A PRIORI ESTIMATES FOR THE YAMABE PROBLEM IN THE NON-LOCALLY CONFORMALLY FLAT CASE

FERNANDO C. MARQUES

Abstract. Given a compact Riemannian manifold (M^{n}, g), with positive Yamabe quotient, not conformally diffeomorphic to the standard sphere, we prove a priori estimates for solutions to the Yamabe problem. We restrict ourselves to the dimensions where the Positive Mass Theorem is known to be true, that is, when n ≤ 7. We also show that, when n ≥ 6, the Weyl tensor has to vanish at a point where solutions to the Yamabe equation blow up.

Contents

1. Introduction 1 2. A Pohozaev-type identity 4 3. Conformal scalar curvature equation 7 4. Isolated and isolated simple blowup points 8 5. Symmetry estimates 17 6. Local blowup analysis 24 7. Compactness theorem 30 References 32

1. Introduction

Let (M^{n}, g) be a compact Riemannian n-manifold, n ≥ 3, without boundary. The classical Yamabe problem consists of finding a metric g̃, conformally related to g, with constant scalar curvature on M. It can be considered as a generalization of the classical Uniformization Theorem on Riemann surfaces to the setting of higher dimensional manifolds.

In analytical terms, the problem is equivalent to show the existence of a positive solution u to the equation

(1.1)  Δ_{g}u − \frac{n − 2}{4(n − 1)}R_{g}u + Ku^{\frac{n+2}{n−2}} = 0 on M,

where Δ_{g} denotes the Laplace-Beltrami operator associated with the metric g, R_{g} denotes the scalar curvature of the metric g, and K is a constant. The linear operator L_{g} = Δ_{g} − \frac{n−2}{4(n−1)}R_{g} is called the conformal Laplacian of the metric g.

The solution of the Yamabe problem was an outstanding achievement since, for the first time, it was given a very satisfactory existence theory to a nonlinear partial differential equation involving a critical exponent. After the initial paper in 1960 by Yamabe [17], which contained an error, contributions made by Trudinger [16], Aubin [1], and finally by Schoen [13] in 1984, solved the problem completely in the affirmative.

In this paper we shall be interested in the set of solutions to the Yamabe problem. When the first eigenvalue of the conformal Laplacian L_{g} is negative, it is not difficult to see that the solution is unique. If this eigenvalue is zero, the equation becomes linear and then solutions are unique up to a constant. Therefore the only interesting case left is the positive one.

We know that when the underlying manifold is the sphere (𝕊^{n}, g_{0}), endowed with the standard metric, M. Obata’s theorem (see [9]) classifies all solutions to the equation, and this set is noncompact in the C^{2} topology. On the other hand, the standard sphere is the only compact manifold with a noncompact group of conformal diffeomorphisms, so one should expect different behavior in the other cases.

In [11], R. Schoen proved the compactness, in the C^{2} topology, of the set of solutions to the Yamabe equation, in the positive case, for every locally conformally flat manifold not conformally diffeomorphic to the sphere. He also suggested a strategy to prove these a priori C^{2,α} estimates in the non-locally conformally flat case, based on Pohozaevtype identities. In [7], Y. Y. Li and M. Zhu followed these lines and proved the theorem in dimension 3, in which case standard estimates on the blowing up solutions are sufficient for a Pohozaev identity to be applied. Other compactness theorems for the Yamabe equation, in the locally conformally flat case, are proved in [3] for manifolds with boundary, and in [10] for singular solutions on the sphere.

Our main result in this paper is the following a priori estimates theorem in the general case:

Theorem 1.1. Let (M^{n}, g) be a smooth closed Riemannian manifold with positive Yamabe quotient, not conformally equivalent to (𝕊^{n}, g_{0}). Assume 4 ≤ n ≤ 7. Then, for every ϵ > 0, there exists a positive constant C = C(ϵ, g) so that

\{ \begin{matrix}1/C ≤ u ≤ C and \\ ||u||_{C^{2,α}(M)} ≤ C \end{matrix}

for every u ∈ ∪_{1+ϵ≤p≤ \frac{n+2}{n−2}} 𝕄_{p}, where 0 < α < 1 and

𝕄_{p} = \{u > 0 : Δ_{g}u − \frac{n − 2}{4(n − 1)}R_{g}u + Ku^{p} = 0 on M\}.

These estimates clearly imply the compactness of the set of solutions to the Yamabe equation in the C^{2} topology. We restrict ourselves to the dimensions covered by the Positive Mass Theorem due to Schoen and Yau [14], i.e., n ≤ 7, since the final global argument in our proof depends essentially on this result. Due to technical problems concerning singularities of minimizing hypersurfaces, the Positive Mass Theorem is still not known to be true for dimensions greater than 7.

Our result will follow from a contradiction between a local restriction coming from a Pohozaev-type identity and a global argument provided by the Positive Mass Theorem. In order to accomplish that we will need a careful blowup analysis of solutions, part of it inspired by the work of C. C. Chen and C. S. Lin [2]. We introduce some new symmetry estimates which allow us to control how close the blowing up solutions get to some specific rotationally symmetric functions. These symmetric functions will be solutions to the corresponding critical (p = \frac{n+2}{n−2}) equation in ℝ^{n}. Since we are also dealing with subcritical equations, estimates on τ = \frac{n+2}{n−2} − p are also given.

We would like to point out that the symmetry estimates (Proposition 5.4), when n ≤ 5, are just as in the conformally flat case. However, when n ≥ 6 we can no longer expect these same estimates to hold. This is because, in general, the asymptotic expansion of the Green function for the conformal Laplacian has additional terms (see [5]).

One important difficulty we must overcome when n ≥ 6, pointed out by Schoen in [11], is to show conformal flatness of the metric to a sufficiently high order at a blowup point. This is needed in order to apply the Positive Mass Theorem when n = 6 or 7. That is the content of our next theorem, where W_{g} denotes the Weyl tensor of the metric g:

Theorem 1.2. Assume n ≥ 6 and let u_{i} be a sequence of positive solutions to (1.1). Suppose x_{i} → \overline{x} is a sequence of points such that u_{i}(x_{i}) → ∞ as i → ∞. Then

W_{g}(\overline{x}) = 0.

In general one should expect, as indicated by Schoen in [12], that at a blowup point \overline{x} we must have :

∇^{k}W_{g}(\overline{x}) = 0

for all 0 ≤ k ≤ \frac{n−6}{2}. The proof of Theorem 1.2 also relies upon the symmetry estimates and a Pohozaev-type identity.

It is not difficult to check that the Theorem 1.1 implies the existence of a solution to the Yamabe problem. This is because solutions u_{i} to the subcritical equations, with p_{i} → \frac{n+2}{n−2} as i → ∞, can be constructed by standard variational methods. Actually one can say more. Another consequence of the compactness theorem is the computation of the total Leray-Schauder degree of all solutions to equation (1.1):

Theorem 1.3. Let (M^{n}, g) be a smooth closed Riemannian manifold with positive Yamabe quotient, not conformally equivalent to (𝕊^{n}, g_{0}), 4 ≤ n ≤ 7. Then, if Λ is sufficiently large,

deg(F, Ω_{Λ}, 0) = −1,

where F (u) = u + L_{g}^{−1}(E(u)u^{\frac{n+2}{n−2}} ), E(u) = − ∫_{M} uL_{g}(u)dv_{g} and

Ω_{Λ} = {u ∈ C^{2,α}(M) : min_{M} u > Λ^{−1}, ||u||_{2,α} < Λ}.

This theorem follows from Theorem 1.1 and arguments given by Schoen in [11].

Acknowledgements. The content of this paper is part of the author’s doctoral thesis [8]. The author would like to dedicate this paper to the memory of his friend and advisor Prof. José F. Escobar. His encouragement was invaluable for the completion of this work. While the author was at Cornell University, he was fully supported by CNPq-Brazil.

2. A Pohozaev-type identity

In this section we will establish a Pohozaev-type identity which will be very useful in the subsequent blowup analysis.

Suppose u : B_{ρ}(0) \ {0} ⊂ ℝ^{n} → ℝ is a positive C^{2} solution to the equation

(2.1)  a^{ij}(x)∂_{ij}u + b^{i}(x)∂_{i}u + c(x)u + K(x)u^{p} = 0,

where p ≠ −1, K ∈ C^{1} and a^{ij}, b^{i}, c are continuous functions, 1 ≤ i, j ≤ n. Here we are using the summation convention.

Define

P (r, u) = (2.2)

∫_{|x|=r}(\frac{n − 2}{2}u\frac{∂u}{∂r} − \frac{|x|}{2}|∇u|^{2} + |x||\frac{∂u}{∂r}|^{2} + \frac{1}{p + 1}K(x)|x|u^{p+1})dσ(r),

whenever 0 < r < ρ.

A PRIORI ESTIMATES FOR THE YAMABE PROBLEM 5

The following lemma gives the Pohozaev-type identity we are interested in. Lemma 2.1. Given 0 < s ≤ r < ρ,

P (r, u) − P (s, u) = − ∫_{s≤|x|≤r}(x^{k}∂_{k}u + \frac{n − 2}{2}u)A(u)dx

+\frac{1}{p + 1} ∫_{s≤|x|≤r}(x^{k}∂_{k}K(x))u^{p+1}dx

+(\frac{n}{p + 1} − \frac{n − 2}{2}) ∫_{s≤|x|≤r} K(x)u^{p+1}dx,

where A(u) = (a^{ij} − δ_{ij})∂_{ij}u + b^{i}∂_{i}u + cu. Proof. Multiplying the equation (2.1) by x^{k}∂_{k}u, and integrating over the set {x : s ≤ |x| ≤ r}, we obtain

(2.3)  ∫_{s≤|x|≤r}(x^{k}∂_{k}u)(Δu + A(u) + K(x)u^{p})dx = 0.

Integration by parts gives:

∫_{s≤|x|≤r}(x^{k}∂_{k}u)∂_{ii}udx =

− ∫_{s≤|x|≤r}(δ_{i}^{k}∂_{k}u∂_{i}u + \frac{1}{2}x^{k}∂_{k}[(∂_{i}u)^{2}])dx

+\frac{1}{r} ∫_{|x|=r}(x^{k}∂_{k}u)(x_{i}∂_{i}u) − \frac{1}{s} ∫_{|x|=s}(x^{k}∂_{k}u)(x_{i}∂_{i}u) =

− ∫_{s≤|x|≤r}(δ_{i}^{k}∂_{k}u∂_{i}u − \frac{n}{2}(∂_{i}u)^{2})dx − \frac{r}{2} ∫_{|x|=r}(∂_{i}u)^{2} + \frac{s}{2} ∫_{|x|=s}(∂_{i}u)^{2}

+\frac{1}{r} ∫_{|x|=r}(x^{k}∂_{k}u)(x_{i}∂_{i}u) − \frac{1}{s} ∫_{|x|=s}(x^{k}∂_{k}u)(x_{i}∂_{i}u),

and summing over i = 1, …, n, we obtain ∫_{s≤|x|≤r}(x^{k}∂_{k}u)Δudx =

\frac{n − 2}{2} ∫_{s≤|x|≤r} |∇u|^{2}dx − \frac{r}{2} ∫_{|x|=r} |∇u|^{2} + \frac{s}{2} ∫_{|x|=s} |∇u|^{2}

+r ∫_{|x|=r}(\frac{∂u}{∂r})^{2} − s ∫_{|x|=s}(\frac{∂u}{∂r})^{2}. (2.4)

Also

∫_{s≤|x|≤r}(x^{k}∂_{k}u)K(x)u^{p}dx =

\frac{1}{p + 1} ∫_{s≤|x|≤r} x^{k}∂_{k}(u^{p+1})K(x)dx =

−\frac{n}{p + 1} ∫_{s≤|x|≤r} K(x)u^{p+1}dx − \frac{1}{p + 1} ∫_{s≤|x|≤r}(x^{k}∂_{k}K(x))u^{p+1}dx

+\frac{r}{p + 1} ∫_{|x|=r} K(x)u^{p+1} − \frac{s}{p + 1} ∫_{|x|=s} K(x)u^{p+1}. (2.5)

On the other hand, multiplying the equation (2.1) by u, and once again integrating by parts, we get

∫_{s≤|x|≤r} |∇u|^{2}dx =

∫_{s≤|x|≤r}(uA(u) + K(x)u^{p+1})dx + ∫_{|x|=r} u\frac{∂u}{∂r} − ∫_{|x|=s} u\frac{∂u}{∂r}. (2.6)

Now we substitute equalities (2.4), (2.5) and equality (2.6) in equality (2.3) and the Pohozaev identity follows by rearranging terms. □

When u is a solution to the equation (2.1) in the entire ball, by taking the limit as s → 0 we get

P (r, u) = − ∫_{|x|≤r}(x^{k}∂_{k}u + \frac{n − 2}{2}u)A(u)dx (2.7)

+\frac{1}{p + 1} ∫_{|x|≤r}(x^{k}∂_{k}K(x))u^{p+1}dx +

(\frac{n}{p + 1} − \frac{n − 2}{2}) ∫_{|x|≤r} K(x)u^{p+1}dx.

Integrating by parts once more, we can also get

P (r, u) = − ∫_{|x|≤r}(x^{k}∂_{k}u + \frac{n − 2}{2}u)((a^{ij} − δ^{ij})∂_{ij}u + b^{i}∂_{i}u)dx (2.8)

+ ∫_{|x|≤r}(\frac{1}{2}x^{k}∂_{k}c + c)u^{2}dx − \frac{r}{2} ∫_{|x|=r} cu^{2}dσ(r)

+\frac{1}{p + 1} ∫_{|x|≤r}(x^{k}∂_{k}K(x))u^{p+1}dx

+(\frac{n}{p + 1} − \frac{n − 2}{2}) ∫_{|x|≤r} K(x)u^{p+1}dx.

3. Conformal scalar curvature equation

In this section we will introduce the partial differential equation we are interested in, and we shall discuss some of its properties related to conformal deformation of metrics.

Let Ω ∈ ℝ^{n} be an open set, and suppose g is a Riemannian metric in Ω. Suppose also f is a positive C^{1} function defined in Ω.

Consider a positive C^{2} function u satisfying

(3.1)  Δ_{g}u − c(n)R_{g}u + Kf^{−τ}u^{p} = 0 in Ω,

where c(n) = \frac{n−2}{4(n−1)}, K = n(n − 2), 1 < p ≤ \frac{n+2}{n−2} and τ = \frac{n+2}{n−2} − p. We will use the notation R_{g} for the scalar curvature of g. The operator L_{g} = Δ_{g} − c(n)R_{g} is called the conformal Laplacian of the metric g.

When p = \frac{n+2}{n−2}, this partial differential equation is intimately related to conformal geometry, particularly when one studies conformally related metrics with constant scalar curvature. More specifically, given a positive solution u, the metric u^{\frac{4}{n−2}} g has constant scalar curvature equal to 4n(n − 1).

Now let us describe an important feature of solutions to that type of equation. Let u be a solution to equation (3.1) and choose \overline{x} ∈ Ω. Given s > 0, define the renormalized function

v(y) = s^{\frac{2}{p−1}} u(exp \overline{x}(sy)).

Then

L_{h}v + Kf̃^{−τ}v^{p} = 0,

where f̃(y) = f(sy) and the components of the metric h in normal coordinates are given by h_{kl}(y) = g_{kl}(sy). The important point here is that v satisfies an equation of the same type.

The equation is also conformally invariant in the following sense. Suppose g̃ = ϕ^{\frac{4}{n−2}} g is a metric conformal to g. Let us recall

(3.2)  Lg̃(ϕ^{−1}u) = ϕ^{− \frac{n+2}{n−2}} L_{g}(u)

for any function u, and

(3.3)  Rg̃ = −c(n)^{−1}ϕ^{− \frac{n+2}{n−2}} L_{g}(ϕ).

(See [5]).

Therefore, if u is a solution to (3.1), then ϕ^{−1}u satisfies

Lg̃(ϕ^{−1}u) + K(ϕf)^{−τ}(ϕ^{−1}u)^{p} = 0,

which is again an equation of the same type.

This will have very important consequences in what follows. We will study sequences of solutions u_{i} to equation (3.1). When what we want to study is conformally invariant, we are allowed to replace u_{i} by another sequence of functions v_{i} = ϕ_{i}^{−1}u_{i}, at the same time replacing

the metric g_{i} by g̃_{i} = ϕi^{\frac{4}{n−2}} g_{i}, as long as we have a uniform control on the conformal factors ϕ_{i}. In this paper, there will be two examples of such a procedure.

First, we can suppose the metric g_{i} has positive scalar curvature in a small ball centered at some fixed point x_{i}. To see this, fix σ > 0 small and let ϕ be the first eigenfunction of Δ_{g} with respect to the Dirichlet condition:

(3.4)  \{ \begin{matrix}Δ_{g}ϕ + λ_{1}ϕ = 0 in B_{2σ}(x) \\ ϕ = 0 on ∂B_{2σ}(x). \end{matrix}

Recall that the corresponding eigenspace is one-dimensional and we can choose ϕ > 0 on B_{2σ}(x). Now, since λ_{1} → ∞ as σ → 0, we can choose σ small enough so that

Δ_{g}ϕ − c(n)R_{g}ϕ < 0

in B_{σ}(x). Defining g̃ = ϕ^{\frac{4}{n−2}} g, relation (3.3) implies Rg̃ > 0 in B_{σ}(x). Moreover, if we take as a conformal factor the solution ψ to:

(3.5)  \{ \begin{matrix}Δ_{g}ψ − c(n)R_{g}ψ = 0 in B_{σ}(x) \\ ψ = 1 on ∂B_{σ}(x), \end{matrix}

we can also have Rg̃ = 0.

The second example is related to the so-called conformal normal coordinates. (See [5]). Given an integer N ≥ 2, there exists a positive function ϕ (which can be constructed explicitly), such that, setting g̃ = ϕ^{\frac{4}{n−2}} g, the volume element satisfies:

det(g̃_{ij}) = 1 + O(r^{N}),

in g̃-normal coordinates around x, where r = dg̃(x, ·). This allows us to simplify the local asymptotic analysis. For example, in conformal normal coordinates around x, Rg̃ = O(r^{2}) and ΔRg̃(x) = −\frac{1}{6}|Wg̃(x)|^{2}, where W stands for the Weyl tensor.

4. Isolated and isolated simple blowup points

In this section we will define isolated and isolated simple blowup points and we shall discuss their basic properties. The results in this section are well-known in the locally conformally flat setting ([6], [15]) and in general when n = 3 ([7]). We will slightly modify their proofs in [7] to make them work in any dimension.

Let Ω ⊂ ℝ^{n} be an open set, and suppose g_{i} is a sequence of Riemannian metrics in Ω converging, in the C_{loc}^{2} topology, to a metric g.

Suppose also that f_{i} is a sequence of positive C^{1} functions converging in the C_{loc}^{1} topology to a positive function f.

We will consider a sequence u_{i} of positive C^{2} functions satisfying

(4.1)  L_{g_{i}}u_{i} + Kf_{i}^{−τ_{i}}u_{i}^{p_{i}} = 0 in Ω,

where c(n) = \frac{n−2}{4(n−1)}, K = n(n − 2), 1 + ϵ_{0} < p_{i} ≤ \frac{n+2}{n−2} for some ϵ_{0} > 0 and τ_{i} = \frac{n+2}{n−2} − p_{i}.

We will sometimes omit the subscript i, for the sake of simplicity, and we will use the symbols c, C to denote various positive constants.

Definition 4.1. We say that \overline{x} ∈ Ω is an isolated blowup point for u_{i} if there exists a sequence x_{i} ∈ Ω, converging to \overline{x}, so that:

(1) x_{i} is a local maximum point of u_{i};

(2) M_{i} := u_{i}(x_{i}) → ∞ as i → ∞;

(3) there exist r, C > 0 such that

(4.2)  u_{i}(x) ≤ Cd_{g_{i}}(x, x_{i})^{− \frac{2}{p_{i}−1}}

for every x ∈ B_{r}(x_{i}) ⊂ Ω. Here B_{r}(x_{i}) denotes the geodesic ball of radius r, centered at x_{i}, with respect to the metric g_{i}.

Remark: In various parts of the text, we will identify x_{i} with the origin, that meaning we are making use of normal coordinates in a small ball around x_{i}. More precisely, we will sometimes write u_{i}(x) instead of u_{i}(exp_{x_{i}}(x)) and |x| instead of d_{g_{i}}(x, x_{i}), and those functions will be defined in balls centered at 0.

Note that the definition of isolated blowup points is invariant under renormalization, which was described in the last section. This follows from the fact that, if v(y) = s^{\frac{2}{p_{i}−1}} u(sy), then

u(x) ≤ C|x|^{− \frac{2}{p_{i}−1}} ⇔ v(y) ≤ C|y|^{− \frac{2}{p_{i}−1}} .

The first result concerning isolated blowup points is the following Harnack inequality.

Proposition 4.2. Suppose that u_{i} is a sequence of positive functions satisfying equation (4.1) and assume x_{i} → \overline{x} is an isolated blowup point. Then there exists a constant C > 0 such that

max_{\frac{s}{2} ≤d_{g_{i}}(x,x_{i})≤2s} u_{i}(x) ≤ C min_{\frac{s}{2} ≤d_{g_{i}}(x,x_{i})≤2s} u_{i}(x),

where 0 < s < \frac{r}{3}.

Proof. Let (x^{1}, …, x^{n}) be normal coordinates with respect to the metric g_{i} on the ball B_{r}(x_{i}). (See remark after Definition 4.1).

Define

v_{i}(y) = s^{\frac{2}{p_{i}−1}} u_{i}(sy),

where |y| < 3.

Then, as discussed in the last section,

L_{h_{i}}v_{i}(y) + Kf̃_{i}^{−τ_{i}}v_{i}^{p_{i}}(y) = 0,

where f̃_{i}(y) = f_{i}(sy) and (h_{i})_{kl}(y) = (g_{i})_{kl}(sy), and we also know that

v_{i}(y) ≤ C|y|^{− \frac{2}{p_{i}−1}} ,

whenever |y| < 3.

It follows from this last inequality that v_{i} is uniformly bounded in compact subsets of B_{3}(0) \ {0}. The Harnack inequality for elliptic linear equations then implies that there exists C > 0 such that

max_{\frac{1}{2} ≤|y|≤2} v_{i}(y) ≤ C min_{\frac{1}{2} ≤|y|≤2} v_{i}(y).

The result now follows directly. □

The Proposition 4.2 clearly implies the so-called spherical Harnack inequality for isolated blowup points. Namely, given 0 < s ≤ \frac{2}{3}r, there exists a positive constant C, not depending on s, such that

(4.3)  max_{d_{g_{i}}(x,x_{i})=s} u_{i}(x) ≤ C min_{d_{g_{i}}(x,x_{i})=s} u_{i}(x).

Define U_{0}(y) = (1 + |y|^{2})^{\frac{2−n}{2}} . It is not difficult to check that

ΔU_{0}(y) + KU0^{\frac{n+2}{n−2}} (y) = 0.

The next proposition says that, in the case of an isolated blowup point, the functions u_{i}, when renormalized, converge in the C^{2} topology to the rotationally symmetric function U_{0}.

Proposition 4.3. Let u_{i} be a sequence of positive functions satisfying the equation (4.1) and x_{i} → \overline{x} be an isolated blowup point. Assume that R_{i} → ∞ and ϵ_{i} → 0 are given. Then p_{i} → \frac{n+2}{n−2} and, after possibly passing to a subsequence,

(4.4)  ||M_{i}^{−1}u_{i}(M_{i}^{− \frac{p_{i}−1}{2}} y) − U_{0}(y)||_{C^{2}(B_{R_{i}}(0))} ≤ ϵ_{i}

and

\frac{R_{i}}{log M_{i}} → 0 as i → ∞.

Here M_{i} is as in Definition 4.1.

Proof. Let (x^{1}, …, x^{n}) be normal coordinates with respect to the metric g_{i} on the ball B_{r}(x_{i}). (See first remark after Definition 4.1).

Define

v_{i}(y) = M_{i}^{−1}u_{i}(M_{i}^{− \frac{p_{i}−1}{2}} y)

for |y| < rMi^{\frac{p_{i}−1}{2}} . Here r is as in Definition 4.1.

Then

L_{h_{i}}v_{i}(y) + Kf̃_{i}^{−τ_{i}}v_{i}^{p_{i}}(y) = 0,

where f̃_{i}(y) = f_{i}(M_{i}^{− \frac{p_{i}−1}{2}} y) and (h_{i})_{kl}(y) = (g_{i})_{kl}(M_{i}^{− \frac{p_{i}−1}{2}} y).

Note also that

(4.5)  \{ \begin{matrix}v_{i}(0) = 1, ∇v_{i}(0) = 0 \\ 0 < v_{i}(y) ≤ C|y|^{− \frac{2}{p_{i}−1}} for |y| < rMi^{\frac{p_{i}−1}{2}} . \end{matrix}

Claim: There exists C > 0 such that v_{i}(y) ≤ C, whenever |y| <

rMi^{\frac{p_{i}−1}{2}} .

Proof of Claim: From properties (4.5), we get

(4.6)  v_{i}(y) ≤ C,

if 1 ≤ |y| ≤ rMi^{\frac{p_{i}−1}{2}} .

Now, from what was discussed in the previous section, up to a conformal deformation we can suppose our metrics have zero scalar curvature in small balls. In particular their conformal Laplacians will satisfy the maximum principle. This implies there exists C > 0 so that

min_{|y|≤r} v_{i}(y) ≥ C^{−1} min_{|y|=r} v_{i}(y) ∀ i,

and 0 < r ≤ 1. The spherical Harnack inequality (4.3) implies

max_{|y|=r} v_{i}(y) ≤ C min_{|y|=r} v_{i}(y) ≤ C min_{|y|≤r} v_{i}(y)
(4.7)  ≤ Cv_{i}(0) = C,

for 0 < r ≤ 1. This and inequality (4.6) imply the claim.

Standard elliptic estimates now imply that, after passing to a subsequence, v_{i} → v > 0 in C_{loc}^{2}(ℝ^{n}), where

(4.8)  \{ \begin{matrix}Δv(y) + Kv^{p}(y) = 0, y ∈ ℝ^{n} \\ v(0) = 1, ∇v(0) = 0, \end{matrix}

where p = lim_{i→∞} p_{i}. Here Δ denotes the Euclidean Laplacian.

A well-known theorem by Caffarelli, Gidas and Spruck [4], states that we necessarily have p = \frac{n+2}{n−2} and v(y) = U_{0}(y).

The Proposition now follows easily. □

Now let us introduce the notion of an isolated simple blowup point. Suppose u_{i} is a sequence of positive functions satisfying equation

(4.1) and x_{i} → \overline{x} is an isolated blowup point. Define

\overline{u}_{i}(r) = \frac{1}{σ_{n−1}r^{n−1}} ∫_{∂B_{r}(x_{i})} u_{i}dσ(r),

where σ_{n−1} denotes the area of a unit sphere in ℝ^{n}. We are using g_{i}-normal coordinates and integrating with respect to the Euclidean volume form.

Definition 4.4. We say x_{i} → \overline{x} is an isolated simple blowup point if there exists a real number 0 < ρ < r such that the functions

û_{i}(r) = r^{\frac{2}{p_{i}−1}} \overline{u}_{i}(r)

have exactly one critical point in the interval (0, ρ), for i large.

It is not difficult to see that Proposition 4.3 implies that û_{i} has exactly one critical point in the interval (0, R_{i}u_{i}(x_{i})^{− \frac{p_{i}−1}{2}} ). Moreover its derivative is negative right after the critical point. As a result, if the blowup is isolated simple, then

û′_{i}(r) < 0

for all R_{i}M_{i}^{− \frac{p_{i}−1}{2}} ≤ r < ρ.

Now we turn to the first estimate on isolated simple blowup points.

Proposition 4.5. Let u_{i} be a sequence of positive functions satisfying equation (4.1) and x_{i} → \overline{x} be an isolated simple blowup point. Then there exists a constant C > 0 and 0 < ρ_{1} < ρ such that, for each i,

(4.9)  M_{i}u_{i}(x) ≤ Cd_{g_{i}}(x, x_{i})^{2−n},

whenever d_{g_{i}}(x, x_{i}) ≤ ρ_{1}. Moreover, if R_{i}M_{i}^{− \frac{p_{i}−1}{2}} ≤ d_{g_{i}}(x, x_{i}) ≤ ρ_{1}, then

(4.10)  M_{i}u_{i}(x) ≥ C^{−1}G_{i}(x_{i}, x),

where G_{i} is the Green function of L_{g_{i}} with respect to the Dirichlet boundary condition on B_{ρ_{1}}(x_{i}).

Proof. We first need a slightly different estimate.

Let δ > 0, and define λ_{i} = (n − 2 − δ)\frac{p_{i}−1}{2} − 1.

Let us apply the Proposition 4.3 to some R_{i} → ∞ and 0 < ϵ_{i} < e^{−R_{i}}.

Claim 1: If δ is sufficiently small, there exist constants 0 < ρ_{1} < ρ and C > 0 such that

(4.11)  M_{i}^{λ_{i}}u_{i}(x) ≤ Cd(x, x_{i})^{2−n+δ},
(4.12)  M_{i}^{λ_{i}}|∇u_{i}(x)| ≤ Cd(x, x_{i})^{1−n+δ},
(4.13)  M_{i}^{λ_{i}}|∇^{2}u_{i}(x)| ≤ Cd(x, x_{i})^{−n+δ},

for every x so that R_{i}M_{i}^{− \frac{p_{i}−1}{2}} ≤ d(x, x_{i}) ≤ ρ_{1}.

The proof of Claim 1 is analogous to the proof of the Lemma 3.3 in [7].

Remark: It is not difficult to see that the previous estimates imply

(4.14)  \{ \begin{matrix}v_{i}(y) & ≤ CM_{i}^{δ \frac{p_{i}−1}{2}} (1 + |y|)^{2−n}, \\ |∇v_{i}(y)| & ≤ CM_{i}^{δ \frac{p_{i}−1}{2}} (1 + |y|)^{1−n}, \\ |∇^{2}v_{i}(y)| & ≤ CM_{i}^{δ \frac{p_{i}−1}{2}} (1 + |y|)^{−n} \end{matrix}

for any |y| ≤ ρ_{1}Mi^{\frac{p_{i}−1}{2}} .

Let us now estimate τ_{i}.

Claim 2: There exists C > 0 such that

τ_{i} ≤ \{ \begin{matrix}CM_{i}^{(−1+δ) \frac{4}{n−2} +o(1)} if n > 4, \\ CM_{i}^{(−1+δ)2+o(1)} log M_{i} if n = 4, \end{matrix}

and, in particular, M_{i}^{τ_{i}} → 1 as i → ∞.

Proof of Claim 2: We will apply the Pohozaev identity (2.7) from Section 2 to u_{i} on the ball of radius \frac{ρ_{1}}{2} :

P (\frac{ρ_{1}}{2} , u_{i}) = − ∫_{|x|≤ \frac{ρ_{1}}{2}} (x^{m}∂_{m}u_{i} + \frac{n − 2}{2}u_{i})A_{i}(x)dx

+(\frac{n}{p_{i} + 1} − \frac{n − 2}{2}) ∫_{|x|≤ \frac{ρ_{1}}{2}} Kf_{i}^{−τ_{i}}u_{i}^{p_{i}+1}dx

−\frac{τ_{i}}{p_{i} + 1} ∫_{|x|≤ \frac{ρ_{1}}{2}} Kf_{i}^{−τ_{i}−1}(x^{m}∂_{m}f_{i})u_{i}^{p_{i}+1}dx, (4.15)

where

A_{i}(x) = (g^{kl} − δ^{kl})(x)∂_{kl}u_{i}(x)
+(∂_{k}g^{kl} + |g|^{− \frac{1}{2}} ∂_{k}(|g|^{\frac{1}{2}} )g^{kl})(x)∂_{l}u_{i}(x) − c(n)R_{g}(x)u_{i}(x)

and recall

P (\frac{ρ_{1}}{2} , u_{i}) = ∫_{|x|= \frac{ρ_{1}}{2}} (\frac{n − 2}{2}u_{i} \frac{∂u_{i}}{∂r} − \frac{|x|}{2}|∇u_{i}|^{2} + |x||\frac{∂u_{i}}{∂r} |^{2})dσ

+\frac{1}{p_{i} + 1} ∫_{|x|= \frac{ρ_{1}}{2}} Kf_{i}^{−τ_{i}}|x|u_{i}^{p_{i}+1}dσ.

From Claim 1 we get

(4.16)  |P (\frac{ρ_{1}}{2} , u_{i})| ≤ cM_{i}^{−2λ_{i}}.

Define

Â_{i}(y) = (g^{kl} − δ^{kl})(M_{i}^{− \frac{p_{i}−1}{2}} y)∂_{kl}v_{i}
+M_{i}^{− \frac{p_{i}−1}{2}} (∂_{k}g^{kl} + |g|^{− \frac{1}{2}} ∂_{k}(|g|^{\frac{1}{2}} )g^{kl})(M_{i}^{− \frac{p_{i}−1}{2}} y)∂_{l}v_{i}

−c(n)M_{i}^{−(p_{i}−1)}R_{g}(M_{i}^{− \frac{p_{i}−1}{2}} y)v_{i}. (4.17)

The change of variables y = M^{\frac{p_{i}−1}{2}} x, the inequalities (4.14) and the fact that the metric is euclidean up to first order in normal coordinates yield

| ∫_{|x|≤ \frac{ρ_{1}}{2}} (x^{m}∂_{m}u_{i} + \frac{n − 2}{2}u_{i})A_{i}(x)dx|

= M_{i}^{p \frac{2−n}{2}} Mi^{\frac{n+2}{2}} | ∫_{|y|≤ \frac{ρ_{1}}{2} M^{\frac{p_{i}−1}{2}}} (y^{m}∂_{m}v_{i} + \frac{n − 2}{2}v_{i})Â_{i}(y)dy|

≤ CM_{i}^{p \frac{2−n}{2}} Mi^{\frac{n+2}{2}} M_{i}^{−(p_{i}−1)}M_{i}^{δ(p_{i}−1)} ∫_{|y|≤ \frac{ρ_{1}}{2} M^{\frac{p_{i}−1}{2}}} (1 + |y|)^{4−2n}dy

≤ \{ \begin{matrix}CM_{i}^{(−1+δ) \frac{4}{n−2} +o(1)} if n > 4, \\ CM_{i}^{(−1+δ)2+o(1)} log M_{i} if n = 4. \end{matrix} (4.18)

Therefore, from the inequalities (4.16) and (4.18) and the identity (4.15),

\frac{(n − 2)τ_{i}}{2(p_{i} + 1)} ∫_{|x|≤ \frac{ρ_{1}}{2}} Kf_{i}^{−τ_{i}}u_{i}(x)^{p_{i}+1}dx

−\frac{τ_{i}}{p_{i} + 1} ∫_{|x|≤ \frac{ρ_{1}}{2}} Kf_{i}^{−τ_{i}−1}(x^{m}∂_{m}f_{i})u_{i}^{p_{i}+1}dx

≤ \{ \begin{matrix}CM_{i}^{(−1+δ) \frac{4}{n−2} +o(1)} if n > 4, \\ CM_{i}^{(−1+δ)2+o(1)} log M_{i} if n = 4. \end{matrix} (4.19)

Since, from the Proposition 4.3,

∫_{|x|≤R_{i}M_{i}^{− \frac{p_{i}−1}{2}}} u_{i}(x)^{p_{i}+1}dx ≥ c > 0,

we conclude that if we choose ρ_{1} sufficiently small, then

\frac{n − 2}{2} ∫_{|x|≤ \frac{ρ_{1}}{2}} Kf_{i}^{−τ_{i}}u_{i}(x)^{p_{i}+1}dx

− ∫_{|x|≤ \frac{ρ_{1}}{2}} Kf_{i}^{−τ_{i}−1}(x^{m}∂_{m}f_{i})u_{i}^{p_{i}+1}dx ≥ c > 0. (4.20)

The result follows immediately from the inequalities (4.19).

Claim 3: Given a small σ > 0, there exists a constant C > 0 such that

∫_{B_{σ}(x_{i})} u_{i}^{p_{i}}(x)dx ≤ CM_{i}^{−1}.

Proof of Claim 3: Set s_{i} = R_{i}M_{i}^{− \frac{p_{i}−1}{2}} .

First note that, changing variables, and then using v_{i}(y) ≤ cU_{0}(y) for |y| ≤ R_{i}, we get

(4.21) ∫_{|x|≤s_{i}} u_{i}^{p_{i}}(x)dx = M_{i}^{− \frac{(p_{i}−1)n}{2}} M_{i}^{p_{i}} ∫_{|y|≤R_{i}} v_{i}^{p_{i}}(y)dy ≤ CM_{i}^{−1}.

On the other hand, by Claim 1,

∫_{s_{i}≤|x|≤σ} u_{i}^{p_{i}}(x)dx ≤ CM_{i}^{−λ_{i}p_{i}} ∫_{s_{i}≤|x|≤σ} |x|^{(2−n+δ)p_{i}}dx
(4.22)  ≤ CM_{i}^{−λ_{i}p_{i}}s_{i}^{(2−n+δ)p_{i}+n} ≤ o(1)M_{i}^{−1}.

Claim 3 now follows from inequalities (4.21) and (4.22).

Claim 4: There exists σ_{1} > 0 such that for all 0 < σ < σ_{1}, there exists a constant C = C(σ) with, for every i,

u_{i}(x_{i})u_{i}(x) ≤ C(σ)

if d(x, x_{i}) = σ.

Proof of Claim 4: From the discussion in Section 3, if we choose σ_{1} > 0 small, we can suppose that R_{g_{i}} ≥ 0.

Choose 0 < σ < σ_{1} small and define

w_{i}(x) = u_{i}(x_{σ})^{−1}u_{i}(x),

where x_{σ} is chosen so that d(x_{σ}, x_{i}) = σ. Note that

L_{g_{i}}w_{i} + Ku_{i}(x_{σ})^{p_{i}−1}f_{i}^{−τ_{i}}w_{i}^{p_{i}} = 0.

The Harnack inequality implies that, for every ϵ > 0, there exists a constant C_{ϵ} > 0 such that

C_{ϵ}^{−1} ≤ w_{i}(x) ≤ C_{ϵ}

if d(x, \overline{x}) > ϵ. From Claim 1 we know that u_{i}(x_{σ})^{p_{i}−1} → 0 as i → ∞, and then standard elliptic theory shows that, after maybe passing to a subsequence,

w_{i} → w in C_{loc}^{2}(B_{σ}(\overline{x})),

and w satisfies

L_{g}w = 0, w > 0.

Since the blowup is isolated simple, the function û_{i}(r) is decreasing

in the interval (R_{i}M_{i}^{− \frac{p_{i}−1}{2}} , ρ). Taking the limit, we conclude that ŵ(r) is decreasing in the whole interval (0, ρ).

As a consequence, w is singular at the origin.

It follows from the results contained in the appendix in [7] that

(4.23) − ∫_{B_{η}(x_{i})} Δ_{g_{i}}w_{i} = − ∫_{∂B_{η}(x_{i})} \frac{∂w_{i}}{∂ν} = −∫_{∂B_{η}(\overline{x})} \frac{∂w}{∂ν} +o(1) > c > 0

for each i, where η > 0 is sufficiently small.

On the other hand,

− ∫_{B_{η}(x_{i})} Δ_{g_{i}}w_{i} = ∫_{B_{η}(x_{i})}[Ku_{i}(x_{σ})^{−1}f_{i}^{−τ_{i}}u_{i}^{p_{i}} − c(n)R_{g_{i}}w_{i}]
(4.24)  ≤ K ∫_{B_{η}(x_{i})} u_{i}(x_{σ})^{−1}f_{i}^{−τ_{i}}u_{i}^{p_{i}} ≤ cu_{i}(x_{σ})^{−1}M_{i}^{−1}.

Here we have used Claim 3. Claim 4 now follows from inequalities (4.23) and (4.24).

Now we are ready to prove Proposition 4.5.

Suppose the inequality (4.9) is not true. Then there exists a sequence x̃_{i}, with d(x̃_{i}, x_{i}) ≤ \frac{ρ_{1}}{2} , ρ_{1} small, such that

(4.25)  u_{i}(x̃_{i})u_{i}(x_{i})d(x̃_{i}, x_{i})^{n−2} → ∞

as i → ∞.

Then, from Proposition 4.3, R_{i}u_{i}(x_{i})^{− \frac{p_{i}−1}{2}} ≤ r̃_{i} ≤ \frac{ρ_{1}}{2} , where r̃_{i} = d(x̃_{i}, x_{i}). Define

ṽ_{i}(y) = r̃i^{\frac{2}{p_{i}−1}} u_{i}(r̃_{i}y), |y| < 2.

Now it is not difficult to see that the origin is an isolated simple blowup point for ṽ_{i}, and Claim 4 implies, together with the Harnack inequality,

max_{|y|=1} ṽ_{i}(0)ṽ_{i}(y) ≤ C.

This contradicts the limit in (4.25) and we finish the proof of inequality (4.9).

For the proof of inequality (4.10), recall that the Green function always exists when ρ_{1} is sufficiently small. Now observe that the inequal-

ity holds where d_{g_{i}}(x, x_{i}) = R_{i}M_{i}^{− \frac{p_{i}−1}{2}} because G_{i}(x_{i}, x) = O(r^{2−n}) and where d_{g_{i}}(x, x_{i}) = ρ_{1} because the Green function vanishes in this case and we are dealing with positive functions. Since

L_{g_{i}}(u_{i}(x_{i})u_{i}) ≤ 0 = L_{g_{i}}G_{i},

we can apply the maximum principle on the region \{x : R_{i}M_{i}^{− \frac{p_{i}−1}{2}} ≤ d_{g_{i}}(x, x_{i}) ≤ ρ_{1}} to get the desired inequality. □

Corollary 4.6. Under the hypotheses of Proposition 4.5, after maybe passing to a subsequence,

M_{i}u_{i}(x) → h in C_{loc}^{2}(B_{ρ_{1}}(\overline{x}) \ \{\overline{x}\}),

where h is a positive solution to the linear equation L_{g}(h) = 0, with a nonremovable singularity at \overline{x}. (Here g stands for the limit metric.)

Proof. Observe that the function M_{i}u_{i}(x) satisfies

L_{g_{i}}(M_{i}u_{i}(x)) + KM_{i}^{1−p_{i}}f_{i}^{−τ_{i}}(x)(M_{i}u_{i}(x))^{p_{i}} = 0.

The previous Proposition implies M_{i}u_{i}(x) is uniformly bounded in compact sets contained in B_{ρ_{1}}(\overline{x}) \ \{\overline{x}\}, and then standard elliptic estimates show that, after extracting a subsequence, M_{i}u_{i}(x) → h in C_{loc}^{2}(B_{ρ_{1}}(\overline{x}) \ \{\overline{x}\}). Since M_{i} → ∞, L_{g}h = 0.

Because of inequality (4.10), taking the limit, one sees that h is singular. This finishes the proof.

□

5. Symmetry estimates

In this section we will estimate the difference between solutions to our equation and standard symmetric functions, which will be solutions to the corresponding critical equation in the Euclidean setting.

The next lemma gives us an estimate on |v_{i} − U_{0}|, depending on M_{i} and τ_{i} = \frac{n+2}{n−2} − p_{i}. This is the first step towards the symmetry estimates, not depending on τ_{i}, we will prove later.

Lemma 5.1. Let u_{i} be a sequence of positive functions satisfying equation (4.1) and x_{i} → \overline{x} be an isolated simple blowup point. Then there exists δ > 0 such that

|v_{i}(y) − U_{0}(y)| ≤ C \{ \begin{matrix}max\{M_{i}^{−2}, τ_{i}\} & if & n = 4, 5, \\ max\{(log M_{i})M_{i}^{−2}, τ_{i}\} & if & n = 6, \\ max\{M_{i}^{−2}Mi^{\frac{2(n−6)}{n−2}} , τ_{i}\} & if & n ≥ 7, \end{matrix}

for |y| ≤ δMi^{\frac{p_{i}−1}{2}} , where τ_{i} = \frac{n+2}{n−2} − p_{i}.

Proof. Set l_{i} = δMi^{\frac{p_{i}−1}{2}} , and Λ_{i} = max_{|y|≤l_{i}} |v_{i} − U_{0}| = v_{i}(y_{i}) − U_{0}(y_{i}),

for a certain |y_{i}| ≤ l_{i}.

We observe that if there exists a constant c > 0 such that |y_{i}| ≥ cl_{i} for every i, then inequality v_{i} ≤ cU_{0} automatically implies the stronger inequality

|(v_{i} − U_{0})(y)| ≤ CM_{i}^{−2},

since

Λ_{i} = |v_{i} − U_{0}|(y_{i}) ≤ C|y_{i}|^{2−n} ≤ Cl_{i}^{2−n} ≤ CM_{i}^{−2}.

So for large i we will have |y_{i}| ≤ \frac{l_{i}}{2} . We are using that M_{i}^{τ_{i}} → 1 as i → ∞.

Define

w_{i}(y) = Λ_{i}^{−1}(v_{i}(y) − U_{0}(y)).

Then w_{i} satisfies

L_{h_{i}}w_{i} + b_{i}w_{i} = Q_{i}(y),

where

b_{i}(y) = Kf̃_{i}^{−τ_{i}} \frac{v_{i}^{p_{i}} − U_{0}^{p_{i}}}{v_{i} − U_{0}} (y),

and

Q_{i}(y) = Λ_{i}^{−1}\{c(n)M_{i}^{−(p_{i}−1)}R_{g_{i}}(M_{i}^{− \frac{p_{i}−1}{2}} y)U_{0}(y) (5.1)

+M_{i}^{−(1+N) \frac{p_{i}−1}{2}} O(|y|^{N})|y|(1 + |y|^{2})^{− \frac{n}{2}} + K(U0^{\frac{n+2}{n−2}} − f̃_{i}^{−τ_{i}}U_{0}^{p_{i}})\},

where f̃_{i}(y) = f_{i}(M_{i}^{− \frac{p_{i}−1}{2}} y), (h_{i})_{kl}(y) = (g_{i})_{kl}(M_{i}^{− \frac{p_{i}−1}{2}} y) and O(|y|^{N}) comes from the expansion of the volume element in conformal normal coordinates and N is as big as we want.

Since the blowup is isolated simple, from inequality v_{i} ≤ cU_{0}, it is easy to check, for example,

(5.2)  b_{i}(y) ≤ c(1 + |y|)^{−3}

for |y| ≤ l_{i}.

We will choose δ small enough to guarantee the existence of the Green’s function for the conformal Laplacian on a ball of radius δ , with respect to a Dirichlet boundary condition.

The Green’s representation formula gives (5.3)

w_{i}(y) = ∫_{B_{i}} G_{i,L}(y, η)(b_{i}(η)w_{i}(η) − Q_{i}(η))dη − ∫_{∂B_{i}} \frac{∂G_{i,L}}{∂ν} (y, η)w_{i}(η)ds

where B_{i} stands for B_{l_{i}}(0) and G_{i,L} is the Green function of L_{h_{i}} in B_{i}. We will need the following lemma proved in [2]:

Lemma 5.2. Suppose w is a solution to the equation

(5.4)  Δw + n(n + 2)U0^{\frac{4}{n−2}} w = 0 in ℝ^{n}.

If lim_{|y|→∞} w(y) = 0, then there exist constants c_{0}, c_{1}, …, c_{n} such that

w(y) = c_{0}(\frac{n − 2}{2}U_{0} + y · ∇U_{0}) + ∑_{j=1}^{n} c_{j} \frac{∂U_{0}}{∂y_{j}} .

Remark: The functions \frac{n−2}{2}U_{0} + y · ∇U_{0} and \frac{∂U_{0}}{∂y_{j}}, j = 1, …, n, are solutions to the equation (5.4).

The proof of the Lemma 5.1 is by contradiction. Set

(5.5)  t_{i} = \{ \begin{matrix}M_{i}^{−2} & if & n = 4, 5, \\ (log M_{i})M_{i}^{−2} & if & n = 6, \\ M_{i}^{−2}Mi^{\frac{2(n−6)}{n−2}} & if & n ≥ 7. \end{matrix}

If the proposition is false, we necessarily have

Λ_{i}^{−1} max{t_{i}, τ_{i}} → 0

as i → ∞, which implies that

Λ_{i}^{−1}t_{i} → 0, Λ_{i}^{−1}τ_{i} → 0.

Since R = O(r^{2}) in conformal normal coordinates, we can get the following estimate:

|Q_{i}(y)| ≤ cΛ_{i}^{−1}\{M_{i}^{− \frac{8}{n−2}} |y|^{2}(1 + |y|)^{2−n}
+ M_{i}^{−(1+N) \frac{p_{i}−1}{2}} O(|y|^{N})|y|(1 + |y|^{2})^{− \frac{n}{2}}
(5.6)  + τ_{i}(| log U_{0}| + | log f̃_{i}|)(1 + |y|)^{−n−2}}.

Using the estimates (5.2) and (5.6), we get from the Green’s representation formula (5.3) that w_{i} is bounded in C_{loc}^{2}, and

(5.7)  |w_{i}(y)| ≤ c[(1 + |y|)^{−1} + cΛ_{i}^{−1}t_{i}]

for |y| ≤ \frac{δ}{2}Mi^{\frac{p_{i}−1}{2}} . We are using that |w_{i}(y)| ≤ CΛ_{i}^{−1}M_{i}^{−2} when |y| =

\frac{δ}{2}Mi^{\frac{p_{i}−1}{2}} , and also that |G_{i,L}(y, η)| ≤ C|y − η|^{2−n} for |y| ≤ \frac{l_{i}}{2} .

Then, by standard elliptic estimates, there exists a subsequence, also denoted w_{i}, converging to w satisfying

\{ \begin{matrix}Δw + n(n + 2)U0^{\frac{4}{n−2}} (y)w = 0 in ℝ^{n}, \\ |w(y)| ≤ c(1 + |y|)^{−1}. \end{matrix}

So, the Lemma 5.2 implies that

w(y) = c_{0}(\frac{n − 2}{2}U_{0} + y · ∇U_{0}) + ∑_{j=1}^{n} c_{j} \frac{∂U_{0}}{∂y_{j}} .

The conditions w(0) = \frac{∂w}{∂y_{j}} (0) = 0 show that c_{j} = 0 for every j, in other words, w(y) ≡ 0. From here we conclude that |y_{i}| → ∞ as i → ∞.

This contradicts the estimate (5.7) since w_{i}(y_{i}) = 1 and Λ_{i}^{−1}t_{i} → 0, and this finishes the proof. □

In the next lemma, we estimate τ_{i}. This result and the Lemma 5.1 give us an estimate on |v_{i} − U_{0}| independent of τ_{i}.

Lemma 5.3. Under the same hypotheses in Lemma 5.1,

τ_{i} ≤ C \{ \begin{matrix}M_{i}^{−2} & if & n = 4, 5, \\ (log M_{i})M_{i}^{−2} & if & n = 6, \\ M_{i}^{−2}Mi^{\frac{2(n−6)}{n−2}} & if & n ≥ 7. \end{matrix}

Proof. The proof will be again by contradiction and recall the definition (5.5). If the lemma is not true, then Lemma 5.1 implies that

|v_{i}(y) − U_{0}(y)| ≤ Cτ_{i}.

Define

w_{i}(y) = τ_{i}^{−1}(v_{i} − U_{0})(y),

so w_{i} is uniformly bounded. The equation satisfied by w_{i} is

L_{h_{i}}w_{i} + b_{i}w_{i} = Q̃_{i}(y),

where

b_{i}(y) = Kf̃_{i}^{−τ_{i}} \frac{v_{i}^{p_{i}} − U_{0}^{p_{i}}}{v_{i} − U_{0}} (y)

and

Q̃_{i}(y) = τ_{i}^{−1}\{c(n)M_{i}^{−(p_{i}−1)}R_{g_{i}}(M_{i}^{− \frac{p−1}{2}} y)U_{0}(y)
+ M_{i}^{−(1+N) \frac{p_{i}−1}{2}} O(|y|^{N})|y|(1 + |y|^{2})^{− \frac{n}{2}}
(5.8)  + K(U0^{\frac{n+2}{n−2}} − f̃_{i}^{−τ_{i}}U_{0}^{p_{i}})\}.

If the lemma is not true, then τ_{i}^{−1}t_{i} → 0 as i → ∞.

We have

|Q̃_{i}(y)| ≤ cτ_{i}^{−1}\{M_{i}^{− \frac{8}{n−2}} |y|^{2}(1 + |y|)^{2−n}
+ M_{i}^{−(1+N) \frac{p_{i}−1}{2}} O(|y|^{N})|y|(1 + |y|^{2})^{− \frac{n}{2}}
(5.9)  + τ_{i}(| log U_{0}| + | log f̃_{i}|)(1 + |y|)^{−n−2}}.

By elliptic linear theory we can suppose w_{i} → w in compact subsets. If ψ(y) = \frac{n−2}{2}U_{0}(y) + y · ∇U_{0}(y), then,

∫_{|y|≤ \frac{l_{i}}{2}} ψ(y)τ_{i}^{−1}(M_{i}^{− \frac{8}{n−2}} |y|^{2}(1 + |y|)^{2−n}

+M_{i}^{−(1+N) \frac{p_{i}−1}{2}} O(|y|^{N})|y|(1 + |y|^{2})^{− \frac{n}{2}} ) → 0. (5.10)

Note that when i → ∞ we have:

τ_{i}^{−1}K(U0^{\frac{n+2}{n−2}} − f̃_{i}^{−τ_{i}}U_{0}^{p_{i}}) → K(log U_{0}(y) + log f(\overline{x}))U0^{\frac{n+2}{n−2}}

pointwise. It is not difficult to check that

∫_{ℝ^{n}} ψ(y)U0^{\frac{n+2}{n−2}} (y)dy = 0.

Therefore we can conclude

lim_{i→∞} ∫_{|y|≤ \frac{l_{i}}{2}} ψ(y)Q̃_{i}(y)dy = n(n − 2) ∫_{ℝ^{n}} ψ(y)(log U_{0}(y))U0^{\frac{n+2}{n−2}} (y)dy.

On the other hand, integration by parts shows that

∫_{|y|≤ \frac{l_{i}}{2}} ψ(y)Q̃_{i}(y)dy = ∫_{|y|≤ \frac{l_{i}}{2}} ψ(y)(L_{h_{i}}w_{i} + b_{i}w_{i})dy

= ∫_{|y|≤ \frac{l_{i}}{2}} (L_{h_{i}}ψ(y) + b_{i}ψ)w_{i}dy + ∫_{|y|= \frac{l_{i}}{2}} (ψ \frac{∂w_{i}}{∂r} − w_{i} \frac{∂ψ}{∂r})dσ. (5.11)

The integral on the boundary goes to zero when i → ∞ because

\{ \begin{matrix}|ψ| = O(r^{2−n}), |∇ψ| = O(r^{1−n}) \\ |w_{i}(\frac{l_{i}}{2} )| ≤ cτ_{i}^{−1}M_{i}^{−2}, |∇w_{i}(\frac{l_{i}}{2} )| ≤ cτ_{i}^{−1}M_{i}^{−2}l_{i}^{−1}. \end{matrix}

Taking the limit when i → ∞, we would have

lim_{i→∞} ∫_{|y|≤ \frac{l_{i}}{2}} ψ(y)Q̃_{i}(y)dy = ∫_{ℝ^{n}}(Δψ(y) + n(n + 2)U0^{\frac{4}{n−2}} ψ)wdy = 0

because Δψ(y) + n(n + 2)U0^{\frac{4}{n−2}} ψ = 0.

This is a contradiction because

(5.12)  n(n − 2) ∫_{ℝ^{n}} ψ(y)(log U_{0}(y))U0^{\frac{n+2}{n−2}} (y)dy > 0.

To see this, first note that

ψ(y) = \frac{n − 2}{2} \frac{1 − r^{2}}{(1 + r^{2})^{\frac{n}{2}}} .

Observe that

∫_{ℝ^{n}} ψ(y)(log U_{0}(y))U0^{\frac{n+2}{n−2}} (y)dy =

−\frac{(n − 2)^{2}}{4} σ_{n−1} ∫_{0}^{∞} \frac{1 − r^{2}}{(1 + r^{2})^{n+1}} r^{n−1} log(1 + r^{2})dr, (5.13)

and after changing variables r = s^{−1}, we get

∫_{0}^{∞} \frac{1 − r^{2}}{(1 + r^{2})^{n+1}} r^{n−1} log(1 + r^{2})dr = 2 ∫_{1}^{∞} \frac{1 − r^{2}}{(1 + r^{2})^{n+1}} r^{n−1} log rdr.

Now inequality (5.12) follows immediately, and that finishes the proof of the lemma. □

The Lemmas 5.1 and 5.3 together imply our symmetry estimate:

Proposition 5.4. Let u_{i} be a sequence of positive functions satisfying equation (4.1) and x_{i} → \overline{x} be an isolated simple blowup point. Then there exists δ > 0 such that

(5.14)  |v_{i}(y) − U_{0}(y)| ≤ C \{ \begin{matrix}M_{i}^{−2} & if & n = 4 or 5, \\ (log M_{i})M_{i}^{−2} & if & n = 6, \\ M_{i}^{−2}Mi^{\frac{2(n−6)}{n−2}} & if & n ≥ 7 \end{matrix}

for |y| ≤ δMi^{\frac{p_{i}−1}{2}} .

When n ≥ 6, by applying the same technique we can also get:

Proposition 5.5. Let u_{i} be a sequence of positive functions satisfying equation (4.1) and x_{i} → \overline{x} be an isolated simple blowup point. Then

(5.15) |v_{i}(y) − U_{0}(y)| ≤ C \{ \begin{matrix}M_{i}^{−2}Mi^{\frac{2}{n−2}} (1 + |y|)^{−1} & if n = 6 \\ M_{i}^{−2}Mi^{\frac{2(n−6)}{n−2}} (1 + |y|)^{6−n} & if n ≥ 7. \end{matrix}

Proof. Set

(5.16)  A_{i} = \{ \begin{matrix}M_{i}^{−2}Mi^{\frac{2}{n−2}} & if n = 6 \\ M_{i}^{−2}Mi^{\frac{2(n−6)}{n−2}} & if n ≥ 7, \end{matrix}

and define

w_{i}(y) = A_{i}^{−1}(v_{i} − U_{0})(y)

for |y| ≤ δMi^{\frac{p_{i}−1}{2}} . Then our previous proposition implies w_{i} is uniformly bounded. The equation satisfied is

L_{h_{i}}w_{i} + b_{i}w_{i} = Q̃_{i}(y),

where

b_{i}(y) = Kf̃_{i}^{−τ_{i}} \frac{v_{i}^{p_{i}} − U_{0}^{p_{i}}}{v_{i} − U_{0}} (y)

and

Q̃_{i}(y) = A_{i}^{−1}\{c(n)M_{i}^{−(p−1)}R_{g}(M_{i}^{− \frac{p−1}{2}} y)U_{0}(y)

+M_{i}^{−(1+N) \frac{p_{i}−1}{2}} O(|y|^{N})|y|(1 + |y|^{2})^{− \frac{n}{2}} + K(U0^{\frac{n+2}{n−2}} − f̃_{i}^{−τ_{i}}U_{0}^{p_{i}})\}. (5.17)

Then

|Q̃_{i}(y)| ≤ cA_{i}^{−1}\{M_{i}^{− \frac{8}{n−2}} |y|^{2}(1 + |y|)^{2−n}
+ M_{i}^{−(1+N) \frac{p_{i}−1}{2}} O(|y|^{N})|y|(1 + |y|^{2})^{− \frac{n}{2}}
(5.18)  + τ_{i}(| log U_{0}| + | log f̃_{i}|)(1 + |y|)^{−n−2}}.

The Green’s representation formula says that (5.19)

w_{i}(y) = ∫_{B_{i}} G_{i,L}(y, η)(b_{i}(η)w_{i}(η) − Q̃_{i}(η))dη − ∫_{∂B_{i}} \frac{∂G_{i,L}}{∂ν} (y, η)w_{i}(η)ds,

where B_{i} stands for B_{l_{i}}(0) and G_{i,L} is the Green function of L_{h_{i}} in B_{i}. Since |G_{i,L}(y, η)| ≤ C|y − η|^{2−n}, for |y| ≤ \frac{l_{i}}{2} , we get

(5.20)  |w_{i}(y)| ≤ c{(1 + |y|)^{−1} + cM_{i}^{−2}A_{i}^{−1}} ≤ c(1 + |y|)^{−1}.

If n = 6 or 7 the result follows multiplying the inequality (5.20) by A_{i}. If n ≥ 8 we plug the estimate (5.20) in the representation formula (5.19) until we reach

|w_{i}(y)| ≤ c(1 + |y|)^{6−n}.

Multiplying by A_{i} we get the result. □

Remark 1: Once we have estimates (5.14), (5.15) on v_{i} − U_{0}, we can also get: (5.21)

|∇(v_{i} − U_{0})(y)| ≤ C \{ \begin{matrix}M_{i}^{−2}(1 + |y|)^{−1} & if & n = 4 or 5, \\ M_{i}^{−2}Mi^{\frac{2}{n−2}} (1 + |y|)^{−2} & if & n = 6 \\ M_{i}^{−2}Mi^{\frac{2(n−6)}{n−2}} (1 + |y|)^{5−n} & if & n ≥ 7 \end{matrix}

and (5.22)

|∇^{2}(v_{i} − U_{0})(y)| ≤ C \{ \begin{matrix}M_{i}^{−2}(1 + |y|)^{−2} & if & n = 4 or 5, \\ M_{i}^{−2}Mi^{\frac{2}{n−2}} (1 + |y|)^{−3} & if & n = 6 \\ M_{i}^{−2}Mi^{\frac{2(n−6)}{n−2}} (1 + |y|)^{4−n} & if & n ≥ 7. \end{matrix}

Remark 2: If h is as in the Corollary 4.6, the estimates (5.14) and (5.15) imply that:

(5.23)  |h(x) − |x|^{2−n}| ≤ C \{ \begin{matrix}1 & if & n = 4, 5, \\ |x|^{−1} & if & n = 6, \\ |x|^{6−n} & if & n ≥ 7. \end{matrix}

Since that gives the asymptotic behavior of the Green function of the conformal Laplacian in conformal normal coordinates (see [5]), in some sense our symmetry estimates cannot be improved.

6. Local blowup analysis

Now let us turn our attention to the applications of these symmetry estimates.

In the first application, we will show that the Weyl tensor of the metric has to vanish at an isolated simple blowup point, when n ≥ 6. This result had been proposed by R. Schoen ([12]). This will allow us to use the Positive Mass Theorem in the proof of compactness of solutions to the Yamabe problem, when the dimension is 6 or 7.

Theorem 6.1. Let u_{i} be a sequence of positive functions satisfying equation (4.1) and x_{i} → \overline{x} be an isolated simple blowup point. If n ≥ 6, then

W_{g}(\overline{x}) = 0.

Proof. We will use the Pohozaev identity (2.8) to u_{i} in a ball of radius r:

P (r, u_{i}) = (6.1)

− ∫_{|x|≤r}(x^{m}∂_{m}u_{i} + \frac{n − 2}{2}u_{i})((g^{kl} − δ^{kl})∂_{kl}u_{i} + ∂_{k}g^{kl}∂_{l}u_{i})dx

−c(n) ∫_{|x|≤r}(\frac{1}{2}x^{k}∂_{k}R + R)u_{i}^{2}dx + c(n)\frac{r}{2} ∫_{|x|=r} Ru_{i}^{2}dσ_{r}

+(\frac{n}{p_{i} + 1} − \frac{n − 2}{2}) ∫_{|x|≤r} Kf_{i}^{−τ_{i}}u_{i}^{p_{i}+1}dx

−\frac{τ_{i}}{p_{i} + 1} ∫_{|x|≤r} Kf_{i}^{−τ_{i}−1}(x^{m}∂_{m}f_{i})u_{i}^{p_{i}+1}dx.

Using conformal normal coordinates we can get rid of the terms involving |g|.

Recall

P (r, u_{i}) = (6.2)

∫_{|x|=r}(\frac{n − 2}{2}u_{i} \frac{∂u_{i}}{∂r} − \frac{r}{2}|∇u_{i}|^{2} + r|\frac{∂u_{i}}{∂r} |^{2} + \frac{1}{p_{i} + 1}Kf_{i}^{−τ_{i}}ru_{i}^{p_{i}+1})dσ_{r}.

Since we have that M_{i}u_{i} → h in the C^{2} topology on compact subsets of B_{ρ}(\overline{x}) \ \{\overline{x}\}), we conclude

(6.3)  M_{i}^{2}|P (r, u_{i})| ≤ c < ∞.

The same holds for

(6.4)  M_{i}^{2}| ∫_{|x|=r} Ru_{i}^{2}dσ(r)| ≤ c < ∞.

The change of variables y = Mi^{\frac{p_{i}−1}{2}} x yields

A_{i}(r) :=

M_{i}^{2}\{− ∫_{|x|≤r}(x^{m}∂_{m}u_{i} + \frac{n − 2}{2}u_{i})((g^{kl} − δ^{kl})∂_{kl}u_{i} + ∂_{k}g^{kl}∂_{l}u_{i})dx

−c(n) ∫_{|x|≤r}(\frac{1}{2}x^{k}∂_{k}R + R)u_{i}^{2}dx\}

= −M_{i}^{2}M_{i}^{2+(2−n) \frac{p_{i}−1}{2}} ∫_{|y|≤rM \frac{p_{i}−1}{2} ; i} \{(y^{m}∂_{m}v_{i} + \frac{n − 2}{2}v_{i})

((g^{kl} − δ^{kl})(M_{i}^{− \frac{p_{i}−1}{2}} y)∂_{kl}v_{i} + M_{i}^{− \frac{p_{i}−1}{2}} ∂_{k}g^{kl}(M_{i}^{− \frac{p_{i}−1}{2}} y)∂_{l}v_{i})

+c(n)M_{i}^{−(p_{i}−1)}(\frac{1}{2}y^{k}∂_{k}R(M_{i}^{− \frac{p_{i}−1}{2}} y) + R(M_{i}^{− \frac{p_{i}−1}{2}} y))v_{i}^{2}\}dy.

Note that M_{i}^{2+(2−n) \frac{p_{i}−1}{2}} → 1. Define

Â_{i}(r) = (6.5)

−M_{i}^{2}M_{i}^{2+(2−n) \frac{p_{i}−1}{2}} ∫_{|y|≤rM \frac{p_{i}−1}{2} ; i} \{(y^{m}∂_{m}U_{0} + \frac{n − 2}{2}U_{0})

((g^{kl} − δ^{kl})(M_{i}^{− \frac{p_{i}−1}{2}} y)∂_{kl}U_{0} + M_{i}^{− \frac{p_{i}−1}{2}} ∂_{k}g^{kl}(M_{i}^{− \frac{p_{i}−1}{2}} y)∂_{l}U_{0})

+c(n)M_{i}^{−(p_{i}−1)}(\frac{1}{2}y^{k}∂_{k}R(M_{i}^{− \frac{p_{i}−1}{2}} y) + R(M_{i}^{− \frac{p_{i}−1}{2}} y))U_{0}^{2}\}dy.

26 FERNANDO C. MARQUES

Then one can check |A_{i}(r) − Â_{i}(r)| (6.6)

≤ cM_{i}^{2}M_{i}^{− \frac{4}{n−2}} ∫_{|y|≤rM \frac{p_{i}−1}{2} ; i} \{|v_{i} − U_{0}|(y)(1 + |y|)^{2−n}

+|∇(v_{i} − U_{0})|(1 + |y|)^{3−n} + |∇^{2}(v_{i} − U_{0})|(1 + |y|)^{4−n}}dy. If n = 6, then

|A_{i}(r) − Â_{i}(r)| ≤ cM_{i}^{− \frac{2}{n−2}} ∫_{|y|≤rM \frac{p_{i}−1}{2} ; i} (1 + |y|)^{1−n}dy.

When n ≥ 7,

|A_{i}(r) − Â_{i}(r)| ≤ cM_{i}^{− \frac{4}{n−2}} M_{i}^{(n−6) \frac{2}{n−2}} ∫_{|y|≤rM \frac{p_{i}−1}{2} ; i} (1 + |y|)^{8−2n}dy.

Then

(6.7)  |A_{i}(r) − Â_{i}(r)| ≤ \{ \begin{matrix}C & if & n = 6, 7, \\ C(log M_{i}) & if & n = 8, \\ CM_{i}^{(n−8) \frac{2}{n−2}} & if & n ≥ 9. \end{matrix}

Also observe that if we choose r sufficiently small,

(\frac{n}{p_{i} + 1} − \frac{n − 2}{2}) ∫_{|x|≤r} Ku_{i}^{p_{i}+1}dx (6.8)

−\frac{τ_{i}}{p_{i} + 1} ∫_{|x|≤r} Kf_{i}^{−τ_{i}−1}(x^{m}∂_{m}f_{i})u_{i}^{p_{i}+1}dx ≥ 0,

so we obtain, from identity (6.1), estimates (6.3), (6.4), (6.7) and inequality (6.8) that

Â_{i}(r) ≤ \{ \begin{matrix}C & if & n = 6, 7, \\ C(log M_{i}) & if & n = 8, \\ CM_{i}^{(n−8) \frac{2}{n−2}} & if & n ≥ 9. \end{matrix}

We will need the Taylor series, for each i,

R(x) = p_{2}(x) + p_{3}(x) + e(x)

(R + \frac{1}{2}x^{k}∂_{k}R)(x) = 2p_{2}(x) + \frac{5}{2}p_{3}(x) + e′(x)

where p_{i} is a homogeneous polynomial of degree i and |e(x)|, |e′(x)| ≤ c|x|^{4}.

Let us denote by

ũ_{i}(x) = M_{i}^{\frac{n−2}{4} τ_{i}} (\frac{M_{i}^{− \frac{p_{i}−1}{2}}}{M_{i}^{−(p_{i}−1)} + |x|^{2}} )^{\frac{n−2}{2}}

and it is not difficult to see that, changing variables, estimates on |v_{i} − U_{0}| yield estimates on |u_{i} − ũ_{i}|.

Note that

M_{i}^{2} ∫_{|x|≤r}(\frac{1}{2}x^{k}∂_{k}e + e)ũ_{i}^{2}dx ≤ \{ \begin{matrix}C & if & n = 6, 7, \\ C(log M_{i}) & if & n = 8, \\ CM_{i}^{(n−8) \frac{2}{n−2}} & if & n ≥ 9. \end{matrix}

Then, using symmetry of ũ_{i}

−c(n) ∫_{|x|≤r} 2p_{2}(x)(M_{i}ũ_{i})^{2}dx ≤ c \{ \begin{matrix}C & if & n = 6, 7, \\ C(log M_{i}) & if & n = 8, \\ CM_{i}^{(n−8) \frac{2}{n−2}} & if & n ≥ 9. \end{matrix}

But, on the other hand, since ΔR(0) = −\frac{1}{6}|W (0)|^{2} in conformal normal coordinates, we obtain

−c(n)M_{i}^{2} ∫_{|x|≤r} 2p_{2}(x)ũ_{i}^{2}dx ≥

\{ \begin{matrix}C|W (x_{i})|^{2}M_{i}^{2}M_{i}^{−4 \frac{2}{n−2}} (log M_{i}) & if & n = 6, \\ C|W (x_{i})|^{2}M_{i}^{2}M_{i}^{−4 \frac{2}{n−2}} & if & n ≥ 7. \end{matrix}

So

|W_{g_{i}}(x_{i})|^{2} ≤ \{ \begin{matrix}c(log M_{i})^{−1} & if & n = 6, \\ cM_{i}^{− \frac{2}{n−2}} & if & n = 7, \\ cM_{i}^{− \frac{4}{n−2}} (log M_{i}) & if & n = 8, \\ cM_{i}^{− \frac{4}{n−2}} & if & n ≥ 9. \end{matrix}

And taking the limit we are done. □

The next result concerns the local asymptotic analysis at a blowup point. It will be used together with the Positive Mass Theorem to exclude the possibility of blowup phenomenon on manifolds not conformally diffeomorphic to the sphere.

Define

(6.9)  P′(r, v) = ∫_{|x|=r}(\frac{n − 2}{2}v \frac{∂v}{∂ν} − \frac{r}{2}|∇v|^{2} + r|\frac{∂v}{∂ν} |^{2})dσ(r).

Theorem 6.2. Let u_{i} be a sequence of positive functions satisfying equation (4.1) and x_{i} → \overline{x} be an isolated simple blowup point. and suppose 4 ≤ n ≤ 7. If u_{i}(x_{i})u_{i} → h away from the origin, then

lim inf_{r→0} P′(r, h) ≥ 0.

Proof. That is another application of the Pohozaev identity (2.7)and the symmetry estimates:

P (r, u_{i}) = − ∫_{|x|≤r}(x^{m}∂_{m}u_{i} + \frac{n − 2}{2}u_{i}) (6.10)

((g^{kl} − δ^{kl})∂_{kl}u_{i} + ∂_{k}g^{kl}∂_{l}u_{i} − c(n)Ru_{i})dx

+(\frac{n}{p_{i} + 1} − \frac{n − 2}{2}) ∫_{|x|≤r} Kf_{i}^{−τ_{i}}u_{i}^{p_{i}+1}dx

−\frac{τ_{i}}{p_{i} + 1} ∫_{|x|≤r} Kf_{i}^{−τ_{i}−1}(x^{m}∂_{m}f_{i})u_{i}^{p_{i}+1}dx.

Firstly observe that

M_{i}^{2}P (r, u_{i}) → P′(r, h)

as i → ∞.

Secondly, as in the previous result, if r is sufficiently small,

(\frac{n}{p_{i} + 1} − \frac{n − 2}{2}) ∫_{|x|≤r} Kf_{i}^{−τ_{i}}u_{i}^{p_{i}+1}dx

−\frac{τ_{i}}{p_{i} + 1} ∫_{|x|≤r} Kf_{i}^{−τ_{i}−1}(x^{m}∂_{m}f_{i})u_{i}^{p_{i}+1}dx ≥ 0. (6.11)

If

A_{i}(r) := M_{i}^{2}\{− ∫_{|x|≤r}(x^{m}∂_{m}u_{i} + \frac{n − 2}{2}u_{i})((g^{kl} − δ^{kl})∂_{kl}u_{i}

+∂_{k}g^{kl}∂_{l}u_{i} − c(n)Ru_{i})dx} (6.12)

and

Â_{i}(r) := M_{i}^{2}\{− ∫_{|x|≤r}(x^{m}∂_{m}ũ_{i} + \frac{n − 2}{2}ũ_{i})((g^{kl} − δ^{kl})∂_{kl}ũ_{i}

+∂_{k}g^{kl}∂_{l}ũ_{i} − c(n)Rũ_{i})dx}, (6.13)

then one can check

|A_{i}(r) − Â_{i}(r)| ≤ \{ \begin{matrix}cr^{2} & if & n = 4, 5, \\ cr & if & n = 6, 7. \end{matrix}

So

lim inf_{r→0} P′(r, h) ≥ (6.14)

c(n)M_{i}^{2} lim inf_{r→0} ∫_{|x|≤r}(x^{m}∂_{m}ũ_{i} + \frac{n − 2}{2}ũ_{i})Rũ_{i}dx.

If n = 4 or 5, since R = O(r^{2}), this automatically gives

lim inf_{r→0} P′(r, h) ≥ 0.

If n = 6 or 7, the fact that R = O(r^{2}) tells us that we only need to consider the second order term in the expansion of R. We are using the symmetry of ũ_{i} to get rid of the third order term.

The change of variables y = Mi^{\frac{p_{i}−1}{2}} x implies that

M_{i}^{2} ∫_{|x|≤r}(x^{m}∂_{m}ũ_{i} + \frac{n − 2}{2}ũ_{i})R_{,ab}x^{a}x^{b}ũ_{i}dx (6.15)

= (1 + o(1))Mi^{\frac{2(n−6)}{n−2}} ∫_{|y|≤rM \frac{p_{i}−1}{2} ; i} (y^{m}∂_{m}U_{0} + \frac{n − 2}{2}U_{0})R_{,ab}y^{a}y^{b}U_{0}dy.

But

∫_{|y|≤rM \frac{p_{i}−1}{2} ; i} (y^{m}∂_{m}U_{0} + \frac{n − 2}{2}U_{0})R_{,ab}y^{a}y^{b}U_{0}dy

= ∫_{0}^{rM \frac{p_{i}−1}{2} ; i} (r∂_{r}U_{0} + \frac{n − 2}{2}U_{0})U_{0}r^{n+1} \frac{ΔR}{n}σ_{n−1}dr

= −\frac{1}{6n}σ_{n−1}|W (0)|^{2} ∫_{0}^{rM \frac{p_{i}−1}{2} ; i} (r∂_{r}U_{0} + \frac{n − 2}{2}U_{0})U_{0}r^{n+1}dr

= −\frac{1}{6n}σ_{n−1}|W (0)|^{2}(−\frac{n − 2}{2} ∫_{0}^{rM \frac{p_{i}−1}{2} ; i} \frac{r^{n+3}}{(1 + r^{2})^{n−1}} dr

+\frac{n − 2}{2} ∫_{0}^{rM \frac{p_{i}−1}{2} ; i} \frac{r^{n+1}}{(1 + r^{2})^{n−1}} dr). (6.16)

When n = 6 and i is large, this last expression is nonnegative because the first integral diverges while the second one is finite. If n = 7 and i is large, again the expression is nonnegative since

(6.17)  ∫_{0}^{∞} \frac{r^{n+3}}{(1 + r^{2})^{n−1}} dr = \frac{n + 2}{n − 6} ∫_{0}^{∞} \frac{r^{n+1}}{(1 + r^{2})^{n−1}} dr.

To see this, first observe that, if m + 1 < 2k, integration by parts gives

∫_{0}^{∞} \frac{t^{m}}{(1 + t^{2})^{k}} dt = \frac{m − 1}{2(k − 1)} ∫_{0}^{∞} \frac{t^{m−2}}{(1 + t^{2})^{k−1}} dt.

Since

∫_{0}^{∞} \frac{t^{m−2}}{(1 + t^{2})^{k−1}} dt = ∫_{0}^{∞} \frac{t^{m−2}}{(1 + t^{2})^{k}} dt + ∫_{0}^{∞} \frac{t^{m}}{(1 + t^{2})^{k}} dt,

one gets

∫_{0}^{∞} \frac{t^{m}}{(1 + t^{2})^{k}} dt = \frac{m − 1}{2k − m − 1} ∫_{0}^{∞} \frac{t^{m−2}}{(1 + t^{2})^{k}} dt.

Just choose m = n + 4 and k = n + 1 to obtain (6.17).

The proof is finished by using the inequality (6.14). □

7. Compactness theorem

In this section we will prove the a priori estimates for the Yamabe problem in the non-locally conformally flat case, for 4 ≤ n ≤ 7.

The next Proposition is fundamental since it allows us to use the symmetry estimates we proved before.

Proposition 7.1. Suppose the blowup x_{i} → \overline{x} is isolated, 4 ≤ n ≤ 7. Then it is also isolated simple.

The proof of this statement is just as in [7], based on Proposition 6.2.

Once we have established Proposition 7.1, again following [7], we have:

Theorem 7.2. Suppose u_{i} ∈ 𝕄_{p_{i}} is a sequence satisfying max_{M} u_{i} → ∞, as i → ∞. Then p_{i} → \frac{n+2}{n−2}. Moreover, after passing to a subsequence,

(1) the set S = {blowup points of u_{i}} is finite;

(2) every blowup point of u_{i} is an isolated simple blowup point.

Now let us turn to the statement and proof of the compactness theorem.

Theorem 7.3. Let (M^{n}, g) be a smooth closed Riemannian manifold with positive Yamabe quotient, not conformally equivalent to (𝕊^{n}, g_{0}). Assume 4 ≤ n ≤ 7. Then, for every ϵ > 0, there exists a positive constant C = C(ϵ, g) so that

\{ \begin{matrix}1/C ≤ u ≤ C and \\ ||u||_{C^{2,α}(M)} ≤ C \end{matrix}

for every u ∈ ∪_{1+ϵ≤p≤ \frac{n+2}{n−2}} 𝕄_{p}.

Proof. Standard elliptic estimates and the Harnack inequality imply that it suffices to estimate ||u||_{C^{0}(M)}. Suppose, by contradiction, that ∪_{1+ϵ≤p≤ \frac{n+2}{n−2}} 𝕄_{p} is not bounded in C^{0}(M). This means that there exist 1 + ϵ ≤ p_{i} ≤ \frac{n+2}{n−2} and u_{i} ∈ 𝕄_{p_{i}} with

max_{M} u_{i} → ∞ as i → ∞.

From Theorem 7.2 we know that this is only possible if p_{i} → \frac{n+2}{n−2}.

Now Theorem 7.2 implies that, after possibly passing to a subsequence, u_{i} has N isolated simple blow-up points x_{i}^{(1)} → x^{(1)}, …, x_{i}^{(N)} → x^{(N)}, for some integer N.

Define w_{i}(x) = u_{i}(x_{i}^{(1)})u_{i}(x). We can suppose, for example, that

u_{i}(x_{i}^{(1)}) = min{u_{i}(x_{i}^{(1)}), …, u_{i}(x_{i}^{(N)})} for all i.

Proposition 4.5 then implies that there exists ρ, c > 0 such that

(7.1)  w_{i}(x) ≤ cd(x, x_{i}^{(j)})^{2−n} when d(x, x_{i}^{(j)}) ≤ ρ, 1 ≤ j ≤ N.

On the other hand, we know that the sequence u_{i} is uniformly bounded in M \ ∪_{j=1}^{j=N}B_{\frac{ρ}{4}} (x^{(j)}), since there is no blowup point in that region. Then the Harnack inequality implies w_{i} is uniformly bounded in M \ ∪_{j=1}^{j=N}B_{\frac{ρ}{2}} (x^{(j)}). This and inequality (7.1) imply, after passing to a subsequence,

u_{i}(x_{i}^{(1)})u_{i}(x) → h(x) = ∑_{j=1}^{N} a_{j}G_{x^{(j)}}(x) + b(x)

in C_{loc}^{2}(M \ {x^{(1)}, …, x^{(N)}}), where a_{1}, …, a_{N} are nonnegative constants, G_{x^{(j)}} is the Green function of the conformal Laplacian with pole at x^{(j)} and b(x) is a regular C^{2} function satisfying L_{g}(b) = 0 in M. Note that, since the first eigenvalue of the conformal Laplacian L_{g} is positive, b ≡ 0.

Since the Green functions considered are positive and a_{1} > 0 because of the Corollary 4.6, we obtain the following expansion ([5]) :

h(x) = ad(x, x^{(1)})^{2−n} + A + O(r), for 4 ≤ n ≤ 6

or

h(x) = ad(x, x^{(1)})^{2−n} − cR_{;ij}x_{i}x_{j}r^{−3} + A + O(r), for n = 7.

The Positive Mass Theorem asserts that A > 0.

We want to compute the limit

lim inf_{r→0} P′(r, h).

Since r^{2−n} is harmonic, definition (6.9) gives

P′(r, h) = ∫_{|x|=r}\{−\frac{(n−2)^{2}}{2} aAr^{1−n} + O(r^{2−n})\}dσ_{r}. (7.2)

When n = 7, we use symmetry and the fact that ΔR(0) = −\frac{1}{6}|W (0)|^{2} = 0.

Then

lim inf_{r→0} P′(r, h) = −\frac{(n − 2)^{2}}{2} aAσ_{n−1} < 0,

by the Positive Mass Theorem.

This contradicts Theorem 6.2 and it finishes the proof.

□

Fernando C. Marques, Instituto de Matemática Pura e Aplicada (IMPA), Estrada Dona Castorina 110, 22460-320, Rio de Janeiro - RJ, Brazil

Email address : coda@impa.br

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