

∪_{i∈J_{ε}}B(x_{i}^{ε}, λε) ⊂ ∪_{i∈J}B(x_{i}^{ε}, hε), λ ≤ h; Card J ≤ Card J_{ε},
|x_{i}^{ε} − x_{j}^{ε}| > 8hε, i, j ∈ J, i ≠ j.

The last condition implies that every two discs in the new family are not intersected. From Proposition 2.4 it is deduced that all the points where |u_{ε3}| = b are contained in these finite, disintersected bad discs.

Proof of Theorem 1.1. Suppose there exists a point x_{0} ∈ Z_{ε} such that x_{0}\overline{∈}B(0, hε). Then all points on the circle S_{0} = \{x ∈ B; |x| = |x_{0}|\} satisfy

u_{ε3}^{2}(x) = b^{2} cos^{2} f_{ε}(|x|) = b^{2} cos^{2} f_{ε}(|x_{0}|) = u_{ε3}^{2}(x_{0}) > γ^{2}.

By virtue of Proposition 2.4 we can see that all points on S_{0} are contained in bad discs. However, since |x_{0}| ≥ hε, S_{0} can not be covered by a single bad disc. As a result, S_{0} has to be covered by at least two bad disintersected discs. This is impossible.

3 Proof of Theorem 1.2

Let u_{ε}(x) = (sin f_{ε}(r)e^{idθ}, b cos f_{ε}(r)) be a symmetric minimizer of E_{ε}(u, B), namely f_{ε} be a minimizer of E_{ε}(f, (0, 1)) in V. From Proposition 2.1, we have

E_{ε}(f_{ε}, (0, 1)) ≤ Cε^{2−p}  (3.1)

for some constant C independent of ε ∈ (0, 1). In this section we further prove that for any η ∈ (0, 1), there exists a constant C(η) such that

E_{ε}(f_{ε}; η) := E_{ε}(f_{ε}, (η, 1)) ≤ C(η)  (3.2)

for ε ∈ (0, ε_{0}) with small ε_{0} > 0. Based on the estimate (3.2) and Theorem 1.1, we may obtain the W_{loc}^{1,p} convergence for minimizers.

To establish (3.2) we first prove

Proposition 3.1 Given η ∈ (0, 1). There exist constants η_{j} ∈ [\frac{(j−1)η}{N+1} , \frac{jη}{N+1}], (N = [p]) and C_{j}, such that

E_{ε}(f_{ε}, η_{j}) ≤ C_{j}ε^{j−p}  (3.3)

for j = 2, ..., N, where ε ∈ (0, ε_{0}).

Proof. For j = 2, the inequality (3.3) is just the one in Proposition 2.1. Suppose that (3.3) holds for all j ≤ n. Then we have, in particular

E_{ε}(f_{ε}; η_{n}) ≤ C_{n}ε^{n−p}.  (3.4)

If n = N then we are done. Suppose n < N. We want to prove (3.3) for j = n + 1.

Obviously (3.4) implies

\frac{1}{4ε^{p}} ∫ \frac{(n+1)η}{N+1} ; \frac{nη}{N+1} b^{2} cos^{2} f_{ε}rdr ≤ C_{n}ε^{n−p}

from which we see by integral mean value theorem that there exists η_{n+1} ∈ [\frac{nη}{N+1}, \frac{(n+1)η}{N+1} ] such that

[\frac{1}{ε^{p}} b^{2} cos^{2} f_{ε}]_{r=η_{n+1}} ≤ C_{n}ε^{n−p}.  (3.5)

Consider the functional

E(ρ, η_{n+1}) = \frac{1}{p} ∫_{η_{n+1}}^{1}(ρ_{r}^{2} + 1)^{p/2}dr + \frac{1}{ε^{p}} ∫_{η_{n+1}}^{1} b^{2} cos^{2} ρdr.

It is easy to prove that the minimizer ρ_{1} of E(ρ, η_{n+1}) in W_{f_{ε}}^{1,p}((η_{n+1}, 1), R^{+}) exists and satisfies

−ε^{p}(v^{(p−2)/2}ρ_{r})_{r} = sin 2ρ, in (η_{n+1}, 1)  (3.6)
ρ|_{r=η_{n+1}} = f_{ε}, ρ|_{r=1} = f_{ε}(1) = \frac{π}{2}  (3.7)

where v = ρ_{r}^{2} + 1. It follows from the maximum principle that ρ_{1} ≤ π/2 and

sin^{2} ρ(r) ≥ sin^{2} ρ(η_{n+1}) = sin^{2} f_{ε}(η_{n+1}) = 1 − cos^{2} f_{ε}(η_{n+1}) ≥ 1 − γ^{2}, (3.8)

the last inequality of which is implied by Theorem 1.1. Noting min{1, b^{2}} ≤ 1 + (b^{2} − 1) sin^{2} f ≤ max{1, b^{2}}, applying (3.4) we see easily that

E(ρ_{1}; η_{n+1}) ≤ E(f_{ε}; η_{n+1}) ≤ C(b)E_{ε}(f_{ε}; η_{n+1}) ≤ C_{n}ε^{n−p}  (3.9)

for ε ∈ (0, ε_{0}) with ε_{0} > 0 sufficiently small.

Now, choosing a smooth function ζ(r) such that ζ = 1 on (0, η), ζ = 0 near r = 1, multiplying (3.6) by ζρ_{r}(ρ = ρ_{1}) and integrating over (η_{n+1}, 1) we obtain

v^{(p−2)/2}ρ_{r}^{2}|_{r=η_{n+1}} + ∫_{η_{n+1}}^{1} v^{(p−2)/2}ρ_{r}(ζ_{r}ρ_{r} + ζρ_{rr})dr = \frac{1}{ε^{p}} ∫_{η_{n+1}}^{1} sin 2ρζρ_{r}dr.

(3.10)

Using (3.9) we have

\begin{matrix}| ∫_{η_{n+1}}^{1} v^{(p−2)/2}ρ_{r}(ζ_{r}ρ_{r} + ζρ_{rr})dr| \\ ≤ ∫_{η_{n+1}}^{1} v^{(p−2)/2}|ζ_{r}|ρ_{r}^{2}dr + \frac{1}{p}| ∫_{η_{n+1}}^{1}(v^{p/2}ζ)_{r}dr − ∫_{η_{n+1}}^{1} v^{p/2}ζ_{r}dr| \\ ≤ C ∫_{η_{n+1}}^{1} v^{p/2}dr + \frac{1}{p}v^{p/2}|_{r=η_{n+1}} + \frac{C}{p} ∫_{η_{n+1}}^{1} v^{p/2}dr \\ ≤ C_{n}ε^{n−p} + \frac{1}{p}v^{p/2}|_{r=η_{n+1}} \end{matrix} (3.11)

and using (3.5)(3.9) we have

|\frac{1}{ε^{p}} ∫_{η_{n+1}}^{1} ζρ_{r} sin 2ρdr| = \frac{1}{ε^{p}} | ∫_{η_{n+1}}^{1} ζ_{r}b^{2} cos^{2} ρdr − ∫_{η_{n+1}}^{1}(ζb^{2} cos^{2} ρ)_{r}dr|

≤ \frac{1}{ε^{p}} b^{2} cos^{2} ρ|_{r=η_{n+1}} + \frac{C}{ε^{p}} ∫_{η_{n+1}}^{1} cos^{2} ρdr ≤ C_{n}ε^{n−p}.
(3.12)

Combining (3.10) with (3.11)(3.12) yields

v^{(p−2)/2}ρ_{r}^{2}|_{r=η_{n+1}} ≤ C_{n}ε^{n−p} + \frac{1}{p}v^{p/2}|_{r=η_{n+1}}.

Hence

v^{p/2}|_{r=η_{n+1}} = v^{(p−2)/2}(ρ_{r}^{2} + 1)|_{r=η_{n+1}}
≤ C_{n}ε^{n−p} + \frac{1}{p}v^{p/2}|_{r=η_{n+1}} + v^{(p−2)/2}|_{r=η_{n+1}}
≤ C_{n}ε^{n−p} + (\frac{1}{p} + δ)v^{p/2}|_{r=η_{n+1}} + C(δ)

from which it follows by choosing δ > 0 small enough that

v^{p/2}|_{r=η_{n+1}} ≤ C_{n}ε^{n−p}.  (3.13)

Noting (3.8), we can see sin ρ > 0. Multiply both sides of (3.6) by cot ρ = \frac{cos ρ}{sin ρ} and integrate. Then

−ε^{p}v^{(p−2)/2}ρ_{r} cot ρ|_{η_{n+1}}^{1} = ε^{p} ∫_{η_{n+1}}^{1} v^{(p−2)/2}ρ_{r}^{2} \frac{1}{sin^{2} ρ}dr + 2 ∫_{η_{n+1}}^{1} cos^{2} ρdr.

Noting cot ρ(1) = 0 (which is implied by (3.7)) and \frac{1}{sin^{2} ρ} ≥ 1, we have

E(ρ_{1}; η_{n+1}) = \frac{1}{p} ∫_{η_{n+1}}^{1} v^{p/2}dr + \frac{1}{ε^{p}} ∫_{η_{n+1}}^{1} cos^{2} ρdr

≤ C[∫_{η_{n+1}}^{1} v^{(p−2)/2}ρ_{r}^{2}dr + \frac{1}{ε^{p}} ∫_{η_{n+1}}^{1} cos^{2} ρdr] ≤ Cv^{(p−2)/2}ρ_{r} cot ρ|_{r=η_{n+1}}.

From this, using(3.13)(3.5) and noticing that n < p, we obtain

\begin{matrix}E(ρ_{1}; η_{n+1}) ≤ Cv^{(p−2)/2}ρ_{r} cot ρ|_{r=η_{n+1}} \\ ≤ Cv^{(p−1)/2} cot ρ|_{r=η_{n+1}} ≤ (C_{n}ε^{n−p})^{(p−1)/p}(\frac{C_{n}ε^{n}}{1−C_{n}ε^{n}})^{1/2} \\ ≤ C_{n+1}ε^{n+1−p+(n/2−n/p)} ≤ C_{n+1}ε^{n+1−p}. \end{matrix}  (3.14)

Define w_{ε} = f_{ε}, for r ∈ (0, η_{n+1}); w_{ε} = ρ_{1}, for r ∈ [η_{n+1}, 1]. Since f_{ε} is a minimizer of E_{ε}(f), we have E_{ε}(f_{ε}) ≤ E_{ε}(w_{ε}), namely,

E_{ε}(f_{ε}; η_{n+1})

≤ \frac{1}{p} ∫_{η_{n+1}}^{1}(ρ_{r}^{2}(1 + (b^{2} − 1) sin^{2} ρ) + d^{2}r^{−2} sin^{2} ρ)^{p/2}rdr + \frac{1}{ε^{p}} ∫_{η_{n+1}}^{1} cos^{2} ρrdr

≤ \frac{C}{p} ∫_{η_{n+1}}^{1}(ρ_{r}^{2} + 1)^{p/2}dr + \frac{C}{2ε^{p}} ∫_{η_{n+1}}^{1} cos^{2} ρdr + C = CE(ρ_{1}; η_{n+1}) + C.

Thus, using (3.14) yields

E_{ε}(f_{ε}; η_{n+1}) ≤ C_{n+1}ε^{n−p+1}

for ε ∈ (0, ε_{0}). This is just (3.3) for j = n + 1. Proposition 3.2 Given η ∈ (0, 1). There exist constants η_{N+1} ∈ [\frac{Nη}{N+1}, η] and C_{N+1} such that

E_{ε}(f_{ε}; η_{N+1}) ≤ C_{N+1}ε^{N−p+1} + \frac{1}{p} ∫_{η_{N+1}}^{1} \frac{d^{p}}{r^{p−1}} dr  (3.15)

where N = [p]. Proof. Similar to the derivation of (3.5) we may obtain from Proposition 3.1 for j = N that there exists η_{N+1} ∈ [\frac{Nη}{N+1}, \frac{(N+1)η}{N+1} ], such that

\frac{1}{ε^{p}} cos^{2} f_{ε}|_{r=η_{N+1}} ≤ C_{N}ε^{N−p}.  (3.16)

Also similarly, consider the functional

E(ρ, η_{N+1}) = \frac{1}{p} ∫_{η_{N+1}}^{1}(ρ_{r}^{2} + 1)^{p/2}dr + \frac{1}{ε^{p}} ∫_{η_{N+1}}^{1} cos^{2} ρdr

whose minimizer ρ_{2} in W_{f_{ε}}^{1,p}((η_{N+1}, 1), R^{+}) exists and satisfies

−ε^{p}(v^{(p−2)/2}ρ_{r})_{r} = sin 2ρ, in (η_{N+1}, 1)

ρ|_{r=η_{N+1}} = f_{ε}, ρ|_{r=1} = f_{ε}(1) = \frac{π}{2}

where v = ρ_{r}^{2} + 1. From (3.4) for n = N it follows immediately that

E(ρ_{2}; η_{N+1}) ≤ E(f_{ε}; η_{N+1}) ≤ C_{N}E_{ε}(f_{ε}; η_{N+1}) ≤ C_{N}E_{ε}(f_{ε}; η_{N}) ≤ C_{N}ε^{N−p}. Similar to the proof of (3.13) and (3.14), we get, from Proposition 3.1 and (3.16),

v^{p/2}|_{r=η_{N+1}} ≤ C_{N}ε^{N−p}, and E(ρ_{2}; η_{N+1}) ≤ C_{N+1}ε^{N+1−p}.  (3.17)

Now we define w_{ε} = f_{ε}, for r ∈ (0, η_{N+1}); w_{ε} = ρ_{2}, for r ∈ [η_{N+1}, 1] and then we have E_{ε}(f_{ε}) ≤ E_{ε}(w_{ε}). Notice that

∫_{η_{N+1}}^{1}(ρ_{r}^{2}(1 + (b^{2} − 1) sin^{2} ρ) + d^{2}r^{−2} sin^{2} ρ)^{p/2}rdr
− ∫_{η_{N+1}}^{1}(d^{2}r^{−2} sin^{2} ρ)^{p/2}rdr
= \frac{p}{2} ∫_{η_{N+1}}^{1} ∫_{0}^{1}[(ρ_{r}^{2}(1 + (b^{2} − 1) sin^{2} ρ) + d^{2}r^{−2} sin^{2} ρ)s
+(d^{2}r^{−2} sin^{2} ρ)(1 − s)]^{(p−2)/2}]dsρ_{r}^{2}rdr
≤ C ∫_{η_{N+1}}^{1}(ρ_{r}^{2} + d^{2}r^{−2} sin^{2} ρ)^{(p−2)/2}ρ_{r}^{2}rdr ∫_{0}^{1} s^{(p−2)/2}ds
+ C ∫_{η_{N+1}}^{1}(d^{2}r^{−2} sin^{2} ρ)^{(p−2)/2}ρ_{r}^{2}rdr ∫_{0}^{1}(1 − s)^{(p−2)/2}ds
≤ C(∫_{η_{N+1}}^{1} ρ_{r}^{p}dr + ∫_{η_{N+1}}^{1} ρ_{r}^{2}dr) ≤ C ∫_{η_{N+1}}^{1}(ρ_{r}^{2} + 1)^{p/2}dr.

Hence

\begin{matrix}E_{ε}(f_{ε}; η_{N+1}) ≤ \frac{1}{p} ∫_{η_{N+1}}^{1}(d^{2}r^{−2} sin^{2} ρ)^{p/2}rdr + \frac{C}{2ε^{p}} ∫_{η_{N+1}}^{1}(cos ρ_{2})^{2}dr \\ +C ∫_{η_{N+1}}^{1}((ρ_{2})_{r}^{2} + 1)^{p/2}dr ≤ \frac{1}{p} ∫_{η_{N+1}}^{1} r(d^{2}r^{−2})^{p/2}dr + CE(ρ_{2}; η_{N+1}). \end{matrix}

Using (3.17) we have

E_{ε}(f_{ε}; η_{N+1}) ≤ \frac{1}{p} ∫_{η_{N+1}}^{1} r(d^{2}r^{−2})^{p/2}dr + C_{N+1}ε^{N−p+1}.

This is my conclusion.

Proof of Theorem 1.2. Without loss of generality, we may assume K = B \ B(0, η_{N+1}). From Proposition 3.2, We have E_{ε}(u_{ε}, K) = 2πE_{ε}(f_{ε}, η_{N+1}) ≤ C where C is independent of ε, namely

∫_{K} |∇u_{ε}|^{p}dx ≤ C,  (3.18)

∫_{K} |u_{ε3}|^{2}dx ≤ Cε^{p}.  (3.19)

(3.18) and |u_{ε}| ≤ max{1, b} imply the existence of a subsequence u_{ε_{k}} of u_{ε} and a function u_{∗} ∈ W^{1,p}(K, R^{3}), such that

lim_{ε_{k}→0} u_{ε_{k}} = u_{∗}, weakly in W^{1,p}(K, R^{3})

lim_{ε_{k}→0} u_{ε_{k}} = u_{∗}, in C^{α}(K, R^{3}), α ∈ (0, 1 − \frac{2}{p}).  (3.20)

(3.19) and (3.20) imply u_{∗} = (e^{idθ}, 0). Noticing that any subsequence of u_{ε} has a convergence subsequence and the limit is always (e^{idθ}, 0), we can assert

lim_{ε→0} u_{ε} = (e^{idθ}, 0), weakly in W^{1,p}(K, R^{3}).  (3.21)

From this and the weakly lower semicontinuity of ∫_{K} |∇u|^{p}, using Proposition 3.2, we have

∫_{K} |∇e^{idθ}|^{p}dx ≤ \underline{lim}_{ε_{k}→0} ∫_{K} |∇u_{ε}|^{p}dx ≤ \overline{lim}_{ε_{k}→0} ∫_{K} |∇u_{ε}|^{p}dx
≤ C lim_{ε→0} ε^{N+1−p} + 2π ∫_{η_{N+1}}^{1}(d^{2}r^{−2})^{p/2}rdr

and hence

lim_{ε→0} ∫_{K} |∇u_{ε}|^{p}dx = ∫_{K} |∇e^{idθ}|^{p}dx

since

∫_{K} |∇e^{idθ}|^{p}dx = 2π ∫_{η_{N+1}}^{1}(d^{2}r^{−2})^{p/2}rdr.

Combining this with (3.21)(3.20) complete the proof.

4 Proof of Theorem 1.3

Firstly, it follows from Jensen’s inequality that

E_{ε}(f_{ε}; η) ≥ \frac{1}{p} ∫_{η}^{1}(f′_{ε})^{p}(1 + (b^{2} − 1) sin^{2} f)^{p/2}rdr
+\frac{1}{2ε^{p}} ∫_{η}^{1} b^{2} cos^{2} f_{ε}rdr + \frac{1}{p} ∫_{η}^{1} \frac{d^{p}}{r^{p}} sin^{p} f_{ε}rdr.

Combining this with (3.15) yields

\frac{1}{p} ∫_{η}^{1}(f′_{ε})^{p}(1 + (b^{2} − 1) sin^{2} f)^{p/2}rdr + \frac{1}{2ε^{p}} ∫_{η}^{1} b^{2} cos^{2} f_{ε}rdr
≤ \frac{1}{p} ∫_{η}^{1} \frac{d^{p}}{r^{p}} (1 − sin^{p} f_{ε})rdr + Cε^{[p]+1−p}.

Noticing that 1 − sin^{p} f_{ε} ≤ C(1 − sin^{2} f_{ε}) = C cos^{2} f_{ε} and (3.19), we obtain

\begin{matrix}∫_{η}^{1}(f′_{ε})^{p}rdr + \frac{1}{ε^{p}} ∫_{η}^{1} b^{2} cos^{2} f_{ε}rdr \\ ≤ C ∫_{η}^{1} \frac{d^{p}}{r^{p}} cos^{2} f_{ε}rdr + Cε^{[p]+1−p} ≤ Cε^{p} + Cε^{[p]+1−p} ≤ Cε^{[p]+1−p}. \end{matrix} (4.1)

Using (4.1) and the integral mean value theorem we can see that there exists η_{1} ∈ [η, η(1 + 1/2)] ⊂ [R/2, R] such that

[\frac{1}{ε^{p}} cos^{2} f_{ε}]_{r=η_{1}} ≤ C_{1}ε^{[p]−p+1}.  (4.2)

Consider the functional

E(ρ, η_{1}) = \frac{1}{p} ∫_{η_{1}}^{1}(ρ_{r}^{2} + 1)^{p/2}dr + \frac{1}{2ε^{p}} ∫_{η_{1}}^{1} cos^{2} ρdr.

It is easy to prove that the minimizer ρ_{3} of E(ρ, η_{1}) in W_{f_{ε}}^{1,p}((η_{1}, 1), R^{+}) exists. By the same way to proof of (3.14), using (3.2) and (4.2) we have

E(ρ_{3}, η_{1}) ≤ v^{\frac{p−2}{2}} ρ_{3r} cot ρ_{3}|_{r=η_{1}} ≤ C_{1} cot ρ_{3}(η_{1}) ≤ Cε^{\frac{[p]+1−p}{2} + \frac{p}{2}} .

Hence, similar to the derivation of (3.15), we obtain

E_{ε}(f_{ε}; η_{1}) ≤ Cε^{\frac{[p]−p+1}{2} + \frac{p}{2}} + \frac{1}{p} ∫_{η_{1}}^{1} \frac{d^{p}}{r^{p−1}} dr.

Thus (4.1) may be rewritten as

∫_{η_{1}}^{1}(f′_{ε})^{p}rdr + \frac{1}{ε^{p}} ∫_{η_{1}}^{1} b^{2} cos^{2} f_{ε}rdr ≤ Cε^{\frac{[p]+1−p}{2} + \frac{p}{2}} + Cε^{p} ≤ C_{2}ε^{\frac{[p]+1−p}{2} + \frac{p}{2}} .

Let η_{m} = R(1 − \frac{1}{2^{m}}) where R < 1. Proceeding in the way above (whose idea is improving the exponent of ε from \frac{[p]+1−p}{2^{k}} + \frac{(2^{k}−1)p}{2^{k}} to \frac{[p]+1−p}{2^{k+1}} + \frac{(2^{k+1}−1)p}{2^{k+1}} step by step), we can get that for any m ∈ N,

∫_{η_{m}}^{1}(f′_{ε})^{p}rdr + \frac{1}{ε^{p}} ∫_{η_{m}}^{1} b^{2} cos^{2} f_{ε}rdr ≤ Cε^{\frac{[p]+1−p}{2^{m}} + \frac{(2^{m}−1)p}{2^{m}}} + Cε^{p}.

Letting m → ∞, we derive (1.2).

From (1.2) we can see that

∫_{K} u_{ε3}^{2}dx ≤ Cε^{2p}.  (4.3)

On the other hand, for any x_{0} ∈ K, we have

|u_{ε3}(x) − u_{ε3}(x_{0})| ≤ Cε^{(2−p)/p}|x − x_{0}|^{1−2/p}, ∀x ∈ B(x_{0}, αε),

by applying Proposition 2.2, where α = (\frac{|u_{ε3}(x_{0})|}{2C})^{\frac{p}{p−2}} . Thus

|u_{ε3}(x)| ≥ |u_{ε3}(x_{0})| − Cα^{1−2/p} ≥ \frac{1}{2}|u_{ε3}(x_{0})|.

Substituting this into (4.3) we obtain

Cε^{2p} ≥ ∫_{K} u_{ε3}^{2}dx ≥ ∫_{B(x_{0},αε)} u_{ε3}^{2}dx ≥ \frac{π}{4} |u_{ε3}(x_{0})|^{2}(αε)^{2},

which implies |u_{ε3}(x_{0})| ≤ Cε^{\frac{p−2}{2}} . Noting x_{0} is an arbitrary point in K, we have

sup_{x∈K} |u_{ε3}(x)| ≤ Cε^{\frac{p−2}{2}} .

Thus (1.3) is derived and the proof of Theorem is complete.

5 Proof of Theorem 1.4

By the method in the calculus of variations we can see the following

Proposition 5.1 The minimizer f_{ε} ∈ V of the functional E_{ε}(f, (0, 1)) satisfies the following equality

∫_{0}^{1} v^{(p−2)/2}[f_{r}ϕ_{r} + \frac{b^{2} − 1}{2}f_{r}^{2}(sin 2f)ϕ + \frac{d^{2}}{2r^{2}} (sin 2f)ϕ]rdr = \frac{1}{2ε^{p}} ∫_{0}^{1}(sin 2f)ϕrdr

for any function ϕ ∈ C_{0}^{∞}[0, 1], where v = f_{r}^{2}(1 + (b^{2} − 1)sin^{2}f) + \frac{d^{2} sin^{2} f}{r^{2}} .

Assume u_{ε}^{τ} = (e^{idθ} sin f_{ε}^{τ} , cos f_{ε}^{τ}) is the minimizer of the regularized functional E_{ε}^{τ} (u, B). It is easy to prove that the minimizer f_{ε}^{τ} is a classical solution of the equation

−(rA^{(p−2)/2}f_{r})_{r} + \frac{r(b^{2} − 1)}{2}A^{(p−2)/2}f_{r}^{2} sin 2f + d^{2}A^{(p−2)/2} \frac{sin 2f}{2r} = \frac{r sin 2f}{2ε^{p}} ,

(5.1)

where A = v + τ. By the same argument of Theorem 1.1 and Proposition 3.2, we can also see that for any compact subset K ∈ (0, 1], there exist constants η ∈ (0, 1/2) and C > 0 which are independent of ε and τ, such that

η ≤ f_{ε}^{τ} (r) ≤ \frac{π}{2} , r ∈ K,  (5.2)
E_{ε}^{τ} (f_{ε}^{τ} , K) ≤ C,  (5.3)

where

E_{ε}^{τ} (f, K) = ∫_{K}[\frac{1}{p}(f_{r}^{2}(1+(b^{2} −1)sin^{2}f)+d^{2}r^{−2} sin^{2} f +τ)^{p/2} + \frac{1}{2ε^{p}} b^{2} cos^{2} f]rdr.

Proposition 5.2 Denote f_{ε}^{τ} = f. Then for any closed subset K ⊂ (0, 1), there exists C > 0 which is independent of ε, τ such that

∥f∥_{C^{1,α}(K,R)} ≤ C, ∀α ≤ 1/2.

Proof. Without loss of the generality, we assume d = 1. Take R > 0 sufficiently small such that K ⊂⊂ (2R, 1 − 2R). Let ζ ∈ C_{0}^{∞}([0, 1], [0, 1]) be a function satisfying ζ = 0 on [0, R] ∪ [1 − R, 1], ζ = 1 on [2R, 1 − 2R] and |ζ_{r}| ≤ C(R) on (0, 1). Differentiating (5.1), multiplying with f_{r}ζ^{2} and integrating, we have

\begin{matrix}− ∫_{0}^{1}(A^{(p−2)/2}f_{r})_{rr}(f_{r}ζ^{2})dr − ∫_{0}^{1}(r^{−1}A^{(p−2)/2}f_{r})_{r}(f_{r}ζ^{2})dr \\ +\frac{1}{2} ∫_{0}^{1}[(r^{−2} + (b^{2} − 1)f_{r}^{2})A^{(p−2)/2} sin 2f]_{r}(f_{r}ζ^{2})dr = \frac{b^{2}}{ε^{p}} ∫_{0}^{1}(cos 2f)f_{r}^{2}ζ^{2}dr. \end{matrix}

Integrating by parts and noting cos 2f = 2 cos^{2} f − 1, we obtain

∫_{0}^{1}(A^{(p−2)/2}f_{r})_{r}(f_{r}ζ^{2})_{r}dr
= ∫_{0}^{1} A^{(p−2)/2}(f_{r}ζ^{2})_{r}[(\frac{1}{2r^{2}} + \frac{b^{2}−1}{2}f_{r}^{2}) sin 2f − r^{−1}f_{r}]dr
+\frac{2}{ε^{p}} ∫_{0}^{1}(b^{2} cos^{2} f)f_{r}^{2}ζ^{2}dr − \frac{1}{ε^{p}} ∫_{0}^{1} f_{r}^{2}ζ^{2}dr.

Denote I = ∫_{R}^{1−R} ζ^{2}(A^{(p−2)/2}f_{rr}^{2} + (p − 2)A^{(p−4)/2}f_{r}^{2}f_{rr}^{2})dr. Then for any δ ∈ (0, 1), there holds

I ≤ δI + C(δ) ∫_{R}^{1−R} A^{p/2}ζ_{r}^{2}dr + \frac{2}{ε^{p}} ∫_{R}^{1−R}(b^{2} cos^{2} f)f_{r}^{2}ζ^{2}dr  (5.4)

by using Young inequality. Noticing that (5.2) implies sin f > 0 as r ∈ [R, 1−R], from (5.1) we can see that

\frac{2}{ε^{p}} (cos f)^{2} = 4r^{−1} cot f[−(A^{(p−2)/2}f_{r})_{r} − r^{−1}A^{(p−2)/2}f_{r}
+A^{(p−2)/2}(\frac{1}{2r} + \frac{r(b^{2}−1)}{2}f_{r}^{2}) sin 2f].

Substituting it into the last term of the right hand side of (5.4) and applying Young inequality again we obtain that for any δ ∈ (0, 1),

\frac{2}{ε^{p}} ∫_{R}^{1−R}(cos^{2} f)f_{r}^{2}ζ^{2}dr ≤ δI + C(δ) ∫_{R}^{1−R} A^{(p+2)/2}ζ^{2}dr.

Combining this with (5.4) and choosing δ sufficiently small, we have

I ≤ C ∫_{R}^{1−R} A^{p/2}ζ_{r}^{2}dr + C ∫_{R}^{1−R} A^{(p+2)/2}ζ^{2}dr.  (5.5)

To estimate the second term of the right hand side of (5.5), we take ϕ = ζ^{2/q}|f_{r}|^{(p+2)/q} in the interpolation inequality (Ch II, Theorem 2.1 in [6])

∥ϕ∥_{L^{q}} ≤ C∥ϕ_{r}∥_{L^{1}}^{1−1/q}∥ϕ∥_{L^{1}}^{1/q}, q ∈ (1 + \frac{2}{p}, 2).  (5.6)

We derive by applying Young inequality that for any δ ∈ (0, 1),

\begin{matrix}∫_{R}^{1−R} |f_{r}|^{p+2}ζ^{2}dr ≤ C(∫_{R}^{1−R} ζ^{2/q}|f_{r}|^{(p+2)/q}dr) \\ ·(∫_{R}^{1−R} ζ^{2/q−1}|ζ_{r}||f_{r}|^{(p+2)/q} + ζ^{2/q}|f_{r}|^{(p+2)/q−1}|f_{rr}|dr)^{q−1} \\ ≤ C(∫_{R}^{1−R} ζ^{2/q}|f_{r}|^{(p+2)/q}dr)(∫_{R}^{1−R} ζ^{2/q−1}|ζ_{r}||f_{r}|^{(p+2)/q} \\ +δI + C(δ) ∫_{R}^{1−R} A^{\frac{p+2}{q} − \frac{p}{2}} ζ^{4/q−2}dr)^{q−1}. \end{matrix}  (5.7)

Noting q ∈ (1 + \frac{2}{p}, 2), we may using Holder inequality to the right hand side of (5.7). Thus, by virtue of (5.3),

∫_{R}^{1−R} |f_{r}|^{p+2}ζ^{2}dr ≤ δI + C(δ).

Substituting this into (5.5) and choosing δ sufficiently small, we obtain

∫_{R}^{1−R} A^{(p−2)/2}f_{rr}^{2}ζ^{2}dr ≤ C,

which, together with (5.3), implies that ∥A^{p/4}ζ∥_{H^{1}(R,1−R)} ≤ C. Noticing ζ = 1 on K, we have ∥A^{p/4}∥_{H^{1}(K)} ≤ C. Using embedding theorem we can see that for any α ≤ 1/2, there holds ∥A^{p/4}∥_{C^{α}(K)} ≤ C. From this it is not difficult to prove our proposition.

Applying the idea above, we also have the estimate near the boundary point r = 1.

Proposition 5.3 Denote f_{ε}^{τ} = f(r). Then for any closed subset K ⊂ (0, 1], there exists C > 0 which is independent of ε, τ such that

∥f∥_{C^{1,α}(K,R)} ≤ C, ∀α ≤ 1/2.

Proof. Without loss of the generality, we assume d = 1. Let g(r) = f(r+1)−1. Define

\begin{matrix}g̃(r) = g(r), as − 1 < r ≤ 0; \\ g̃(r) = −g(−r) as 0 < r ≤ \frac{1}{2}. \end{matrix}

If still denote f(r) = g̃(r −1)+1 on (0, \frac{3}{2}), then f(r) solves (5.1) on (0, \frac{3}{2}). Take R < \frac{1}{4} sufficiently small, and set ζ ∈ C^{∞}[0, 1], ζ = 1 as r ≥ 1 − R, ζ = 0 as r ≤ 2R. Differentiating (5.1), multiplying with f_{r}ζ^{2} and integrating over [R, 1], we have

\begin{matrix}− ∫_{R}^{1}(A^{(p−2)/2}f_{r})_{rr}(f_{r}ζ^{2})dr − ∫_{R}^{1}(r^{−1}A^{(p−2)/2}f_{r})_{r}(f_{r}ζ^{2})dr \\ + ∫_{R}^{1}[(\frac{1}{2r^{2}} + \frac{b^{2}−1}{2}f_{r}^{2})A^{(p−2)/2} sin 2f]_{r}(f_{r}ζ^{2})dr = \frac{1}{ε^{p}} ∫_{R}^{1}(b^{2} cos 2f)f_{r}^{2}ζ^{2}dr. \end{matrix}

Integrating by parts yields

∫_{R}^{1}(A^{(p−2)/2}f_{r})_{r}(f_{r}ζ^{2})_{r}dr
≤ | ∫_{R}^{1}[A^{(p−2)/2}((\frac{1}{2r^{2}} + \frac{b^{2}−1}{2}f_{r}^{2}) sin 2f − r^{−1}f_{r}]_{r}(f_{r}ζ^{2})dr|
+\frac{2}{ε^{p}} ∫_{R}^{1}(b^{2} cos^{2} f)f_{r}^{2}ζ^{2}dr + |I(1) − I(R)|,

where I(r) = −[(A^{(p−2)/2}f_{r})_{r} + \frac{1}{r}A^{(p−2)/2}f_{r} − \frac{1}{2r^{2}} A^{(p−2)/2} sin 2f]f_{r}ζ^{2}. The second term of the right hand side of the inequality above can be handled similar to the proof of Proposition 5.2. Computing the first term of the right hand side yields

\begin{matrix}| ∫_{R}^{1}[A^{(p−2)/2}((\frac{1}{2r^{2}} + \frac{b^{2}−1}{2}f_{r}^{2}) sin 2f − r^{−1}f_{r}]_{r}(f_{r}ζ^{2})dr| \\ ≤ δ ∫_{R}^{1} A^{(p−2)/2}f_{rr}^{2}ζ^{2}dr + C(δ) ∫_{R}^{1} A^{(p+2)/2}dr \end{matrix}

with any δ ∈ (0, 1) by using Young inequality. In view of (5.1), we have I(r) = \frac{1}{2ε^{p}} (sin 2f)f_{r}ζ^{2}. Hence, I(1) = I(R) = 0 since sin 2f(1) = 0 and ζ(R) = 0. Hence, we may also obtain the result as (5.5)

∫_{R}^{1} A^{(p−2)/2}f_{rr}^{2}ζ^{2}dr ≤ C ∫_{R}^{1}(A^{p/2} + A^{(p+2)/2}ζ^{2})dr.

Now, if we take ϕ = ζ^{2/q}|f_{r}|^{(p+2)/q}, then the interpolation inequality (5.6) is invalid since ϕ ≠ 0 near r = 1. Thus, we apply a new interpolation inequality [6, (2.19) in Chapter 2]

∥ϕ∥_{L^{q}} ≤ C(∥ϕ_{r}∥_{L^{1}} + ∥ϕ∥_{L^{1}})^{1−1/q}∥ϕ∥_{L^{1}}^{1/q}, q ∈ (1 + \frac{2}{p}, 2).

Then it still follows the same result as (5.7). The rest of the proof is similar to the proof of Proposition 5.2.

Proof of Theorem 1.4. For every compact subset K ⊂ B \ {0}, applying Propositions 5.2 and 5.3 yields that for α ∈ (0, 1/2] one has

∥u_{ε}^{τ}∥_{C^{1,α}(K)} ≤ C = C(K),  (5.8)

where the constant does not depend on ε, τ.

Applying (5.8) and the embedding theorem we know that for any ε and β_{1} < α, there exist w_{ε}^{∗} ∈ C^{1,β_{1}}(K, E(b)) and a subsequence of τ_{k} of τ such that as k → ∞,

u_{ε}^{τ_{k}} → w_{ε}^{∗}, in C^{1,β_{1}}(K, E(b)).  (5.9)

Combining this with (1.4) we know that w_{ε}^{∗} = ũ_{ε}.

Applying (5.8) and the embedding theorem again we can see that for any β_{2} < α, there exist w^{∗} ∈ C^{1,β_{2}}(K, E(b)) and a subsequence of τ_{k} which can be denoted by τ_{m} such that as m → ∞,

u_{ε_{m}}^{τ_{m}} → w^{∗}, in C^{1,β_{2}}(K, E(b)).  (5.10)

Noticing (1.2), we know that w^{∗} = (e^{idθ}, 0). Denote γ = min(β_{1}, β_{2}). Then as m → ∞, we have

\begin{aligned}∥ũ_{ε_{m}} − (e^{idθ}, 0)∥_{C^{1,γ}(K,E(b))} & ≤ ∥ũ_{ε_{m}} − u_{ε_{m}}^{τ_{m}}∥_{C^{1,γ}(K,E(b))} \\ & +∥u_{ε_{m}}^{τ_{m}} − (e^{idθ}, 0)∥_{C^{1,γ}(K,E(b))} ≤ o(1) \end{aligned} (5.11)

by applying (5.9) and (5.10).

Noting the limit (e^{idθ}, 0) is unique, we can see that the convergence (5.11) holds not only for some subsequence but for all ũ_{ε}. Theorem is proved.

6 Proof of Theorem 1.5

Without loss of the generality, we assume d = 1. Denote f = f_{ε}^{τ}. Set ψ = \frac{cos f}{ε^{p}}. Multiplying (5.1) by sin f we obtain

−(rA^{(p−2)/2}(sin f)f_{r})_{r} + r cos fA^{(p−2)/2}(A − τ) = r(sin f)^{2}ψ.  (6.1)

Substituting ψ_{r} = \frac{− sin f}{ε^{p}} f_{r} into (6.1) we have

ε^{p}(rA^{(p−2)/2}ψ_{r})_{r} + r cos fA^{(p−2)/2}(A − τ) = r(sin f)^{2}ψ.

Suppose ψ(r) achieves its maximum at the point r_{0} in K, where K is an arbitrary open interval in any compact subset of (0, 1). Then ψ_{r}(r_{0}) = 0, ψ_{rr}(r_{0}) ≤ 0. And (sin f)^{2} ≥ C_{1} > 0 with the constant C_{1} independent of ε and τ which is implied by (5.2). Thus, it is deduced that, from Proposition 5.2,

ψ(r) ≤ ψ(r_{0}) ≤ \frac{1}{C_{1}} A^{(p−2)/2}(A − τ)|_{r=r_{0}} ≤ C,

which implies sup_{K} | cos f| ≤ Cε^{p} with the constant C > 0 independent of ε and τ, where K is any compact subset of (0, 1). Letting τ → 0 and using (5.9) we may see the conclusion

sup_{K} | cos f_{ε}| ≤ Cε^{p}.

To derive estimate near the boundary r = 1, we use the idea of Pohozaev’s equality. Choose R ∈ (0, \frac{1}{4}). Set ζ(r) ∈ C^{∞}[0, 1], ζ = 0 as r ∈ [0, 2R], ζ = 1 as r ∈ [1 − R, 1]. Then ζ_{r} ≤ C(R). Multiplying (5.1) with f_{r}ζ and integrating over [R, T] with T being an arbitrary constant in (1 − R, 1), we have

\begin{matrix}− ∫_{R}^{T} (rA^{(p−2)/2}f_{r})_{r}f_{r}ζdr + ∫_{R}^{T} A^{(p−2)/2}(sin 2f)[\frac{1}{2r} + \frac{r(b^{2}−1)}{2}f_{r}^{2}]f_{r}ζdr \\ = \frac{1}{2ε^{p}} ∫_{R}^{T} rf_{r}(sin 2f)ζdr. \end{matrix}

(6.2)

Integrating the right hand side of (6.2) by parts yields

\frac{1}{2ε^{p}} ∫_{R}^{T} rf_{r}(sin 2f)ζdr = −\frac{1}{2ε^{p}} ∫_{R}^{T} r(cos^{2} f)_{r}ζdr
= −\frac{1}{2ε^{p}} r(cos f)^{2}|_{r=T} + \frac{1}{2ε^{p}} ∫_{R}^{T} (cos^{2} f)(rζ)_{r}dr.

(6.3)

Similarly, the first term of the left hand side of (6.2) may be written as

\begin{matrix}− ∫_{R}^{T} (rA^{(p−2)/2}f_{r})_{r}f_{r}ζdr = −rA^{(p−2)/2}f_{r}^{2}|_{r=T} \\ + ∫_{R}^{T} rA^{(p−2)/2}f_{r}f_{rr}ζdr + ∫_{R}^{T} rA^{(p−2)/2}f_{r}^{2}ζ_{r}dr = Σ_{i=1}^{3}. \end{matrix}  (6.4)

Combining I_{2} with the second term of the left hand side of (6.2) we have

I_{2} + ∫_{R}^{T} A^{(p−2)/2} \frac{sin 2f}{2r} f_{r}ζdr
= ∫_{R}^{T} rA^{(p−2)/2}[f_{r}f_{rr} + (\frac{1}{2r^{2}} + \frac{b^{2}−1}{2}f_{r}^{2})f_{r} sin 2f)ζdr
= \frac{1}{2} ∫_{R}^{T} rA^{(p−2)/2}ζ(A − τ)_{r}dr + \frac{1}{2} ∫_{R}^{T} r^{−2}A^{(p−2)/2}ζ sin^{2} fdr
= \frac{1}{p}rA^{p/2}|_{r=T} − \frac{1}{p} ∫_{R}^{T} A^{p/2}(rζ)_{r}dr + \frac{1}{2} ∫_{R}^{T} r^{−2}A^{(p−2)/2}ζ sin^{2} fdr.

Substituting this and (6.3),(6.4) into (6.2) yields

\frac{1}{2ε^{p}} r(cos f)^{2}|_{r=T} + \frac{1}{p}rA^{p/2}|_{r=T} + ∫_{R}^{T} rA^{(p−2)/2}f_{r}^{2}ζ_{r}dr
+\frac{1}{2} ∫_{R}^{T} A^{(p−2)/2}r^{−2}(sin f)^{2}ζdr
= \frac{1}{2ε^{p}} ∫_{R}^{T} (cos f)^{2}(rζ)_{r}dr + \frac{1}{p} ∫_{R}^{T} A^{p/2}(rζ)_{r}dr + rA^{(p−2)/2}f_{r}^{2}|_{r=T} .

Applying Proposition 5.3 and (5.3) we obtain \frac{1}{2ε^{p}} T cos^{2} f(T) ≤ C, with C > 0 independent of ε and τ. Letting τ → 0 and using (5.9) we derive

\frac{1}{2ε^{p}} cos^{2} f_{ε}(T) ≤ C.

By virtue of the arbitrary of the point T, it is not difficult to get our Theorem.

Acknowledgements. The research was supported by NSF (19271086) and Tianyuan Fund of Mathematics (A0324628)(China).

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