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81(2009) 225-250

BLOW-UP PHENOMENA FOR THE YAMABE EQUATION II

Simon Brendle & Fernando C. Marques

Abstract

Let n be an integer such that 25 n 51. We construct a smooth metricg onSn with the property that the set of constant scalar curvature metrics in the conformal class ofgis not compact.

1. Introduction

Let (M, g) be a compact Riemannian manifold of dimension n 3. The Yamabe problem is concerned with finding metrics of constant scalar curvature in the conformal class of g. This problem leads to a semi-linear elliptic PDE for the conformal factor. More precisely, a conformal metric of the form un−24 g has constant scalar curvature c if and only if

(1) 4(n1)

n−2 ∆gu−Rgu+c un+2n−2 = 0,

where ∆g is the Laplace operator with respect to g andRg denotes the scalar curvature of g. Every solution of (1) is a critical point of the functional

(2) Eg(u) =

R

M

¡4(n−1)

n−2 |du|2g+Rgu2¢ dvolg

¡ R

Mun−22n dvolg¢n−2

n

.

In this paper, we address the question whether the set of all solutions to the Yamabe PDE is compact in theC2-topology. It has been conjectured that this should be true unless (M, g) is conformally equivalent to the round sphere (see [15], [16], [17]). The case of the round sphere Sn is special in that (1) is invariant under the action of the conformal group on Sn, which is non-compact. It follows from a theorem of Obata [14]

that every solution of the Yamabe PDE on Sn is minimizing, and the space of all solutions to the Yamabe PDE on Sncan be identified with the unit ball Bn+1. Note that the round sphere is the only compact manifold for which the set of minimizing solutions is non-compact.

The Compactness Conjecture has been verified in low dimensions and in the locally conformally flat case. If (M, g) is locally conformally

225

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flat, compactness follows from work of R. Schoen [15], [16]. Moreover, Schoen proposed a strategy, based on the Pohozaev identity, for proving the conjecture in the non-locally conformally flat case. In [12], Y.Y. Li and M. Zhu followed this strategy to prove compactness in dimension 3. O. Druet [6] proved the conjecture in dimensions 4 and 5.

The case n≥6 is more subtle, and requires a careful analysis of the local properties of the background metric g near a blow-up point. The Compactness Conjecture is closely related to the Weyl Vanishing Con- jecture, which asserts that the Weyl tensor should vanish to an order greater than n−62 at a blow-up point (see [17]). The Weyl Vanishing Conjecture has been verified in dimensions 6 and 7 by F. Marques [13]

and, independently, by Y.Y. Li and L. Zhang [10]. Using these re- sults and the positive mass theorem, these authors were able to prove compactness for n≤7. Moreover, Li and Zhang showed that compact- ness holds in all dimensions provided that |Wg(p)|+|∇Wg(p)|>0 for all p M. In dimensions 10 and 11, it is sufficient to assume that

|Wg(p)|+|∇Wg(p)|+|∇2Wg(p)|>0 for allp∈M (see [11]).

Very recently, M. Khuri, F. Marques, and R. Schoen [9] proved the Weyl Vanishing Conjecture up to dimension 24. This result, combined with the positive mass theorem, implies the Compactness Conjecture for those dimensions. After proving sharp pointwise estimates, they reduce these questions to showing a certain quadratic form is positive definite.

It turns out the quadratic form has negative eigenvalues ifn≥25.

In a recent paper [4], it was shown that the Compactness Conjecture fails for n≥52. More precisely, given any integer n≥52, there exists a smooth Riemannian metric g on Sn such that set of constant scalar curvature metrics in the conformal class ofgis non-compact. Moreover, the blowing-up sequences obtained in [4] form exactly one bubble. The construction relies on a gluing procedure based on some local model metric. These local models are directions in which the quadratic form of [9] is negative definite. We refer to [5] for a survey of this and related results.

In the present paper, we extend these counterexamples to the dimen- sions 25≤n≤51. Our main theorem is:

Theorem. Assume that 25≤n≤51. Then there exists a Riemann- ian metric g on Sn (of class C) and a sequence of positive functions vν ∈C(Sn) (νN) with the following properties:

(i) g is not conformally flat,

(ii) vν is a solution of the Yamabe PDE (1) for all ν N,

(iii) Eg(vν)< Y(Sn) for allν N, and Eg(vν)→Y(Sn) as ν → ∞, (iv) supSnvν → ∞ as ν→ ∞.

(Here, Y(Sn) denotes the Yamabe energy of the round metric on Sn.)

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We note that O. Druet and E. Hebey [7] have constructed blow-up examples for perturbations of (1) (see also [8]).

In Section 2, we describe how the problem can be reduced to find- ing critical points of a certain function Fg(ξ, ε), where ξ is a vector in Rn and ε is a positive real number. This idea has been used by many authors (see, e.g., [1], [2], [3], [4]). In Section 3, we show that the func- tion Fg(ξ, ε) can be approximated by an auxiliary function F(ξ, ε). In Section 4, we prove that the functionF(ξ, ε) has a critical point, which is a strict local minimum. Finally, in Section 5, we use a perturbation argument to construct critical points of the functionFg(ξ, ε). From this the non-compactness result follows.

The authors would like to thank Professor Richard Schoen for con- stant support and encouragement. The first author was supported by the Alfred P. Sloan foundation and by the National Science Founda- tion under grant DMS-0605223. The second author was supported by CNPq-Brazil, FAPERJ and the Stanford Mathematics Department.

2. Lyapunov-Schmidt reduction

In this section, we collect some basic results established in [4]. Let E =

½

w∈Ln−22n (Rn)∩Wloc1,2(Rn) : Z

Rn

|dw|2 <∞

¾ .

By Sobolev’s inequality, there exists a constant K, depending only on n, such that

µ Z

Rn

|w|n−22n

n−2

n ≤K Z

Rn

|dw|2 for all w ∈ E. We define a norm on E by kwk2E =R

Rn|dw|2. It is easy to see that E, equipped with this norm, is complete.

Given any pair (ξ, ε)Rn×(0,∞), we define a functionu(ξ,ε) :Rn Rby

u(ξ,ε)(x) =

³ ε

ε2+|x−ξ|2

´n−2

2 .

The function u(ξ,ε) satisfies the elliptic PDE

∆u(ξ,ε)+n(n−2)u

n+2 n−2

(ξ,ε)= 0.

It is well known that Z

Rn

u

2n n−2

(ξ,ε)=

³ Y(Sn) 4n(n1)

´n

2

for all (ξ, ε)Rn×(0,∞). We next define ϕ(ξ,ε,0)(x) =

³ ε

ε2+|x−ξ|2

´n+2

2 ε2− |x−ξ|2 ε2+|x−ξ|2

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and

ϕ(ξ,ε,k)(x) =

³ ε

ε2+|x−ξ|2

´n+2

2 2ε(xk−ξk) ε2+|x−ξ|2

fork= 1, . . . , n. Finally, we define a closed subspace E(ξ,ε)⊂ E by E(ξ,ε)=

½

w∈ E : Z

Rn

ϕ(ξ,ε,k)w= 0 for k= 0,1, . . . , n

¾ . Clearly,u(ξ,ε)∈ E(ξ,ε).

Proposition 1. Consider a Riemannian metric on Rn of the form g(x) = exp(h(x)), where h(x) is a trace-free symmetric two-tensor on Rn satisfying h(x) = 0 for |x| ≥ 1. There exists a positive constant α01, depending only onn, with the following significance: if|h(x)|+

|∂h(x)|+|∂2h(x)| ≤ α0 for all x Rn, then, given any pair (ξ, ε) Rn ×(0,∞) and any function f Ln+22n (Rn), there exists a unique function w=G(ξ,ε)(f)∈ E(ξ,ε) such that

Z

Rn

³

hdw, dψig+ n−2

4(n1)Rgwψ−n(n+ 2)u

4 n−2

(ξ,ε)

´

= Z

Rn

f ψ for all test functions ψ∈ E(ξ,ε).

Moreover, we have kwkE Ckfk

Ln+22n (Rn), where C is a constant that depends only on n.

Proposition 2. Consider a Riemannian metric on Rn of the form g(x) = exp(h(x)), where h(x) is a trace-free symmetric two-tensor on Rn satisfying h(x) = 0 for |x| ≥1. Moreover, let (ξ, ε)Rn×(0,∞).

There exists a positive constant α1 ≤α0, depending only on n, with the following significance: if |h(x)|+|∂h(x)|+|∂2h(x)| ≤α1 for allx∈Rn, then there exists a functionv(ξ,ε)∈ E such that v(ξ,ε)−u(ξ,ε) ∈ E(ξ,ε) and Z

Rn

³

hdv(ξ,ε), dψig+ n−2

4(n1)Rgv(ξ,ε)ψ−n(n−2)|v(ξ,ε)|n−24 v(ξ,ε)ψ

´

= 0 for all test functions ψ∈ E(ξ,ε). Moreover, we have the estimate

kv(ξ,ε)−u(ξ,ε)kE

≤C

°°

°∆gu(ξ,ε) n−2

4(n1)Rgu(ξ,ε)+n(n−2)u

n+2 n−2

(ξ,ε)

°°

°Ln+22n (Rn), where C is a constant that depends only on n.

We next define a functionFg :Rn×(0,∞)→Rby Fg(ξ, ε) =

Z

Rn

³

|dv(ξ,ε)|2g+ n−2

4(n1)Rgv(ξ,ε)2 (n2)2|v(ξ,ε)|n−22n

´

2(n2)

³ Y(Sn) 4n(n1)

´n

2.

If we choose α1 small enough, then we obtain the following result:

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Proposition 3. The function Fg is continuously differentiable.

Moreover, ifξ,ε)¯ is a critical point of the functionFg, then the function vξ,¯ε) is a non-negative weak solution of the equation

gvξ,¯ε) n−2

4(n1)Rgv( ¯ξ,¯ε)+n(n−2)v

n+2 n−2

ξ,¯ε)= 0.

3. An estimate for the energy of a “bubble”

Throughout this paper, we fix a real number τ and a multi-linear form W :Rn×Rn×Rn×RnR. The number τ depends only on the dimension n. The exact choice of τ will be postponed until Section 4.

We assume that Wijkl satisfies all the algebraic properties of the Weyl tensor. Moreover, we assume that some components ofW are non-zero, so that

Xn i,j,k,l=1

(Wijkl+Wilkj)2 >0.

For abbreviation, we put Hik(x) =

Xn p,q=1

Wipkqxpxq and

Hik(x) =f(|x|2)Hik(x),

wheref(s) =τ+5s−s2+201 s3. It is easy to see thatHik(x) is trace-free, Pn

i=1xiHik(x) = 0, andPn

i=1iHik(x) = 0 for allx∈Rn.

We consider a Riemannian metric of the form g(x) = exp(h(x)), whereh(x) is a trace-free symmetric two-tensor onRnsatisfyingh(x) = 0 for |x| ≥1,

|h(x)|+|∂h(x)|+|∂2h(x)| ≤α1 for all x∈Rn, and

hik(x) =µ λ6f−2|x|2)Hik(x)

for |x| ≤ ρ. We assume that the parameters λ, µ, and ρ are chosen such that µ 1 and λ ρ 1. Note that Pn

i=1xihik(x) = 0 and Pn

i=1ihik(x) = 0 for |x| ≤ρ.

Given any pair (ξ, ε) Rn×(0,∞), there exists a unique function v(ξ,ε) such thatv(ξ,ε)−u(ξ,ε)∈ E(ξ,ε) and

Z

Rn

³

hdv(ξ,ε), dψig+ n−2

4(n1)Rgv(ξ,ε)ψ−n(n−2)|v(ξ,ε)|n−24 v(ξ,ε)ψ

´

= 0 for all test functionsψ∈ E(ξ,ε) (see Proposition 2).

For abbreviation, let Ω =

½

(ξ, ε)Rn×R:|ξ|<1, 1

2 < ε <2

¾ .

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The following result is proved in the Appendix A of [4]. A similar formula is derived in [2].

Proposition 4. Consider a Riemannian metric on Rn of the form g(x) = exp(h(x)), where h(x) is a trace-free symmetric two-tensor on Rn satisfying |h(x)| ≤1 for all x Rn. Let Rg be the scalar curvature of g. There exists a constant C, depending only on n, such that

¯¯

¯Rg−∂ikhik+i(hilkhkl)1

2ihilkhkl+1

4lhiklhik

¯¯

¯

≤C|h|2|∂2h|+C|h| |∂h|2.

Proposition 5. Assume that (ξ, ε)∈λΩ. Then we have

°°

°°∆gu(ξ,ε) n−2

4(n1)Rgu(ξ,ε)+n(n−2)u

n+2 n−2

(ξ,ε)

°°

°°

Ln+22n (Rn)

≤C λ8µ+C µλ

ρ

n−2

2

and

°°

°°∆gu(ξ,ε) n−2

4(n1)Rgu(ξ,ε)+n(n−2)u

n+2 n−2

(ξ,ε)

+ Xn i,k=1

µ λ6f(λ−2|x|2)Hik(x)iku(ξ,ε)

°°

°°

Ln+22n (Rn)

≤C λ8(n+2)n−2 µ2+C µλ

ρ

n−2

2 .

Proof. Note that Pn

i=1ihik(x) = 0 for |x| ≤ ρ. Hence, it follows from Proposition 4 that

|Rg(x)| ≤C|h(x)|2|∂2h(x)|+C|∂h(x)|2≤C µ2(λ+|x|)14 for|x| ≤ρ. This implies

¯¯

¯¯∆gu(ξ,ε) n−2

4(n1)Rgu(ξ,ε)+n(n−2)u

n+2 n−2

(ξ,ε)

¯¯

¯¯

=

¯¯

¯¯

¯¯ Xn i,k=1

i h

(gik−δik)ku(ξ,ε) i

n−2

4(n1)Rgu(ξ,ε)

¯¯

¯¯

¯¯

≤C λn−22 µ(λ+|x|)8−n

(7)

and

¯¯

¯∆gu(ξ,ε) n−2

4(n1)Rgu(ξ,ε)+n(n−2)u

n+2 n−2

(ξ,ε)+ Xn i,k=1

hikiku(ξ,ε)

¯¯

¯

=

¯¯

¯ Xn i,k=1

i£

(gik−δik+hik)ku(ξ,ε)¤

n−2

4(n1)Rgu(ξ,ε)

¯¯

¯

≤C λn−22 µ2(λ+|x|)16−n

≤C λn−22 µ2(λ+|x|)8(n+2)n−2 −n

for|x| ≤ρ. From this the assertion follows.

Corollary 6. The function v(ξ,ε)−u(ξ,ε) satisfies the estimate kv(ξ,ε)−u(ξ,ε)k

Ln−22n (Rn)≤C λ8µ+C

³λ ρ

´n−2

2

whenever (ξ, ε)∈λΩ.

Proof. It follows from Proposition 2 that kv(ξ,ε)−u(ξ,ε)k

Ln−22n (Rn)

≤C

°°

°∆gu(ξ,ε) n−2

4(n1)Rgu(ξ,ε)+n(n−2)u

n+2 n−2

(ξ,ε)

°°

°Ln+22n (Rn), where C is a constant that depends only on n. Hence, the assertion follows from Proposition 5.

We next establish a more precise estimate for the function v(ξ,ε) u(ξ,ε). Applying Proposition 1 withh= 0, we conclude that there exists a unique function w(ξ,ε)∈ E(ξ,ε) such that

Z

Rn

³

hdw(ξ,ε), dψi −n(n+ 2)u

4 n−2

(ξ,ε)w(ξ,ε)ψ

´

= Z

Rn

Xn i,k=1

µ λ6f−2|x|2)Hik(x)iku(ξ,ε)ψ for all test functionsψ∈ E(ξ,ε).

Proposition 7. The function w(ξ,ε) is smooth. Moreover, if (ξ, ε) λΩ, then the function w(ξ,ε) satisfies the estimates

|w(ξ,ε)(x)| ≤C λn−22 µ(λ+|x|)10−n

|∂w(ξ,ε)(x)| ≤C λn−22 µ(λ+|x|)9−n

|∂2w(ξ,ε)(x)| ≤C λn−22 µ(λ+|x|)8−n for all x∈Rn.

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Proof. There exist real numbersbk(ξ, ε) such that Z

Rn

³

hdw(ξ,ε), dψi −n(n+ 2)u

4 n−2

(ξ,ε)w(ξ,ε)ψ

´

= Z

Rn

Xn i,k=1

µ λ6f−2|x|2)Hik(x)iku(ξ,ε)ψ

+ Xn k=0

bk(ξ, ε) Z

Rn

ϕ(ξ,ε,k)ψ

for all test functions ψ ∈ E. Hence, standard elliptic regularity theory implies thatw(ξ,ε) is smooth.

It remains to prove quantitative estimates forw(ξ,ε). To that end, we consider a pair (ξ, ε)∈λΩ. One readily verifies that

°°

° Xn i,k=1

µ λ6f−2|x|2)Hik(x)iku(ξ,ε)

°°

°Ln+22n (Rn) ≤C λ8µ.

As a consequence, the functionw(ξ,ε)satisfieskw(ξ,ε)k

Ln−22n (Rn)≤C λ8µ.

Moreover, we havePn

k=0|bk(ξ, ε)| ≤C λ8µ. This implies

¯¯∆w(ξ,ε)+n(n+ 2)u

4 n−2

(ξ,ε)w(ξ,ε)¯

¯

=

¯¯

¯¯ Xn i,k=1

µ λ6f−2|x|2)Hik(x)iku(ξ,ε) Xn k=0

bk(ξ, ε) Z

Rn

ϕ(ξ,ε,k)

¯¯

¯¯

≤C λn−22 µ(λ+|x|)8−n for all x∈Rn. We claim that

sup

x∈Rn

(λ+|x|)n−22 |w(ξ,ε)(x)| ≤C λ8µ.

To show this, we fix a point x0 Rn. Letr= 12(λ+|x0|). Then u(ξ,ε)(x)n−24 ≤C r−2

and ¯¯∆w(ξ,ε)+n(n+ 2)u

4 n−2

(ξ,ε)w(ξ,ε)¯¯≤C λn−22 µ r8−n

for all x Br(x0). Hence, it follows from standard interior estimates that

rn−22 |w(ξ,ε)(x0)| ≤Ckw(ξ,ε)k

Ln−22n (Br(x0))

+C rn+22 °

°∆w(ξ,ε)+n(n+ 2)u

4 n−2

(ξ,ε)w(ξ,ε)°

°L(Br(x0))

≤C λ8µ+C λn−22 µ rn−182

≤C λ8µ.

(9)

Therefore, we have sup

x∈Rn(λ+|x|)n−22 |w(ξ,ε)(x)| ≤C λ8µ,

as claimed. Since supx∈Rn|x|n−22 |w(ξ,ε)(x)| < ∞, we can express the functionw(ξ,ε) in the form

(3) w(ξ,ε)(x) = 1

(n2)|Sn−1| Z

Rn

|x−y|2−n∆w(ξ,ε)(y)dy for all x∈Rn.

We are now able to use a bootstrap argument to prove the desired estimate for w(ξ,ε). It follows from (3) that

sup

x∈Rn

(λ+|x|)β|w(ξ,ε)(x)| ≤C sup

x∈Rn

(λ+|x|)β+2|∆w(ξ,ε)(x)|

for all 0< β < n−2. Since

|∆w(ξ,ε)(x)| ≤n(n+ 2)u(ξ,ε)(x)n−24 |w(ξ,ε)(x)|

+C λn−22 µ(λ+|x|)8−n for all x∈Rn, we conclude that

sup

x∈Rn

(λ+|x|)β|w(ξ,ε)(x)| ≤C λ2 sup

x∈Rn

(λ+|x|)β−2|w(ξ,ε)(x)|

+C λβ−n−182 µ

for all 0< β≤n−10. Iterating this inequality, we obtain sup

x∈Rn

(λ+|x|)n−10|w(ξ,ε)(x)| ≤C λn−22 µ.

The estimates for the first and second derivatives of w(ξ,ε) follow now from standard interior estimates.

Corollary 8. The functionv(ξ,ε)−u(ξ,ε)−w(ξ,ε) satisfies the estimate kv(ξ,ε)−u(ξ,ε)−w(ξ,ε)k

Ln−22n (Rn)≤C λ8(n+2)n−2 µn+2n−2 +C

³λ ρ

´n−2

2

whenever (ξ, ε)∈λΩ.

Proof. Consider the functions B1=

Xn i,k=1

i£

(gik−δik)kw(ξ,ε)¤

n−2

4(n1)Rgw(ξ,ε) and

B2= Xn i,k=1

µ λ6f−2|x|2)Hik(x)iku(ξ,ε).

(10)

By definition ofw(ξ,ε), we have Z

Rn

³

hdw(ξ,ε), dψig+ n−2

4(n1)Rgw(ξ,ε)ψ−n(n+ 2)u

4 n−2

(ξ,ε)w(ξ,ε)ψ

´

= Z

Rn

(B1+B2)ψ

for all functions ψ∈ E(ξ,ε). Since w(ξ,ε)∈ E(ξ,ε), it follows that w(ξ,ε) =−G(ξ,ε)(B1+B2).

Moreover, we have

v(ξ,ε)−u(ξ,ε)=G(ξ,ε)¡

B3+n(n−2)B4¢ , where

B3= ∆gu(ξ,ε) n−2

4(n1)Rgu(ξ,ε)+n(n−2)u

n+2 n−2

(ξ,ε)

and

B4=|v(ξ,ε)|n−24 v(ξ,ε)−u

n+2 n−2

(ξ,ε)−n+ 2 n−2u

4 n−2

(ξ,ε)(v(ξ,ε)−u(ξ,ε)).

Thus, we conclude that

v(ξ,ε)−u(ξ,ε)−w(ξ,ε)=G(ξ,ε)¡

B1+B2+B3+n(n−2)B4¢ , whereG(ξ,ε)denotes the solution operator constructed in Proposition 1.

As a consequence, we obtain kv(ξ,ε)−u(ξ,ε)−w(ξ,ε)k

Ln−22n (Rn)≤C°

°B1+B2+B3+n(n−2)B4°

°

Ln+22n (Rn). It follows from Proposition 7 that

|B1(x)| ≤C λn−22 µ2(λ+|x|)16−n≤C λn−22 µ2(λ+|x|)8(n+2)n−2 −n for|x| ≤ρand

|B1(x)| ≤C λn−22 µ|x|8−n for|x| ≥ρ. This implies

kB1k

Ln+22n (Rn)≤C λ8(n+2)n−2 µ2+C ρ8µ

³λ ρ

´n−2

2 .

Moreover, observe that kB2+B3k

Ln+22n (Rn)≤C λ8(n+2)n−2 µ2+C

³λ ρ

´n−2

2

by Proposition 5. Finally, Corollary 6 implies that kB4k

Ln+22n (Rn)≤Ckv(ξ,ε)−u(ξ,ε)k

n+2 n−2

Ln−22n (Rn)

≤C λ8(n+2)n−2 µn+2n−2 +C

³λ ρ

´n+2

2 .

(11)

Putting these facts together, we obtain kv(ξ,ε)−u(ξ,ε)−w(ξ,ε)k

Ln−22n (Rn) ≤C λ8(n+2)n−2 µn+2n−2 +C

³λ ρ

´n−2

2 . This completes the proof.

Proposition 9. We have

¯¯

¯¯ Z

Rn

³

|dv(ξ,ε)|2g− |du(ξ,ε)|2g+ n−2

4(n1)Rg(v(ξ,ε)2 −u2(ξ,ε))

´

+ Z

Rn

n(n−2) (|v(ξ,ε)|n−24 −u

4 n−2

(ξ,ε))u(ξ,ε)v(ξ,ε)

Z

Rn

n(n−2) (|v(ξ,ε)|n−22n −u

2n n−2

(ξ,ε))

Z

Rn

Xn i,k=1

µ λ6f−2|x|2)Hik(x)iku(ξ,ε)w(ξ,ε)

¯¯

¯¯

≤C λn−216n µn−22n +C λ8µ

³λ ρ

´n−2

2 +C

³λ ρ

´n−2

whenever (ξ, ε)∈λΩ.

Proof. By definition ofv(ξ,ε), we have Z

Rn

³

|dv(ξ,ε)|2g− hdu(ξ,ε), dv(ξ,ε)ig+ n−2

4(n1)Rgv(ξ,ε)(v(ξ,ε)−u(ξ,ε))

´

Z

Rn

n(n−2)|v(ξ,ε)|n−24 v(ξ,ε)(v(ξ,ε)−u(ξ,ε)) = 0.

Using Proposition 5 and Corollary 6, we obtain

¯¯

¯¯ Z

Rn

³

hdu(ξ,ε), dv(ξ,ε)ig− |du(ξ,ε)|2g+ n−2

4(n1)Rgu(ξ,ε)(v(ξ,ε)−u(ξ,ε))

´

Z

Rn

n(n−2)u

n+2 n−2

(ξ,ε)(v(ξ,ε)−u(ξ,ε))

Z

Rn

Xn i,k=1

µ λ6f−2|x|2)Hik(x)iku(ξ,ε)(v(ξ,ε)−u(ξ,ε))

¯¯

¯¯

°°

°∆gu(ξ,ε) n−2

4(n1)Rgu(ξ,ε)+n(n−2)u

n+2 n−2

(ξ,ε)

+ Xn i,k=1

µ λ6f−2|x|2)Hik(x)iku(ξ,ε)

°°

°Ln+22n (Rn)

· kv(ξ,ε)−u(ξ,ε)k

Ln−22n (Rn)

≤C λn−216n µ3+C λ8µ

³λ ρ

´n−2

2 +C

³λ ρ

´n−2 .

(12)

Moreover, we have

¯¯

¯¯ Z

Rn

Xn i,k=1

µ λ6f−2|x|2)Hik(x)iku(ξ,ε)(v(ξ,ε)−u(ξ,ε)−w(ξ,ε))

¯¯

¯¯

≤C λ8µkv(ξ,ε)−u(ξ,ε)−w(ξ,ε)k

Ln−22n (Rn)

≤C λn−216n µn−22n +C λ8µ

³λ ρ

´n−2

2

by Corollary 8. Putting these facts together, the assertion follows.

Proposition 10. We have

¯¯

¯¯ Z

Rn

(|v(ξ,ε)|n−24 −u

4 n−2

(ξ,ε))u(ξ,ε)v(ξ,ε) 2 n

Z

Rn

(|v(ξ,ε)|n−22n −u

2n n−2

(ξ,ε))

¯¯

¯¯

≤C λn−216n µn−22n +C

³λ ρ

´n

whenever (ξ, ε)∈λΩ.

Proof. We have the pointwise estimate

¯¯

¯(|v(ξ,ε)|n−24 −u

4 n−2

(ξ,ε))u(ξ,ε)v(ξ,ε) 2

n(|v(ξ,ε)|n−22n −u

2n n−2

(ξ,ε))

¯¯

¯

≤C|v(ξ,ε)−u(ξ,ε)|n−22n ,

where C is a constant that depends only onn. This implies

¯¯

¯¯ Z

Rn

(|v(ξ,ε)|n−24 −u

4 n−2

(ξ,ε))u(ξ,ε)v(ξ,ε) 2 n

Z

Rn

(|v(ξ,ε)|n−22n −u

2n n−2

(ξ,ε))

¯¯

¯¯

≤Ckv(ξ,ε)−u(ξ,ε)k

2n n−2

Ln−22n (Rn)

≤C λn−216n µn−22n +C

³λ ρ

´n

by Corollary 6.

Proposition 11. We have

¯¯

¯¯ Z

Rn

³

|du(ξ,ε)|2g+ n−2

4(n1)Rgu2(ξ,ε)−n(n−2)u

2n n−2

(ξ,ε)

´

Z

Bρ(0)

1 2

Xn i,k,l=1

hilhkliu(ξ,ε)ku(ξ,ε)

+ Z

Bρ(0)

n−2 16(n1)

Xn i,k,l=1

(∂lhik)2u2(ξ,ε)

¯¯

¯¯

≤C λn−216n µ3+C

³λ ρ

´n−2 whenever (ξ, ε)∈λΩ.

(13)

Proof. Note that

¯¯

¯gik(x)−δik+hik(x)1 2

Xn l=1

hil(x)hkl(x)

¯¯

¯

≤C|h(x)|3 ≤C µ3(λ+|x|)24≤C µ3(λ+|x|)n−216n for|x| ≤ρ. This implies

¯¯

¯¯ Z

Rn

¡|du(ξ,ε)|2g− |du(ξ,ε)|2¢ +

Z

Rn

Xn i,k=1

hikiu(ξ,ε)ku(ξ,ε)

Z

Bρ(0)

1 2

Xn i,k,l=1

hilhkliu(ξ,ε)ku(ξ,ε)

¯¯

¯¯

≤C λn−2µ3 Z

Bρ(0)

(λ+|x|)n−216n+2−2n+C λn−2 Z

Rn\Bρ(0)

(λ+|x|)2−2n

≤C λn−216n µ3+C

³λ ρ

´n−2 .

By Proposition 4, the scalar curvature of g satisfies the estimate

¯¯

¯Rg(x) + 1 4

Xn i,k,l=1

(∂lhik(x))2

¯¯

¯

≤C|h(x)|2|∂2h(x)|+C|h(x)| |∂h(x)|2

≤C µ3(λ+|x|)22≤C µ3(λ+|x|)n−216n−2 for|x| ≤ρ. This implies

¯¯

¯¯ Z

Rn

Rgu2(ξ,ε)+ Z

Bρ(0)

1 4

Xn i,k,l=1

(∂lhik)2u2(ξ,ε)

¯¯

¯¯

≤C λn−2µ3 Z

Bρ(0)

(λ+|x|)n−216n+2−2n+C λn−2 Z

Rn\Bρ(0)

(λ+|x|)4−2n

≤C λn−216n µ3+C ρ2

³λ ρ

´n−2 . At this point, we use the formula

iu(ξ,ε)ku(ξ,ε) n−2

4(n1)ik(u2(ξ,ε))

= 1 n

³

|du(ξ,ε)|2 n−2

4(n1)∆(u2(ξ,ε))

´ δik. Since hik is trace-free, we obtain

Xn i,k=1

hikiu(ξ,ε)ku(ξ,ε)= n−2 4(n1)

Xn i,k=1

hikik(u2(ξ,ε)),

(14)

hence Z

Rn

Xn i,k=1

hikiu(ξ,ε)ku(ξ,ε)= Z

Rn

n−2 4(n1)

Xn i,k=1

ikhiku2(ξ,ε). Since Pn

i=1ihik(x) = 0 for |x| ≤ρ, it follows that

¯¯

¯¯ Z

Rn

Xn i,k=1

hikiu(ξ,ε)ku(ξ,ε)

¯¯

¯¯≤C Z

Rn\Bρ(0)

u2(ξ,ε)≤C ρ2

³λ ρ

´n−2 . Putting these facts together, the assertion follows.

Corollary 12. The functionFg(ξ, ε) satisfies the estimate

¯¯

¯¯Fg(ξ, ε) Z

Bρ(0)

1 2

Xn i,k,l=1

hilhkliu(ξ,ε)ku(ξ,ε)

+ Z

Bρ(0)

n−2 16(n1)

Xn i,k,l=1

(∂lhik)2u2(ξ,ε)

Z

Rn

Xn i,k=1

µ λ6f(λ−2|x|2)Hik(x)iku(ξ,ε)w(ξ,ε)

¯¯

¯¯

≤C λn−216n µn−22n +C λ8µ

³λ ρ

´n−2

2 +C

³λ ρ

´n−2

whenever (ξ, ε)∈λΩ.

4. Finding a critical point of an auxiliary function We define a functionF :Rn×(0,∞)→R as follows: given any pair (ξ, ε)Rn×(0,∞), we define

F(ξ, ε) = Z

Rn

1 2

Xn i,k,l=1

Hil(x)Hkl(x)iu(ξ,ε)(x)ku(ξ,ε)(x)

Z

Rn

n−2 16(n1)

Xn i,k,l=1

(∂lHik(x))2u(ξ,ε)(x)2

+ Z

Rn

Xn i,k=1

Hik(x)iku(ξ,ε)(x)z(ξ,ε)(x), where z(ξ,ε)∈ E(ξ,ε) satisfies the relation

Z

Rn

³

hdz(ξ,ε), dψi −n(n+ 2)u(ξ,ε)(x)n−24 z(ξ,ε)ψ

´

= Z

Rn

Xn i,k=1

Hikiku(ξ,ε)ψ

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