F R Éd

Éric Robert

J Ér Ôme

V Étois

Frédéric Robert
email: frederic.robert@univ-lorraine.fr

Jérôme Vétois
email: vetois@unice.fr

SIGN-CHANGING BLOW-UP FOR SCALAR CURVATURE TYPE EQUATIONS

Given (M, g) a compact Riemannian manifold of dimension n ≥ 3, we are interested in the existence of blowing-up sign-changing families (uε) ε>0 ∈ C 2,θ (M ), θ ∈ (0, 1), of solutions to

.

Introduction

Let (M, g) be a smooth compact Riemannian manifold of dimension n ≥ 3. Given θ ∈ (0, 1), we consider solutions u ∈ C 2,θ (M ) to the equation [START_REF] Ambrosetti | Perturbation methods and semilinear elliptic problems on R n[END_REF] ∆

g u + hu = |u| 2 ⋆ -2 u in M ,
where h ∈ C 0,θ (M ), ∆ g := -div g (∇) is the Laplace-Beltrami operator, and 2 ⋆ := 2n n-2 . When h ≡ n-2 4(n-1) Scal g (Scal g being the scalar curvature of (M, g)), [START_REF] Ambrosetti | Perturbation methods and semilinear elliptic problems on R n[END_REF] is the Yamabe equation and rewrites [START_REF] Th | Problèmes isopérimétriques et espaces de Sobolev[END_REF] ∆ g u + c n Scal g u = |u| 2 * -2 u in M , where c n := n-2 4(n-1) . The conformal invariance of the Yamabe equation induces a dynamic that makes equations ( 1) and (2) unstable. Taking inspiration from the terminology introduced by R. Schoen [START_REF] Schoen | Variational theory for the total scalar curvature functional for Riemannian metrics and related topics[END_REF], we say that equation [START_REF] Ambrosetti | Perturbation methods and semilinear elliptic problems on R n[END_REF] is compact (resp. positively compact) if for any family (q ε ) ε ∈ (2, 2 ⋆ ] such that q ε → 2 ⋆ when ε → 0 and for any family of functions (resp. positive functions) (u ε ) ε ∈ C 2,θ (M ) of solutions to

(3) ∆ g u ε + hu ε = |u ε | qε-2 u ε in M
for ε > 0 small, then a uniform bound on the Dirichlet energy ( ∇u ε 2 ) ε implies the relative compactness of (u ε ) ε in C 2 (M ), and therefore the convergence of a subfamily of (u ε ) ε in C 2 (M ). Otherwise, we say that equation (1) is noncompact (resp. non positively-compact). A basic example of non compact equation is [START_REF] Th | Problèmes isopérimétriques et espaces de Sobolev[END_REF] on the canonical sphere (S n , can): we refer to the second part of this section for (positive) compactness results for equations like [START_REF] Ambrosetti | Perturbation methods and semilinear elliptic problems on R n[END_REF].

Date: October 4th, 2012.

The authors are partially supported by the ANR grant ANR-08-BLAN-0335-01.

We say that a family (u ε ) ε>0 ∈ C 2,θ (M ) blows-up when ε → 0 if lim ε→0 u ε ∞ = +∞. It is now well-known (see Struwe [START_REF] Struwe | A global compactness result for elliptic boundary value problems involving limiting nonlinearities[END_REF] for a description in Sobolev spaces and Druet-Hebey-Robert [START_REF] Druet | Blow-up theory for elliptic PDEs in Riemannian geometry[END_REF] for a description in C 0 ) that noncompactness is described by bubbles. In the present paper, we investigate the existence of families (u ε ) ε ∈ C 2,θ (M ) of sign-changing blowing-up solutions to the equation ( 4)

∆ g u ε + hu ε = |u ε | 2 ⋆ -2-ε u ε in M , ε > 0.
In the sequel, we say that a blowing-up family (u ε ) ε ∈ C 2,θ (M ) is of type (u 0 -B) if there exists u 0 ∈ C 2,θ (M ) and a bubble (B ε ) ε (see definition [START_REF] Brendle | Blow-up phenomena for the Yamabe equation. II[END_REF] below) such that ( 5)

u ε = u 0 -B ε + o(1),
where lim ε→0 o(1) = 0 in H 2 1 (M ), the completion of C ∞ (M ) for the norm u → u 2 + ∇u 2 . Our first result is the following: Theorem 1.1 (dimensions 3 ≤ n ≤ 6 and arbitrary potential). Let (M, g) be a smooth compact Riemannian manifold of dimension 3 ≤ n ≤ 6 and let h ∈ C 0,θ (M ) (θ ∈ (0, 1)) be such that ∆ g +h is coercive. Assume that there exists a nondegenerate solution u 0 ∈ C 2,θ (M ) to equation [START_REF] Ambrosetti | Perturbation methods and semilinear elliptic problems on R n[END_REF]. In case n = 6, assume in addition that c n Scal g -h < 2u 0 in M . Then for ε > 0 small, equation (4) admits a sign-changing solution u ε of type (u 0 -B). In particular, the family (u ε ) ε>0 blows up as ε → 0 and (1) is noncompact.

In full generality, it is not possible to construct positive blowing-up solutions to equation [START_REF] Bianchi | A note on the Sobolev inequality[END_REF]. Indeed, in addition to the assumptions of Theorem 1.1, if we assume that h < c n Scal g , then (1) is positively compact (Druet [START_REF] Druet | From one bubble to several bubbles: the low-dimensional case[END_REF] and the discussion below), and therefore any family of blowing-up solutions to (4) must be sign-changing. In the early reference [START_REF] Ding | On a conformally invariant elliptic equation on R n[END_REF], Ding proved the existence of infinitely many nonequivalent solutions to (2) on the canonical sphere, highlighting the diversity of the behavior of solutions to (1) depending on whether they are positive or negative.

The nondegeneracy assumption in Theorem 1.1 is necessary. We refer to Proposition 3.1 in Section 3 for the proof of necessity. However, the nondegeneracy assumption of Theorem 1.1 is generic in the sense that any degenerate solution to (1) can be approached by a solution of a slight perturbation of [START_REF] Ambrosetti | Perturbation methods and semilinear elliptic problems on R n[END_REF]. We refer to Proposition 3.2 of Section 3 for the precise genericity statement.

The above theorem outlines a role of the geometry in dimension n = 6. In higher dimension n ≥ 7, the geometry of (M, g) is more present. When the potential is strictly below the scalar curvature (that is h < c n Scal g ), equation ( 1) is compact for n ≥ 7, at least in the locally conformally flat case (Vétois [36]). Conversely, if h(x 0 ) > c n Scal g (x 0 ) for some x 0 ∈ M , then under some additional nondegeneracy assumption, equation ( 1) is non-compact when n ≥ 7 (see Pistoia-Vétois [START_REF] Pistoia | Sign-changing bubble towers for asymptotically critical elliptic equations on Riemannian manifolds[END_REF] for general results). Our second result is in the case h ≡ c n Scal g : Theorem 1.2 (dimensions 3 ≤ n ≤ 9 and h ≡ c n Scal g ). Let (M, g) be a smooth compact Riemannian manifold of dimension 3 ≤ n ≤ 9 with positive Yamabe invariant. Assume that there exists a nondegenerate positive solution u 0 > 0 to the Yamabe equation [START_REF] Th | Problèmes isopérimétriques et espaces de Sobolev[END_REF]. Assume that h ≡ c n Scal g . Then for ε > 0 small, equation (4) admits a sign-changing solution u ε of type (u 0 -B). In particular, the family (u ε ) ε>0 blows up as ε → 0 and (2) is noncompact.

Constructing positive blowing-up solutions is not possible in this context. Indeed, for 3 ≤ n ≤ 9 and except for the canonical sphere, the scalar curvature equation ( 2) is positively compact (see , Druet [START_REF]Compactness for Yamabe metrics in low dimensions[END_REF], Marques [START_REF] Marques | A priori estimates for the Yamabe problem in the non-locally conformally flat case[END_REF], Li-Zhang [START_REF] Li | A Harnack type inequality for the Yamabe equation in low dimensions[END_REF][START_REF]Compactness of solutions to the Yamabe problem. II[END_REF], Khuri-Marques-Schoen [START_REF] Khuri | A compactness theorem for the Yamabe problem[END_REF], and the discussion below). We refer also to Druet-Hebey [START_REF] Druet | Stability for strongly coupled critical elliptic systems in a fully inhomogeneous medium[END_REF] and Druet-Hebey-Vétois [START_REF] Druet | Bounded stability for strongly coupled critical elliptic systems below the geometric threshold of the conformal Laplacian[END_REF] for the extension of compactness issues to stability issues.

The restriction of the dimensions in Theorem 1.2 is due to the geometry of the manifold. We refer to Subsection 2.1 in Section 2 for the extension of Theorem 1.2 to dimension n = 10 in general and in any dimension in the locally conformally flat case.

Here again, it is natural to ask about the nondegeneracy assumption of a solution to the limit equation in Theorem 1.2: actually, it is both a necessary and a generic assumption. Concerning necessity, on the standard sphere (where all positive solutions to (2) are degenerate), it is not possible to construct blowing-up solutions of type (u 0 -B), see Proposition 3.1 in Section 3. However, it is proved in Khuri-Marques-Schoen [START_REF] Khuri | A compactness theorem for the Yamabe problem[END_REF] that the nondegeneracy assumption is generic for the Yamabe equation ( 2), at least in dimensions n ≤ 24, see Proposition 3.3 in Section 3.

Here is a brief overview of the positive compactness results known so far for equations like [START_REF] Ambrosetti | Perturbation methods and semilinear elliptic problems on R n[END_REF].

In 1987, Schoen [START_REF] Schoen | Variational theory for the total scalar curvature functional for Riemannian metrics and related topics[END_REF] adressed the question of positive compactness of equation ( 2) for manifolds non conformally diffeomorphic to the canonical sphere (S n , can) (say aspherical manifolds). The known results are the following: positive compactness holds for aspherical locally conformally flat manifolds (Schoen [START_REF] Schoen | Variational theory for the total scalar curvature functional for Riemannian metrics and related topics[END_REF][START_REF]On the number of constant scalar curvature metrics in a conformal class[END_REF]) and for arbitrary aspherical manifolds of dimension 3 ≤ n ≤ 24 (Li-Zhu [START_REF] Li | Yamabe type equations on three-dimensional Riemannian manifolds[END_REF] (n = 3), Druet [START_REF]Compactness for Yamabe metrics in low dimensions[END_REF] (n ≤ 5), Marques [START_REF] Marques | A priori estimates for the Yamabe problem in the non-locally conformally flat case[END_REF] (n ≤ 7), Li-Zhang [START_REF] Li | A Harnack type inequality for the Yamabe equation in low dimensions[END_REF][START_REF]Compactness of solutions to the Yamabe problem. II[END_REF][START_REF] Li | Compactness of solutions to the Yamabe problem. III[END_REF] (n ≤ 11), Kuhri-Marques-Schoen [START_REF] Khuri | A compactness theorem for the Yamabe problem[END_REF] (n ≤ 24)). But positive compactness does not hold in general in dimension n ≥ 25 (There are blowing-up examples by Brendle [START_REF] Brendle | Blow-up phenomena for the Yamabe equation[END_REF] and Brendle-Marques [START_REF] Brendle | Blow-up phenomena for the Yamabe equation. II[END_REF]). Combining these results with Theorem 1.1, we get that equation ( 2) is positively compact, but not compact, at least when 3 ≤ n ≤ 9.

When h ≡ c n Scal g , the situation is different. When h < c n Scal g , Druet [START_REF] Druet | From one bubble to several bubbles: the low-dimensional case[END_REF] proved that (3) is positively compact in dimension n ≥ 3 (see also and Druet-Hebey-Vétois [START_REF] Druet | Bounded stability for strongly coupled critical elliptic systems below the geometric threshold of the conformal Laplacian[END_REF] for n = 3). Conversely, in dimension n ≥ 4, Micheletti-Pistoia-Vétois [START_REF] Micheletti | Blow-up solutions for asymptotically critical elliptic equations[END_REF] and Pistoia-Vétois [START_REF] Pistoia | Sign-changing bubble towers for asymptotically critical elliptic equations on Riemannian manifolds[END_REF] proved that if h is above c n Scal g somewhere, then, under some some nondegeneracy assumption, equation ( 1) is not positively compact. On the canonical sphere, there are blowing-up positive solutions with arbitrarily high energy when h ≡ Cte > c n Scal can (Chen-Wei-Yan [START_REF] Chen | Infinitely many solutions for the Schrödinger equations in R N with critical growth[END_REF] for n ≥ 5). We refer to Esposito-Pistoia-Vétois [START_REF] Esposito | The effect of linear perturbations on the Yamabe problem[END_REF] for blowing-up positive solutions in case of a potential h depending on ε and approaching c n Scal g , and to Hebey-Wei [START_REF] Hebey | Resonant states for the static Klein-Gordon-Maxwell-Proca system[END_REF] for the construction of multi-peak solutions on the three-sphere with a potential approaching constants arbitrarily larger than the scalar curvature. Here again, combining Druet [START_REF] Druet | From one bubble to several bubbles: the low-dimensional case[END_REF] and Theorem 1.2 yields the following: when h < c n Scal g and 3 ≤ n ≤ 5, equation ( 1) is positively compact, but not compact.

The proofs of Theorems 1.1 and 1.2 rely on a Lyapunov-Schmidt reduction. Over the past two decades, there has been intensive developments in Lyapunov-Schmidt reductions applied to critical elliptic equations. In addition to the references in the geometric context of a Riemannian manifold cited above, an early reference for single-bubble solutions is Rey [START_REF] Rey | The role of the Green's function in a nonlinear elliptic equation involving the critical Sobolev exponent[END_REF]. Possible references on the construction of blowing-up solutions to equations like (4) by glueing a fixed function to bubbles are del Pino-Musso-Pacard-Pistoia [START_REF] Del Pino | Large energy entire solutions for the Yamabe equation[END_REF][START_REF]Torus action on S n and sign changing solutions for conformally invariant equations[END_REF] and Guo-Li-Wei [START_REF] Guo | Large energy entire solutions for the Yamabe-type problem of polyharmonic operator[END_REF] (for the Yamabe equation on the canonical sphere) and Wei-Yan [START_REF] Wei | On a stronger Lazer-McKenna conjecture for Ambrosetti-Prodi type problems[END_REF] (for a Lazer-McKenna type problem). The list of constributions above does not pretend to exhaustivity: we refer to the references of the above papers and also to the monograph [START_REF] Ambrosetti | Perturbation methods and semilinear elliptic problems on R n[END_REF] by Ambrosetti-Malchiodi for further bibliographic complements. Our paper is organized as follows.

In Section 2, we discuss extensions and generalizations of the above theorems. In Section 3, we discuss the nondegeneracy assumption. The finite dimensional reduction is performed in Section 4. The reduced problem is studied in Section 5. Theorems 1.1 and 1.2 are proved in Section 6. The proof of the error estimate is postponed to Section 7.

Acknowledgements:

The authors express their gratitude to Emmanuel Hebey and Lionel Bérard-Bergery for fruitful comments on this work.

Miscellaneous extensions

2.1. About the critical dimension n = 10 in Theorem 1.2. As mentioned in the introduction, the method developed here fails to produce blowing-up solutions to (4) in higher dimension. Indeed, in dimensions n ≥ 7, a term involving the Weyl tensor appear in the Taylor expansion (75) of the Lyapunov-Schmidt functional. In dimension n < 10, this term is dominated by the contribution of u 0 . In dimension n = 10, there is a competition between the Weyl tensor and u 0 , and one gets the following result: Theorem 2.1 (dimension n = 10 and h ≡ c n Scal g ). Let (M, g) be a smooth compact Riemannian manifold of dimension n = 10 with positive Yamabe invariant. Assume that there exists a nondegenerate positive solution to the Yamabe equation [START_REF] Th | Problèmes isopérimétriques et espaces de Sobolev[END_REF]. Assume that h ≡ c n Scal g and that u 0 > 5 567 | Weyl g | 2 g . Then for ε > 0 small, equation (4) admits a sign-changing solution u ε of type (u 0 -B). In particular, the family (u ε ) ε>0 blows up as ε → 0.

In dimension n > 10, the Weyl tensor dominates the contribution of u 0 , and it is not possible to produce blowing-up solutions in general (see the explicit Taylor expansion (75) in Section 6). However, in the locally conformally flat case, that is when the Weyl tensor vanishes (at least in dimension larger than four), one gets the following result: Theorem 2.2 (the locally conformally flat case in any dimension). Let (M, g) be a smooth compact Riemannian manifold of dimension n ≥ 3 with positive Yamabe invariant. Assume that there exists a nondegenerate positive solution to the Yamabe equation [START_REF] Th | Problèmes isopérimétriques et espaces de Sobolev[END_REF]. Assume that (M, g) is locally conformally flat and that h ≡ c n Scal g . Then for ε > 0 small, equation (4) admits a sign-changing solution u ε of type (u 0 -B). In particular, the family (u ε ) ε>0 blows up as ε → 0.

Examples of manifolds and metrics satisfying the hypothesis of Theorem 2.2 are in Proposition 3.4. As stated in Theorems 2.1 and 2.2, the solutions we construct change sign. Here again, it is natural to ask if there exist positive blowing-up solutions to (4) under the assumptions of Theorems 2.1 and 2.2. The answer is negative. Indeed, it follows from positive compactness theorems (Khuri-Marques-Schoen [START_REF] Khuri | A compactness theorem for the Yamabe problem[END_REF] and Schoen [START_REF] Schoen | Variational theory for the total scalar curvature functional for Riemannian metrics and related topics[END_REF][START_REF]On the number of constant scalar curvature metrics in a conformal class[END_REF]) that positive blowing-up solutions to equation [START_REF] Bianchi | A note on the Sobolev inequality[END_REF] do not exist in the locally conformally flat and aspherical case. A consequence of the above results is that the Yamabe equation ( 2) is positively compact, but not compact in the context of Theorems 2.1 and 2.2.

2.2.

Positive blowing-up solutions in dimension n = 6. In this subsection, we focus on positive solutions to [START_REF] Bianchi | A note on the Sobolev inequality[END_REF]. A direct offshot of the techniques developed for the proof of Theorem 1.1 yields the following result: Theorem 2.3 (positive solutions in dimension n = 6). Let (M, g) be a smooth compact Riemannian manifold of dimension n = 6 and let h ∈ C 0,θ (M ) (θ ∈ (0, 1)) be such that ∆ g + h is coercive. Assume that there exists a nondegenerate solution u 0 ∈ C 2,θ (M ) to equation [START_REF] Ambrosetti | Perturbation methods and semilinear elliptic problems on R n[END_REF] and that

h -c 6 Scal g > 2u 0 > 0 in M .
Then for ε > 0 small, equation (4) admits a positive solution u ε > 0 such that

u ε = u 0 + B ε + o(1), where (B ε ) ε is a bubble and lim ε→0 o(1) = 0 in H 2 1 (M ).
This result is a complement to a specific 6-dimensional result: Druet ([11] and private communication) showed that blow-up for positive solutions to (4) with bounded energy necessarily occurs at points x ∈ M such that (h -c 6 Scal g )(x) ≥ 2u 0 (x). Dimension six is critical when considering positive blowing-up solutions with nontrivial weak limit u 0 > 0. More precisely, the blow-up analysis shows that there is balance between the contributions of u 0 and h -c n Scal g : one of the terms dominates the other when n = 6, and they compete at the same growth when n = 6. We refer to the Taylor expansion (45) and to [START_REF] Druet | From one bubble to several bubbles: the low-dimensional case[END_REF] to outline this phenomenon. We refer to Druet-Hebey [START_REF] Druet | Stability for strongly coupled critical elliptic systems in a fully inhomogeneous medium[END_REF] for an extensive discussion on dimension six.

Prescription of the blow-up point.

The above theorems show the existence of blowing-up families of solutions, but the blow-up point (that is the point where the bubble is centered) is not prescribed. The only information obtained from the construction is that blow-up occurs at a minimum point of u 0 (for Theorems 1.1 and 1.2 when n = 6) or of u 0 -(c n Scal g -h)/2 (for Theorem 1.1 when n = 6). Prescribing the location of the blow-up point of the bubbles requires additional informations. We define Φ : M → R as follows:

Φ :=    u 0 + 1 2 (h -c n Scal g )1 n=6 in Theorems 1.1, 1.2 and 2.2 u 0 -5 567 | Weyl g | 2 g in Theorem 2.1 1 2 (h -c 6 Scal g ) -u 0 in Theorem 2.3.
Our prescription result is the following:

Theorem 2.4 (Prescription of the blow-up point). In addition to the hypothesis in Theorems 1.1, 1.2, 2.1, 2.2, and 2.3, assume that there exists ξ 0 ∈ M which is a strict local minimum point of Φ with Φ(ξ 0 ) > 0. Then the conclusions of the above theorems hold with the extra information that the bubbles are centered at a family

(ξ ε ) ε ∈ M such that lim ε→0 ξ ε = ξ 0 .
In case h ∈ C 1 (M ) and there exists ξ 0 ∈ M which is a C 1 -stable critical point of Φ with Φ(ξ 0 ) > 0, the same conclusion holds with the convergence (45) holding in C 1 .

Discussion on the degenerate case

In the sequel, we say that (B ε ) ε is a bubble if there exists a family (x ε ) ε ∈ M and a family (µ ε ) ε ∈ R >0 such that lim ε→0 µ ε = 0 and (6)

B ε (x) := n(n -2)µ ε µ 2 ε + d g (x, x ε ) 2 n-2 2
for all x ∈ M .

In this situation, we say that the bubble is centered at (x ε ) ε . We say that a solution

u 0 ∈ C 2,θ (M ) to (7) ∆ g u 0 + hu 0 = u 2 ⋆ -1 0 in M is nondegenerate if the linearization of the equation has a trivial kernel, that is (8) K h,u0 := ϕ ∈ C 2,θ (M )/ ∆ g ϕ + hϕ = (2 ⋆ -1)|u 0 | 2 ⋆ -2 ϕ = {0}.
Theorems 1.1 and 1.2 require the assumption that u 0 is a nondegenerate solution to [START_REF] Chen | Infinitely many solutions for the Schrödinger equations in R N with critical growth[END_REF]. In this section, we prove that this is a necessary assumption, and that it is generic.

3.1. The conformal geometric equation and necessity of the nondegeneracy assumption. Let (M, g) be a compact Riemannian manifold of dimension n ≥ 3 with positive Yamabe invariant. Up to a conformal change of metric, it follows from the resolution of the Yamabe problem that we can assume that the scalar curvature Scal g is a positive constant, and we consider u 0,g :

= (c n Scal g ) 1/(2 ⋆ -2)
the only positive constant solution to the Yamabe equation

(9) ∆ g u 0,g + c n Scal g u 0,g = u 2 ⋆ -1 0,g in M .
As is easily checked, in this situation,

K cn Scalg,u0,g = ϕ ∈ C 2 (M )/ ∆ g ϕ = Scal g n -1 ϕ ,
where the kernel is defined in [START_REF] Del Pino | Large energy entire solutions for the Yamabe equation[END_REF]. Therefore

u 0,g is a nondegenerate solution to (9) ⇔ Scal g n -1 ∈ Spec (∆ g ),
where Spec (∆ g ) is the nonnegative spectrum of ∆ g . We define the Yamabe invariant by ( 10)

µ [g] (M ) := inf g ′ ∈[g] M Scal g ′ dv g ′ Vol g ′ (M ) n-2 n
, where [g] is the conformal class of g and dv g is the Riemannian element of volume.

The Yamabe invariant µ

[g] (M ) is positive iff the operator ∆ g ′ + c n Scal g ′ is coercive for all g ′ ∈ [g].
It is well known that if g is a Yamabe metric (that is a minimizer of the Yamabe functional [START_REF] Ding | On a conformally invariant elliptic equation on R n[END_REF]), one has that Scalg n-1 ≤ λ 1 (∆ g ), the first nonzero eigenvalue of ∆ g . Note that equality is achieved on the canonical sphere (S n , can). More generally, any positive solution to the Yamabe equation on the canonical sphere is a Yamabe metric and is degenerate.

The following result shows that the conclusion of Theorems 1.1 and 1.2 do not hold on the standard sphere (where positive solutions to the Yamabe equations are all degenerate): Proposition 3.1. There does not exist any family of functions

(u ε ) ε ∈ C 2,θ (S n ) of type (u 0 -B) to the equation (11) ∆ can u ε + c n Scal can u ε = |u ε | 2 ⋆ -2-ε u ε in M for all ε ∈ (0, 2 ⋆ ).
Proof. We argue by contradiction and assume the existence of a family (u ε ) ε ∈ C 2,θ (S n ) of the form (5) of solutions to equation [START_REF] Druet | From one bubble to several bubbles: the low-dimensional case[END_REF]. Multiplying [START_REF] Druet | From one bubble to several bubbles: the low-dimensional case[END_REF] by the bubble B ε and integrating by parts yield µ ε ε → 1 when ε → 0. Fix φ ∈ Λ 1 (S n , can), the set of eigenfunctions of λ 1 (∆ can ) = n, the first nonzero eigenvalue of ∆ can on S n : indeed, see for instance Berger-Gauduchon-Mazet [START_REF] Berger | Le spectre d'une variété riemannienne[END_REF], we have that Λ 1 (S n , can) = {l |S n / l : R n+1 → R linear}. It follows from Kazdan-Warner [START_REF] Kazdan | Scalar curvature and conformal deformation of Riemannian structure[END_REF] that

S n ∆ can u ε ∇φ, ∇u ε can dv can = n -2 2n S n ∆ can φ|∇u ε | 2 can dv can .
Using equation [START_REF] Druet | From one bubble to several bubbles: the low-dimensional case[END_REF] and integrating by parts yields

ε S n φ|u ε | 2 ⋆ -ε dv can = 0 .
Letting ε → 0 and using ( 6), [START_REF] Brendle | Blow-up phenomena for the Yamabe equation[END_REF], and (15) yields, up to a subsequence,

S n φu 2 ⋆ 0 dv can + R n U 2 ⋆ dx φ(x 0 ) = 0 ,
where x 0 ∈ S n . Passing to the weak limit in [START_REF] Druet | From one bubble to several bubbles: the low-dimensional case[END_REF] when ε → 0 implies that u 0 is a positive solutions to the Yamabe equation on S n . It then follows from Obata [START_REF] Obata | The conjectures on conformal transformations of Riemannian manifolds[END_REF] that

S n u 2 ⋆ 0 dv can = R n U 2 ⋆ dx. Taking φ ∈ Λ 1 (S n
, can) such that min φ = φ(-x 0 ) = 0 in the above equation yields a contradiction since φ ≡ 0. This ends the proof of Proposition 3.1.

Genericity of the nondegeneracy assumption.

The following proposition shows that the nondegeneracy hypothesis of Theorem 1.1 is generic:

Proposition 3.2. Let h ∈ C 0,θ (M ) and let u 0 ∈ C 2,θ (M ) be a positive solution to ∆ g u 0 +hu 0 = u 2 ⋆ -1 0 . Fix ν > 0. Then there exist hν ∈ C 0,θ (M ) and ũ0,ν ∈ C 2,θ (M ) such that h-hν C 0,θ + u 0 -ũ0,ν C 2,θ < ν and ũ0,ν > 0 is a nondegenerate solution to ∆ g ũ0,ν + hν ũ0,ν = ũ2 ⋆ -1 0,ν in M .
Proof. We define

µ η := inf u∈H 2 1 (M )\{0} M (|∇u| 2 g + (h -(2 ⋆ -1)u 2 ⋆ -2 0 -η)u 2 ) dv g M |u| 2 ⋆ dv g 2 2 ⋆
for all η ≥ 0. Testing the functional on u 0 yields µ η < 0 for all η ≥ 0. As is easily checked, lim η→0 µ η = µ 0 < 0. Standard variational arguments yield the existence of a positive minimizer

w η ∈ C 2,θ (M ) for µ η such that ∆ g w η + (h - (2 ⋆ -1)u 2 ⋆ -2 0 -η)w η = -(2 ⋆ -2)w 2 ⋆ -1 η
in M for all η > 0; moreover, the family (w η ) η≥0 is relatively compact in C 2 (M ). Since u 0 is the only positive solution to the equation

∆ g v + (h -(2 ⋆ -1)u 2 ⋆ -2 0 )v = -(2 ⋆ -2)v 2 ⋆ -1 (
let w be the quotient of two positive solutions and estimate ∆ g w at extremal points of w), one gets that lim η→0 w η = u 0 in C 2 (M ), and then C 2,θ (M ) by elliptic regularity. One defines

h η := h-(2 ⋆ -1)(u 2 ⋆ -2 0 -w 2 ⋆ -2 η )-η. Then ∆ g w η +h η w η = w 2 ⋆ -1
η in M and spectral theory yields the existence of η 0 > 0 such that K hη,wη = {0} for all η ∈ (0, η 0 ).

The conclusion of the proposition follows from taking ũ0,ν := w η and hν := h η for η > 0 small enough.

We now focus on the geometric case, that is Theorem 1.2. We adopt the terminology of Khuri-Marques-Schoen [START_REF] Khuri | A compactness theorem for the Yamabe problem[END_REF]: given M a compact manifold, g 0 a background Riemannian metric on M , and ω a volume form on M , to each class c ∈ C := {Conformal classes of Riemannian metrics on M }, we associate the unique metric g ∈ c for which the Riemannian n-volume form is ω. The C k,θ -distance between two classes is defined as the C k,θ -distance between their representatives via this analogy. The following result holds: [START_REF] Khuri | A compactness theorem for the Yamabe problem[END_REF]). There exists O ⊂ C an open dense set such that for all c ∈ O, there exists exactly a finite nonzero number of metrics g ∈ c (up to homothetic transformations) such that the constant positive function u 0,g is nondegenerate.

Proposition 3.3 (Khuri-Marques-Schoen
In other words, up to a perturbation in the conformal class, the hypothesis of Theorem 1.2 holds.

3.3.

A family of nondegenerate geometric solutions in the locally conformally flat case. We exhibit here a situation in which the nondegeneracy assumption is satisfied for the geometric equation in the locally conformally flat case (see Theorem 2.2). For all k ≥ 1 and t > 0, (S k (t), can) is the canonical sphere ot radius t in R k+1 : Proposition 3.4. Let M r := S 1 (r) × S n-1 be endowed with its canonical product metric g r , where r > 0. Then (M r , g r ) is locally conformally flat with positive constant scalar curvature. Moreover, for any r ∈ { i √ n-2 / i ∈ Z >0 }, the positive constant solution to the Yamabe equation is nondegenerate.

Proof. Recall that on the canonical sphere S k (t), the spectrum of the Laplacian is i(k+i-1) t 2 / i ∈ Z ≥0 (see [START_REF] Berger | Le spectre d'une variété riemannienne[END_REF]). Then the spectrum of ∆ gr is { i 2 r 2 + j(n -2 + j)/ i, j ∈ Z ≥0 }. Independently, the scalar curvature is Scal gr := (n -1)(n -2). As a consequence,

Scal gr n -1 ∈ Spec(∆ gr ) ⇔ r ∈ i √ n -2 / i ∈ Z >0 .
In addition, it is standard that the product of a one-dimensional circle with a space form is locally conformally flat.

Finite dimensional reduction

Since the operator ∆ g + h is coercive, the Sobolev space H 2 1 (M ) is endowed with the scalar product •, • h defined by [START_REF]Compactness for Yamabe metrics in low dimensions[END_REF] u, v h = M ∇u, ∇v g dv g + M huvdv g for all u, v ∈ H 2 1 (M ). We let • h be the norm induced by •, • h : this norm is equivalent to the standard norm on H 2 1 (M ). We let i * :

L 2n n+2 (M ) → H 2 1 (M ) be such that for any w in L 2n n+2 (M ), the function u = i * (w) in H 2 1 (M )
is the unique solution of the equation ∆ g u + hu = w in M . We then rewrite equation ( 4) as ( 13)

u = i * (f ε (u)) , u ∈ H 2 1 (M ) , where f ε (u) := |u| 2 * -2-ε u.
In case (M, g) is locally conformally flat, it follows from the compactness of M that there exists r 0 ∈ (0, i g (M )) (where i g (M ) > 0 is the injectivity radius of (M, g)) such that for any point ξ in M , there exists Λ ξ ∈ C ∞ (M ) such that the conformal metric g ξ = Λ

4/(n-2) ξ g is flat in B ξ (r 0 ) and i g ξ (M ) > r 0 . As is easily seen, the functions Λ ξ can be chosen smooth with respect to ξ and such that Λ ξ (ξ) = 1. If the manifold is not locally conformally flat, then we let Λ ξ (x) = 1 for all points x and ξ in M , and we fix r 0 ∈ (0, i g (M )) arbitrarily. We let χ be a smooth cutoff function such that 0 ≤ χ ≤ 1 in R, χ = 1 in [-r 0 /2, r 0 /2], and χ = 0 in R\ (-r 0 , r 0 ). For any positive real number δ and any point ξ in M , we define the function W δ,ξ on M by ( 14)

W δ,ξ (x) := χ d g ξ (x, ξ) Λ ξ (x) δ 2-n 2 U δ -1 exp -1 ξ (x) ,
where d g ξ is the geodesic distance on M associated with the metric g ξ , the exponential map is taken with respect to the same metric g ξ and ( 15)

U (x) := n (n -2) 1 + |x| 2 n-2 2
for all x ∈ R n . For any positive real number δ, the function

U δ (x) = δ 2-n 2 U δ -1 x satisfies the equation ∆ Eucl U δ = U 2 ⋆ -1 δ
where ∆ Eucl is the Laplace operator associated with the Euclidean metric. Moreover, by Bianchi-Egnell [START_REF] Bianchi | A note on the Sobolev inequality[END_REF], any solution

in v ∈ D 2 1 (R n ) (the completion of C ∞ c (R n ) for the norm u D 2 1 := ∇u 2 ) of the linear equation ∆ Eucl v = (2 * -1) U 2 * -2 v is a linear combination of the functions (16) V 0 (x) := |x| 2 -1 1 + |x| 2 n 2 and V i (x) := x i 1 + |x| 2 n 2
for all i = 1, . . . , n and x ∈ R n . For any positive real number δ and any point (ξ, ω) in T M , we define the functions Z δ,ξ and Z δ,ξ,ω in M by

Z δ,ξ (x) := χ d g ξ (x, ξ) Λ ξ (x)δ n-2 2 d g ξ (x, ξ) 2 -δ 2 δ 2 + d g ξ (x, ξ) 2 n 2 , (17)
Z δ,ξ,ω (x) := χ d g ξ (x, ξ) Λ ξ (x)δ n 2 exp -1 ξ x, ω g ξ δ 2 + d g ξ (x, ξ) 2 n 2 (18)
for all x ∈ M . We then let Π δ,ξ and Π ⊥ δ,ξ be the projections of the Sobolev space H 2 1 (M ) onto the respective closed subspaces

K δ,ξ := { λZ δ,ξ + Z δ,ξ,ω / λ ∈ R and ω ∈ T ξ M } and (19) K ⊥ δ,ξ := φ ∈ H 2 1 (M )
/ φ, Z δ,ξ h = 0 and φ, Z δ,ξ,ω h = 0 for all ω ∈ T ξ M , where the scalar product •, • h is as in [START_REF]Compactness for Yamabe metrics in low dimensions[END_REF].

Recall that u 0 ∈ C 2,θ (M ) is a nondegenerate positive solution to equation [START_REF] Chen | Infinitely many solutions for the Schrödinger equations in R N with critical growth[END_REF]. We construct solutions of type (u 0 -B) to equations (4), or equivalently [START_REF] Druet | Stability for strongly coupled critical elliptic systems in a fully inhomogeneous medium[END_REF], like

u ε := u 0 -W δε(tε),ξε + φ δε(tε),ξε , with δ ε (t ε ) := t ε ε 2 n-2 ,
where W δε(tε),ξε is as in [START_REF] Druet | Blow-up theory for elliptic PDEs in Riemannian geometry[END_REF], φ δε(tε),ξε is a function in K ⊥ δε(tε),ξε , and t ε > 0. As easily checked, if φ δε(tε),ξε → 0 in H 2 1 (M ) when ε → 0, then (u ε ) is of type u 0 -B. We rewrite equation ( 4) as the couple of equations ( 20)

Π δε(t),ξ u 0 -W δε(t),ξ + φ δε(t),ξ -i * f ε u 0 -W δε(t),ξ + φ δε(t),ξ = 0 and (21) Π ⊥ δε(t),ξ u 0 -W δε(t),ξ + φ δε(t),ξ -i * f ε u 0 -W δε(t),ξ + φ δε(t),ξ = 0 .
We begin with solving equation [START_REF] Lee | The Yamabe problem[END_REF]

φ δε(t),ξ h ≤ C a,b        ε |ln ε| if 3 ≤ n ≤ 6 ε 4 n-2 if n ≥ 7 ε n+2 2(n-2)
if n ≥ 7, h ≡ c n Scal g , and (M, g) is loc. conf. flat. Moreover, φ δε(t),ξ is continuously differentiable with respect to t and ξ.

The sequel of this section is devoted to the proof of Proposition 4.1. For ε small, for any positive real number δ, and any point ξ in M , we let the map L ε,δ,ξ : K ⊥ δ,ξ → K ⊥ δ,ξ be defined by ( 23)

L ε,δ,ξ (φ) := Π ⊥ δ,ξ (φ -i * (f ′ ε (u 0 -W δ,ξ ) φ)) for all φ ∈ K ⊥ δ,ξ
, where u 0 ∈ C 2,θ (M ) is a nondegenerate positive solution to (7), W δ,ξ is as in ( 14) and K ⊥ δ,ξ is as in [START_REF] Khuri | A compactness theorem for the Yamabe problem[END_REF]. Clearly, we get that L ε,δ,ξ is linear and continuous. In Lemma 4.2 below, we prove the invertibility of L ε,δ,ξ for δ and ε small. Lemma 4.2. Given two positive real numbers a < b, for ε small, the map L ε,δε(t),ξ is invertible for all real numbers t in [a, b] and all points ξ in M , where δ ε (t) = tε 2/(n-2) and L ε,δε(t),ξ is as in [START_REF]Compactness of solutions to the Yamabe problem. II[END_REF]. Moreover, there exists a positive constant C a,b such that for ε small, for any real number t ∈ [a, b], any point ξ ∈ M , and any function φ ∈ K ⊥ δε(t),ξ , there holds

(24) L ε,δε(t),ξ (φ) h ≥ C a,b φ h .
In particular, the inverse map L -1 ε,δε(t),ξ is continuous. Proof. We prove [START_REF] Li | Compactness of solutions to the Yamabe problem. III[END_REF]. We proceed by contradiction. We assume that there exist two sequences of positive real numbers (ε α ) α and (t α ) α such that ε α → 0 for all α → +∞ and a ≤ t α ≤ b, a sequence of points (ξ α ) α in M , and a sequence of functions (φ α ) α such that [START_REF] Li | Yamabe type equations on three-dimensional Riemannian manifolds[END_REF] φ α ∈ K ⊥ δε α (tα),ξα , φ α h = 1 , and L εα,δε α (tα),ξα (φ α ) h -→ 0 as α → +∞. We define W α := W δε α (tα),ξα . First, we claim that

(26) φ α -i * f ′ εα (u 0 -W α ) φ α h -→ 0 as α → +∞.
Passing if necessary to a subsequence, we may assume that all the points ξ α belong to a small open subset Ω in M on which there exists a smooth orthonormal frame. Thanks to this frame, we identify the tangent space T ξ M with R n for all points ξ in Ω, so that exp ξ is in fact the composition of the standard exponential map with a linear isometry Ψ ξ : R n → T ξ M which is smooth with respect to ξ. We define [START_REF] Micheletti | Blow-up solutions for asymptotically critical elliptic equations[END_REF] Z 0,α := Z δε α (tα),ξα and Z i,α = Z δε α (tα),ξα,ei for all i = 1, . . . , n, where e i is the i-th vector in the canonical basis of R n and the functions Z δε α (tα),ξα and Z δε α (tα),ξα,ei are as in ( 17)- [START_REF] Hebey | Resonant states for the static Klein-Gordon-Maxwell-Proca system[END_REF]. For any α, by definition of L ε,δε α (tα),ξα , we get that

(28) φ α -i * f ′ εα (u 0 -W α ) φ α -L εα,δε α (tα),ξα (φ α ) = n i=0 λ i,α Z i,α
for some real numbers λ i,α , where the functions Z i,α are as in [START_REF] Micheletti | Blow-up solutions for asymptotically critical elliptic equations[END_REF]. Taking into account ( 25) and ( 28), one sees that in order to get [START_REF] Marques | A priori estimates for the Yamabe problem in the non-locally conformally flat case[END_REF], it suffices to prove that λ i,α → 0 as α → +∞ for all i = 0, . . . , n. As is easily checked, for any i, j = 0, . . . , n, there holds

(29) Z i,α , Z j,α h -→ ∇V i 2 2
δ ij as α → +∞ where the function V i is as in ( 16) and the real numbers δ ij are the Kronecker symbols. By [START_REF] Obata | The conjectures on conformal transformations of Riemannian manifolds[END_REF], [START_REF] Pistoia | Sign-changing bubble towers for asymptotically critical elliptic equations on Riemannian manifolds[END_REF], and since the functions φ α and L εα,δε α (tα),ξα (φ α ) belong to K ⊥ δε α (tα),ξα , for any i = 0, . . . , n, we get that (30)

M f ′ εα (u 0 -W α ) Z i,α φ α dv g = -λ i,α ∇V i 2 2 + o   n j=0 |λ j,α |   as α → +∞.
As is easily checked, we get that

M f ′ εα (u 0 -W α ) Z i,α φ α dv g = M f ′ εα (W α ) Z i,α φ α dv g + o (1) (31)
as α → +∞. We find

M f ′ εα (W α ) Z i,α φ α dv g 2 ⋆ -1 -ε α = δ εα (t α ) n-2 2 εα R n χ α Λ 2 ⋆ -1-ǫα α U 2 * -2-εα V i φ α dv gα (32)
as α → +∞, where the functions U and V i are as in ( 15) and ( 16), the cutoff function χ is as in Section 4 and

χ α := χ (δ εα (t α ) |x|) 2 * -2-εα , Λ α := Λ ξα (exp ξα (δ εα (t α ) x)), φ α (x) := δ εα (t α ) n-2 2 χ (δ εα (t α ) |x|) φ α exp ξα (δ εα (t α ) x) , (33)
g α (x) := exp * ξα g (δ εα (t α ) x) (34)
for any α → +∞ and x ∈ R n small enough. In the definitions above, the exponential map is taken with respect to the metric g ξα . Since (φ α ) α is bounded in H 2 1 (M ), we get that φ α α is bounded in D 1,2 (R n ). Passing to a subsequence, we may assume that φ α α converges weakly to some function φ in D 1,2 (R n ). Passing to the limit into (32) yields ( 35)

M f ′ εα (W α ) Z i,α φ α dv g -→ (2 * -1) R n U 2 * -2 V i φdx as α → +∞. Since the function V i satisfies the equation ∆ Eucl V i = (2 * -1) U 2 * -2 V i in R n
, and since, for any α, the function φ α belongs to K ⊥ δε α (tα),ξα , passing to the limit as α → +∞ into the equation Z i,α , φ α h = 0, we get that (36)

R n ∇V i , ∇ φ dx = (2 * -1) R n U 2 * -2 V i φdx = 0 .
By ( 30), ( 31), [START_REF] Talenti | Best constant in Sobolev inequality[END_REF], and ( 36), we get that

λ i,α = o (1) + o   n j=0 |λ j,α |  
as α → +∞. It follows that λ i,α → 0 as α → +∞ for all i = 0, . . . , n. The claim [START_REF] Marques | A priori estimates for the Yamabe problem in the non-locally conformally flat case[END_REF] then follows from ( 25) and [START_REF] Obata | The conjectures on conformal transformations of Riemannian manifolds[END_REF].

For any sequence (ϕ α ) α in H 2 1 (M ), and by ( 26), we get that

φ α , ϕ α h - M f ′ εα (u 0 -W α ) φ α ϕ α dv g (37) = φ α -i * f ′ εα (u 0 -W α ) φ α , ϕ α h ≤ φ α -i * f ′ εα (u 0 -W α ) φ α h ϕ α h = o ( ϕ α h ) as α → +∞.
We claim that φ α ⇀ 0 weakly in H 2 1 (M ) when α → +∞. We prove the claim. Since (φ α ) is bounded in H 2 1 (M ), up to a subsequence, there exists φ ∈ H 2 1 (M ) such that (φ α ) ⇀ φ weakly in H 2 1 (M ) when α → +∞. Then for any ϕ ∈ H 2 1 (M ), taking ϕ α ≡ ϕ in [START_REF] Wei | On a stronger Lazer-McKenna conjecture for Ambrosetti-Prodi type problems[END_REF] and letting α → +∞ yields

φ, ϕ h = M (2 ⋆ -1)u 2 ⋆ -2 0 φϕ dv g for all ϕ ∈ H 2 1 (M ), and then ∆ g φ + hφ = (2 ⋆ -1)u 2 ⋆ -2 0
φ, which implies φ ≡ 0 since u 0 is nondegenerate. This proves the claim.

We claim that φα ⇀ 0 weakly in D 2 1 (R n ) when α → +∞, where φα has been defined in [START_REF]On the number of constant scalar curvature metrics in a conformal class[END_REF]. We prove the claim. Given a smooth function ϕ with compact support in R n , we define

ϕ α (x) := χ d g ξα (x, ξ α ) δ εα (t α ) 2-n 2 ϕ δ εα (t α ) -1 exp -1 ξα (x)
for all x ∈ M . It follows from [START_REF] Wei | On a stronger Lazer-McKenna conjecture for Ambrosetti-Prodi type problems[END_REF] together with a change of variable that

R n Λ -2 α ∇ φ α , ∇ϕ gα dv gα + δ εα (t α ) 2 R n Λ -2 ⋆ α h exp ξα (δ εα (t α ) x) φ α ϕdv gα (38) = δ εα (t α ) 2 R n Λ -2 ⋆ α f ′ εα u 0,α (x) -W α exp ξα (δ εα (t α ) x) φ α ϕdv gα + o (1)
as α → +∞, where u 0,α (•) := u 0 exp ξα (δ εα (t α ) •) , φ α and g α are as in [START_REF]On the number of constant scalar curvature metrics in a conformal class[END_REF] and [START_REF] Struwe | A global compactness result for elliptic boundary value problems involving limiting nonlinearities[END_REF]. One easily checks that

Λ -2 ⋆ α δ εα (t α ) 2 f ′ εα u 0 exp ξα (δ εα (t α ) •) -W α exp ξα (δ εα (t α ) •) goes to (2 * -1) U 2 * -2 as α → +∞ in C 0 loc (R n ).
Moreover, since φ α α converges weakly to φ in D 2 1 (R n ), passing to the limit into (38) as α → +∞ yields (39)

R n ∇ φ, ∇ϕ dx = (2 * -1) R n U 2 * -2 φϕdx .
Since (39) holds for all smooth functions ϕ with compact support in R n , we get that the function φ satisfies the equation ∆ Eucl φ = (2 * -1) U 2 * -2 φ in R n . By Bianchi-Egnell [START_REF] Bianchi | A note on the Sobolev inequality[END_REF], it follows that φ = n i=0 λ i V i for some real numbers λ i . It then follows from the orthogonality condition (36) that φ ≡ 0 is identically zero. This proves the claim.

Letting ϕ α := φ α and using [START_REF] Wei | On a stronger Lazer-McKenna conjecture for Ambrosetti-Prodi type problems[END_REF] together with a change of variable, we get

φ α 2 h = (2 ⋆ -1 -ε α ) M |u 0 -W α | 2 ⋆ -2-εα φ 2 α dv g + o (1) ≤ C M φ 2 α dv g + C M |W α | 2 ⋆ -2-εα φ 2 α dv g + o (1) ≤ C M φ 2 α dv g + C M |U | 2 ⋆ -2-εα φ2 α dv gα + o (1)
as α → +∞, where φ α and g α are as in [START_REF]On the number of constant scalar curvature metrics in a conformal class[END_REF] and [START_REF] Struwe | A global compactness result for elliptic boundary value problems involving limiting nonlinearities[END_REF]. Since φ α → 0 strongly in L 2 (M ), φ 2 α α is bounded in L n n-2 (R n ) and converges almost everywhere to 0, standard elliptic theory yields φ α → 0 as α → +∞ in H 2 1 (M ). This is a contradiction with [START_REF] Li | Yamabe type equations on three-dimensional Riemannian manifolds[END_REF]. This ends the proof of [START_REF] Li | Compactness of solutions to the Yamabe problem. III[END_REF].

The invertibility of L ε,δε(t),ξ follows from the Fredholm alternative. This ends the proof of Lemma 4.2. Now, we prove Proposition 4.1 by using Lemma 4.2 together with the error estimate in Section 7.

Proof of Proposition 4.1. We let a and b be two positive real numbers such that a < b. For ε small, for any real number t in [a, b], and any point ξ in M , equation ( 21) is equivalent to

(40) L ε,δε(t),ξ (φ) = N ε,δε(t),ξ (φ) + R ε,δε(t),ξ ,
where δ ε (t) = tε 2/(n-2) , L ε,δε(t),ξ is as in [START_REF]Compactness of solutions to the Yamabe problem. II[END_REF], and

N ε,δε(t),ξ (φ) := Π ⊥ δε(t),ξ i * f ε u 0 -W δε(t),ξ + φ (41) -f ε u 0 -W δε(t),ξ -f ′ ε u 0 -W δε(t),ξ φ and (42) R ε,δε(t),ξ := Π ⊥ δε(t),ξ i * f ε u 0 -W δε(t),ξ -u 0 + W δε(t),ξ
. By Lemma 4.2, for ε small, we get that the map L ε,δε(t),ξ is invertible for all real numbers t in [a, b] and all points ξ in M . We then let the map T ε,δε(t),ξ : K ⊥ δε(t),ξ → K ⊥ δε(t),ξ be defined for all φ ∈ K ⊥ δε(t),ξ by

T ε,δε(t),ξ (φ) := L -1 ε,δε(t),ξ N ε,δε(t),ξ (φ) + R ε,δε(t),ξ ,
where N ε,δε(t),ξ (φ) and R ε,δε(t),ξ are as in ( 41) and (42). For any positive real number Ξ, we let B ε,δε(t),ξ (Ξ) be the closed ball defined by

B ε,δε(t),ξ (Ξ) := φ ∈ K ⊥ δε(t),ξ / φ h ≤ Ξν ε ,
where ν ε > 0 is the error obtained in Lemma 7.1 of Section 7, namely

(43) ν ε :=        ε |ln ε| if n ≤ 6 ε 4 n-2 if n ≥ 7 ε n+2 2(n-2)
if n ≥ 7, h ≡ c n Scal g , and (M, g) loc. conformally flat.

We fix

θ 0 ∈ (0, min{1, 2 ⋆ -2}), so that u → f ε (u) is locally in C 1,θ0 on H 2 1 (M )
uniformly with respect to ε > 0 small. By Lemma 4.2 and by continuity of i * , for ε small, for any real number t in [a, b], any point ξ in M , and any functions φ, φ 1 , and φ 2 in H 2 1 (M ), we get that

T ε,δε(t),ξ (φ 1 ) -T ε,δε(t),ξ (φ 2 ) h ≤ C • max{ φ 1 θ0 h , φ 2 θ0 h } • φ 1 -φ 2 h
for some positive constant C independent of Ξ, ε, t, ξ, φ, φ 1 , and φ 2 , where ν ε is as in (43). By Lemma 7.1, we have that T ε,δε(t),ξ (0) ≤ Cν ε . We then get that for Ξ > 0 large enough, and then for ε small, for any real number t in [a, b], and any point ξ in M , then the map T ε,δε(t),ξ is a contraction map from the closed ball B ε,δε(t),ξ (Ξ) into itself . We then get that the map T ε,δε(t),ξ admits a unique fixed point φ δε(t),ξ in the ball B ε,δε(t),ξ (Ξ). In other words, the function φ δε(t),ξ is the unique solution of equation ( 40), or equivalently [START_REF] Lee | The Yamabe problem[END_REF], which satisfies [START_REF] Li | A Harnack type inequality for the Yamabe equation in low dimensions[END_REF] with

C a,b = Ξ.
The continuous differentiability of (t, ξ) → φ δε(t),ξ on (a, b) × M is standard. This ends the proof of Proposition 4.1.

The reduced problem

For ε small, we introduce the functional J ε defined on H 2 1 (M ) by

J ε (u) := 1 2 M |∇u| 2 g dv g + 1 2 M hu 2 dv g - M F ε (u) dv g , where F ε (u) := u 0 f ε (s) ds.
The critical points of J ε are the solutions of equation [START_REF] Druet | Stability for strongly coupled critical elliptic systems in a fully inhomogeneous medium[END_REF]. For any positive real number t and any point ξ in M , we define

(44) J ε (t, ξ) := J ε u 0 -W δε(t),ξ + φ δε(t),ξ ,
where W δε(t),ξ is as in ( 14) and φ δε(t),ξ is given by Proposition 4.1. We solve equation [START_REF] Kazdan | Scalar curvature and conformal deformation of Riemannian structure[END_REF] in Proposition 5.1 below:

Proposition 5.1. Let u 0 ∈ C 2,θ (M ) be a positive nondegenerate solution to [START_REF] Chen | Infinitely many solutions for the Schrödinger equations in R N with critical growth[END_REF].

Assume that either {h ∈ C 0,θ (M ) and 3 ≤ n ≤ 6} or {h ∈ C 2 (M ) and 3 ≤ n ≤ 9} or {(M, g) is locally conformally flat and h ≡ c n Scal g }. Then

(45)

J ε (t, ξ) = c 1 (n, u 0 ) + c 2 (n, u 0 )ε + c 3 (n)ε ln ε + c 4 (n)ε ln 1 t + c 5 (n) εt n-2 2 u 0 (ξ) + n n-2 4
(n -2)

n-6 4 (n -1) ω n ε 4 n-2 t 2 2 n-1 (n -4) ω n-1 • (h (ξ) -c n Scal g (ξ)) 1 n≥6 + o (ε)
as ε → 0, uniformly with respect to t in compact subsets of R >0 and with respect to the point ξ in M , ω n (resp. ω n-1 ) is the volume of the unit n-sphere (resp.

(n -1)-sphere), c i (n, u 0 ) (i = 1, 2) are positive constants depending only on n, u 0 , and the manifold, c i (n) (i = 3, 4, 5) depend only on n, and c 4 (n), c 5 (n) > 0.

Moreover, given two positive real numbers a < b, for ε small, if

(t ε , ξ ε ) ∈ (a, b) × M is a critical point of J ε , then the function u 0 -W δε(tε),ξε + φ δε(tε)
,ξε is a solution to equation [START_REF] Druet | Stability for strongly coupled critical elliptic systems in a fully inhomogeneous medium[END_REF], or equivalently (4).

This section is devoted to the proof of Proposition 5.1. We define the optimal Sobolev constant K n by (

n := inf u∈D 2 1 (R n )\{0} ∇u 2 u 2 ⋆ = n (n -2) ω 2/n n 4 , 46) 1 K
where ω n is the volume of the unit n-sphere: see Aubin [START_REF] Th | Problèmes isopérimétriques et espaces de Sobolev[END_REF], Talenti [START_REF] Talenti | Best constant in Sobolev inequality[END_REF], Rodemich [START_REF] Rodemich | The Sobolev inequalities with best possible constants[END_REF]. The infimum in (46) is achieved by the function U defined in [START_REF] Druet | Bounded stability for strongly coupled critical elliptic systems below the geometric threshold of the conformal Laplacian[END_REF].

Lemma 5.2. Let u 0 ∈ C 2 (M ) be a positive solution to [START_REF] Druet | Blow-up theory for elliptic PDEs in Riemannian geometry[END_REF]. Assume that either {h ∈ C 0,θ (M ) and 3 ≤ n ≤ 6} or {h ∈ C 2 (M ) and 3 ≤ n ≤ 9} or {(M, g) is locally conformally flat and h ≡ c n Scal g }. Then

(47) J ε u 0 -W δε(t),ξ = 1 n M u 2 * 0 dv g + ε 2 * M u 2 * 0 ln u 0 - 1 2 * dv g + K -n n n 1 -β n ε - n -2 2 ε ln ε - (n -2) 2 4 ε ln t + 2 n ω n-1 εt n-2 2 u 0 (ξ) (n (n -2)) n-2 4 ω n + 2 (n -1) ε 4 n-2 t 2 (n -2) (n -4) (h (ξ) -c n Scal g (ξ)) 1 n≥6 + o (ε)
as ε → 0, uniformly with respect to t in compact subsets of R >0 and with respect to the point ξ in M , where ω n (resp. ω n-1 ) is the volume of the unit n-sphere (resp.

(n -1)-sphere), K n is as in (46), and

β n = 2 n-3 (n -2) 2 ω n-1 ω n +∞ 0 r n-2 2 ln (1 + r) (1 + r) n dr + (n -2) 2 4n 1 -n ln n (n -2) . (48)
Proof. All our estimates in this proof are uniform with respect to t in compact subsets of R >0 , with respect to the point ξ in M , and with respect to ε in (0, ε 0 ) for some fixed positive real number ε 0 . Expanding J ε u 0 -W δε(t),ξ , using that u 0 is a solution to [START_REF] Chen | Infinitely many solutions for the Schrödinger equations in R N with critical growth[END_REF] and rough estimates yield

(49) J ε u 0 -W δε(t),ξ = 1 n M u 2 * 0 dv g + ε 2 * M u 2 * 0 ln u 0 - 1 2 * dv g + I 1,ε,t,ξ + I 2,ε,t,ξ -I 3,ε,t,ξ + O ε 2
when ε → 0 where

I 1,ε,t,ξ := 1 2 M ∇W δε(t),ξ 2
g dv g + 1 2 M hW 2 δε(t),ξ dv g - 1 2 * -ε M W 2 * -ε δε(t),ξ dv g , I 2,ε,t,ξ := M u 0 W 2 * -1-ε δε(t),ξ dv g , I 3,ε,t,ξ := M F ε u 0 -W δε(t),ξ -F ε (u 0 ) -F ε W δε(t),ξ + f ε (u 0 ) W δε(t),ξ + f ε W δε(t),ξ u 0 dv g .
We estimate these terms separately.

Step 1: Estimate of I 1,ε,t,ξ in the locally conformally case when h ≡ c n Scal g .

In case h ≡ c n Scal g and the manifold is locally conformally flat, the conformal change of metric

g ξ = Λ 4/(n-2) ξ g yields 1 2 M ∇W δε(t),ξ 2
g dv g + 1 2 M hW 2 δε(t),ξ dv g - 1 2 * -ε M W 2 * -ε δε(t),ξ dv g = 1 2 M ∇ W δε(t),ξ 2 g ξ dv g ξ - 1 2 * -ε M Λ -ε ξ W 2 * -ε δε(t),ξ dv g ξ ,
where W δε(t),ξ = W δε(t),ξ /Λ ξ . In this case, since the metric g ξ is flat in B ξ (r 0 ), we find

M ∇ W δε(t),ξ 2 g ξ dv g ξ = R n |∇U | 2 dx + O δ ε (t) n-2 = K -n n + O ε 2
when ε → 0. Moreover, since g ξ is flat around ξ, we get that

1 2 * -ε M Λ -ε ξ W 2 * -ε δε(t),ξ dv g ξ = (n (n -2)) n-2 4 (2 * -ε) 2 * -ε ω n-1 δ ε (t) n-2 2 ε r 0 2δε (t) 0 r n-1 dr (1 + r 2 ) n-2 2 (2 * -ε) + O (δ ε (t) n ) + O (εδ ε (t)) = n -2 2n K -n n 1 + 2β n n -2 ε + ε ln ε + n -2 2 ε ln t + O ε 2 |ln ε| 2 + O (εδ ε (t)) ,
where K n is as in (46), and β n is as in (48). Therefore, we get that (50)

I 1,ε,t,ξ = K -n n n 1 -β n ε - n -2 2 ε ln ε - (n -2) 2 4 ε ln t + o(ε)
when ε → 0 uniformly for all ξ ∈ M and t in a compact of (0, +∞) when h ≡ c n Scal g and (M, g) is locally conformally flat.

The asymptotic expansion (45) follows from Lemmas 5.2 and 5.3. Now, we prove the second part of Proposition 5.1.

End of proof of Proposition 5.1. Given two positive real numbers a < b, it remains to prove that for ε small, if (t ε , ξ ε ) ∈ [a, b] × M is a critical point of J ε , then the function u 0 -W δε(tε),ξε + φ δε(tε),ξε is a solution of equation [START_REF] Druet | Stability for strongly coupled critical elliptic systems in a fully inhomogeneous medium[END_REF]. In order to prove this claim, we consider a sequence of points (ξ α ) α in M and two sequences of positive real numbers (ε α ) α and (t α ) α such that ε α → 0 as α → +∞, a ≤ t α ≤ b, and (t α , ξ α ) is a critical point of J εα for all α. It is enough to show that for α large, the function u 0 -W δε α (tα),ξα + φ δε α (tα),ξα is a solution of equation [START_REF] Druet | Stability for strongly coupled critical elliptic systems in a fully inhomogeneous medium[END_REF]. As in the proof of Lemma 4.2, up to a subsequence, we identify the tangent space with R n around the ξ α 's. We define (64) Z 0,δε α (tα),ξα := Z δε α (tα),ξα and Z i,δε α (tα),ξα := Z δε α (tα),ξα,ei for all i = 1, . . . , n, where e i is the i-th vector in the canonical basis of R n and the functions Z δε α (tα),ξα and Z δε α (tα),ξα,ei are as in ( 17) and [START_REF] Hebey | Resonant states for the static Klein-Gordon-Maxwell-Proca system[END_REF]. By Proposition 4. , where the exponential map is taken with respect to the metric g ξα . On the one hand, direct computations yield

d dy i W δε α (tα),exp ξα y y=0 = n n-2 4 (n -2) n+2 4 δ εα (t) Z i,δε α (tα),ξα + R i,δε α (tα),ξα ,
where R i,δε α (tα),ξα → 0 as α → +∞ in H 2 1 (M ). For any i = 1, . . . , n, j = 0, . . . , n, and any α, since the function φ δε α (tα),ξα ∈ K ⊥ δε α (tα),ξα , differentiating the equation Z j,δε α (tα),exp ξα y , φ δε α (tα),exp ξα y h = 0 with respect to y i at 0 yields (n -2)

Z j,δε α (tα),
n+2 4 λ i,α ∇V i 2 + o   n j=0 |λ j,α |  
as α → +∞, where the function V i is as in [START_REF] Esposito | The effect of linear perturbations on the Yamabe problem[END_REF]. If (t α , ξ α ) is a critical point of J εα for all α, then it follows from (70) and (71) that for any i = 0, . . . , n, there λ i,α = 0 for all i = 0, . . . , n. By (65), if follows that for α large, the function u 0 -W δε α (tα),ξα + φ δε α (tα),ξα is a critical point of the functional J εα , and therefore a solution of equation ( 13). This ends the proof of Proposition 5.1.

Proof of the theorems

Proof of Theorems 1.2 and 2.2. We let G be the function defined on

R >0 × M by (72) G (t, ξ) := c 4 (n) ln 1 t + c 5 (n)t n-2 2 u 0 (ξ) ,
where c 4 (n) and c 5 (n) are as in (45). Since u 0 is positive and M is compact, we get

1 ε (J ε (t, ξ) -c 1 (n, u 0 ) -c 2 (n, u 0 )ε -c 3 (n)ε ln ε) = G (t, ξ)
uniformly with respect to t in compact subsets of R >0 and with respect to the point ξ in M . For ε small, by (73), (74), and by continuity of J ε and G, we get the existence of a family of points (t ε , ξ ε ) which realize the minimum values of the functions J ε in (a, b)×M for some positive real numbers a < b independent of ε. By Proposition 5.1, it follows that for ε small, the function u ε = u 0 -W δε(tε),ξε + φ δε(tε),ξε is a solution of equation ( 2), where W δε(t),ξ is as in ( 14) and φ δε(t),ξ is given by Proposition 4.1.

We get that lim ε→0 u ε = u 0 in H 2 1,loc. (M \ {ξ 0 }) where ξ 0 := lim ε→0 ξ ε (up to a subsequence): it then follows from standard elliptic theory that lim ε→0 u

ε = u 0 in C 2 loc (M \ {ξ 0 }). Independently, lim ε→0 δ ε (t ε ) n-2 2 u ε (exp ξǫ •) = -U in H 2 1,loc. (R n
), and still by elliptic theory, one then gets the convergence in C 2 loc (R n ). This proves that (u ε ) ε>0 changes sign and blows-up when ε → 0. This ends the proof of Theorems 1.2 and 2.2.

Proof of Theorems 1.1 and 2.3. In dimensions 3 ≤ n ≤ 5, the proof of Theorem 1.1 is similar to the proof of Theorem 1.2. The specificity of dimension n = 6, is that the function G in (72) is replaced by

G (t, ξ) := c 4 (6) ln 1 t + c 5 (6) u 0 (ξ) + 1 2 (h (ξ) -c 6 Scal g (ξ)) t 2 ,
where c 4 (6), c 5 (6) > 0 are as in (45): therefore (73) holds with the hypothesis of Theorem 1.1 and the proof of Theorem 1.1 goes as for Theorem 1.2. We focus on Theorem 2.3. In dimension n = 6, computations similar to (45) yield

J + ε u 0 + W δε(t),ξ = c 1 (6, u 0 ) + c 2 (6, u 0 )ε + c 3 (6)ε ln ε + c 4 (6) ln 1 t + c 5 (6) 1 2 (h (ξ) -c 6 Scal g (ξ)) -u 0 (ξ) t 2 ε + o (ε) as ε → 0, where J + ε (u) := 1 2 M |∇u| 2 g dv g + 1 2 M hu 2 dv g -1 2 ⋆ -ε M u 2 ⋆ -ε + dv g .
The proof then is similar to the proof of Theorem 1.1.

Proof of Theorem 2.1. The introduction of another type of model for blow-up is required here. It follows from Lee-Parker [START_REF] Lee | The Yamabe problem[END_REF] that for any ξ ∈ M , there exists

Λ ξ ∈ C ∞ (M ) positive such that g ξ := Λ 4 n-2 ξ g satisfies dv g ξ = (1+O(d g ξ (ξ, •) n )) dx in a ge- odesic normal chart. An immediate consequence is that Scal g ξ (ξ) = |∇ Scal g ξ (ξ)| = 0 and ∆ g ξ Scal g ξ (ξ) = 1 6 | Weyl g (ξ)| 2
g . Moreover, we can assume that (ξ, x) → Λ ξ (x) is C ∞ and ∇Λ ξ (ξ) = 0. We define W δ,ξ in ( 14 when ε → 0 for n ≥ 7. When n < 10, the term involving the Weyl tensor is neglictible. When n = 10, it competes with the one involving u 0 : arguing as in the proofs above, we get the existence of a blowing-up family when u 0 > 5 567 | Weyl g | 2 g , which proves Theorem 2.1 since the additional terms involving φ δε(t),ξ are neglictible when n ≤ 17. When n > 10, the Weyl tensor dominates but the negative sign does not allow to construct a critical point for the reduced functional.

Proof of Theorem 2.4. If ξ 0 ∈ M is a strict minimizer of Φ on B ξ0 (ν 0 ) ⊂ M with ν 0 > 0, the arguments above extend by minimizing G on (0, +∞) × B ξ0 (ν 0 ).

Error estimate

This section is devoted to the error estimate used in previous sections. All notations refer to Section 4. The estimate is as follows: i * f ε u 0 -W δε(t),ξ -u 0 + W δε(t),ξ h

≤ C ′ a,b        ε |ln ε| if n ≤ 6 ε 4 n-2 if n ≥ 7 ε n+2 2(n-2)
if n ≥ 7, h ≡ c n Scal g , and (M, g) loc. conformally flat, where δ ε (t) = tε 2/(n-2) and W δε(t),ξ is as in [START_REF] Druet | Blow-up theory for elliptic PDEs in Riemannian geometry[END_REF].

Proof. All our estimates in this proof are uniform with respect to t in [a, b], ξ in M , and ε in (0, ε 0 ) for some fixed positive real number ε 0 . The continuity of i * yields i * f ε u 0 -W δε(t),ξ -u + W δε(t),ξ h = O f ε u 0 -W δε(t),ξ -(∆ g + h) u 0 -W δε(t),ξ 2n n+2 .

It follows that (77)

i * f ε u 0 -W δε(t),ξ -u 0 + W δε(t),ξ h = O Ĩ1,ε,t,ξ + Ĩ2,ε,t,ξ + Ĩ3,ε,t,ξ , where Ĩ1,ε,t,ξ := f ε u 0 -W δε(t),ξ -f ε (u 0 ) + f ε W δε(t),ξ 2n n+2 , Ĩ2,ε,t,ξ := f ε (u 0 ) -∆ g u 0 -hu 0 2n n+2 , Ĩ3,ε,t,ξ := f ε W δε(t),ξ -∆ g W δε(t),ξ -hW δε(t),ξ 2n n+2 .

We estimate these terms separately.

Step if n ≥ 7 when ε → 0.

Step 2: Estimate of Ĩ2,ε,t,ξ . Since u 0 is a solution of (7), we get that (79) Ĩ2,ε,t,ξ = f ε (u 0 ) -f 0 (u 0 ) 2n n+2 = O (ε) .

Step 3: Estimate of Ĩ3,ε,t,ξ . We define χ ξ (•) = χ d g ξ (•, ξ) , U δ,ξ (•) = δ 2-n 2 U (δ -1 exp -1 ξ (•)), where the function U is as in [START_REF] Druet | Bounded stability for strongly coupled critical elliptic systems below the geometric threshold of the conformal Laplacian[END_REF] and the exponential map is taken with respect to the metric g ξ .

Step 3.1: Estimate of I 3,ε,t,ξ when (M, g) is locally conformally flat and h ≡ c n Scal g . Since g ξ = Λ 4/(n-2) ξ g is flat, we get that

f ε W δε(t),ξ -∆ g W δε(t),ξ -hW δε(t),ξ = Λ 2 * -1 ξ Λ -ε ξ f ε W δε(t),ξ -∆ g ξ W δε(t),ξ ,
where W δε(t),ξ = W δε(t),ξ /Λ ξ . In this case, since the metric g ξ is flat in B ξ (r 0 ) and since the function U is a solution of the equation ∆ Eucl U = U 2 * -1 in R n , we get that (80) I 3,ε,t,ξ ≤ (χ ξ Λ ξ ) 2 * -1-ε U 2 * -1-ε δε(t),ξ -U 2 * -1 δε(t),ξ .

2n n+2 + χ 2 * -1-ε ξ Λ -ε ξ -χ ξ Λ 2 * -1 ξ U

G

  (t, ξ) = +∞ and lim t→+∞ G (t, ξ) = +∞ uniformly with respect to ξ ∈ M . Since h ≡ c n Scal g and either {3 ≤ n ≤ 9} or {(M, g) is locally conformally flat}, it follows from Proposition 5.1 that (74) lim ε→0

+ 2 n

 2 ) with the function Λ ξ above. When h ≡ c n Scal g , the conformal law of change of metric yields the Taylor expansion (75)J ε u 0 -W δε(t),ξ = c 1 (n, u 0 ) + c 2 (n, u 0 )ε + c 3 (n)ε ln ε + K ω n-1 ω n (n(n -2)) (n-2)/4 u(ξ)εt

Lemma 7 . 1 .

 71 Given two positive real numbers a < b, there exists a positive constant C ′ a,b such that for ε small, for any real number t in [a, b], and any point ξ in M , there holds (76)

  DJ εα u 0 -W δε α (tα),ξα + φ δε α (tα),ξα = Z i,δε α (tα),ξα , • h for some real numbers λ i,α , where the functions Z i,δε α (tα),ξα are as in (64).On the other hand, for any i = 0, . . . , n and any α, since the function φ δε α (tα),ξα belongs to K ⊥ δε α (tα),ξα , differentiating Z i,δε α (t),ξα , φ δε α (t),ξα h = 0 with respect to +∞, where the function V 0 is as in[START_REF] Esposito | The effect of linear perturbations on the Yamabe problem[END_REF]. For any i = 1, . . . , n, by (65), we get that

	as α → d dy i J εα t α , exp ξα y		y=0
		=	n j=0	λ j,α Z j,δε α (tα),ξα ,	d dy i	-W δε α (tα),exp ξα y + φ δε α (tα),exp ξα y	y=0 h
													1,
	we get that
													n
	(65)												i=0	λ i,α It
	follows from (65) that
	(66)	∂J εα ∂t	(t α , ξ α ) =	n i=0	λ i,α Z i,δε α (tα),ξα ,	d dt	-W δε α (t),ξα + φ δε α (t),ξα	t=tα	h .
	On the one hand, we find
	(67)				d dt	W δε α (t),ξα	t=tα	=	n	n-2 4	(n -2) 2t α	n+2 4	Z 0,δε α (tα),ξα .
	t yields
	(68)	Z i,δε α (tα),ξα ,	d dt	φ δε α (t),ξα	t=tα h	= -	d dt	Z i,δε α (t),ξα	t=tα	, φ δε α (tα),ξα	h	.
	Moreover, one easily checks
	(69)										d dt	Z i,δε α (t),ξα	t=tα h	= O (1)
	as α → +∞. Proposition 4.1, (68), (69), (29), (66), and (67) yield
	(70)		∂J εα ∂t	(t α , ξ α ) = -	n	n-2 4	(n -2) 2t α	n+2 4	λ 0,α ∇V 0	2 2 + o	n i=0	|λ i,α |

  1: Estimate of Ĩ1,ε,t,ξ . We get Ĩ1,ε,t,ξ ≤ f ε u 0 -W δε(t),ξ + f ε W δε(t),ξ 1 B ξ ( √ δε(t)) 2n n+2 + f ε u 0 -W δε(t),ξ -f ε (u 0 ) 1 M \B ξ ( √As is easily checked, Taylor's expansion for f ε u 0 -W δε(t),ξ yieldsf ε u 0 -W δε(t),ξ + f ε W δε(t),ξ 1 B ξ ( √ -W δε(t),ξ -f ε (u 0 ) 1 M \B ξ ( √

										δε(t)) 2n
										n+2
		≤ C	u 0 W 2 * -2-ε δε(t),ξ 1 B ξ (	√	δε(t)) 2n n+2	+ u 2 * -1-ε 0	1 B ξ (	√	δε(t)) 2n n+2
	and
	f ε u 0 δε(t)) 2n
										n+2
	≤ C	u 2 * -2-ε 0	W δε(t),ξ 1 M \B ξ (	√	δε(t)) 2n n+2	+ W 2 * -1-ε δε(t),ξ 1 M \B ξ (	√	n+2 δε(t)) 2n	.
	Estimating roughly these terms yields
	(78)			Ĩ1,ε,t,ξ =	        	O (ε) O ε |ln ε| n+2 O ε 2(n-2) 2 3	if n ≤ 5 if n = 6
										δε(t)) 2n

n+2 + f ε W δε(t),ξ 1 M \B ξ ( √ δε(t)) 2n n+2 + f ε (u 0 ) 1 B ξ ( √ δε(t)) 2n

n+2

.

Step 2: Estimate of I 1,ε,t,ξ in the general case. Cartan's expansion of the metric in geodesic normal coordinates yields for any α, β = 1, . . . , n and for y close to 0, there holds (51)

R µν (ξ) y µ y ν + P 3 (y) + O |y| 4 ,

where the function |g| is the determinant of the metric, the functions R µν are the components of the Ricci curvature tensor in geodesic normal coordinates associated with the map exp ξ and P 3 (y) is a homogenous polynomial of degree three. Using (51) together with expression of the gradient of a radially symmetrical function in geodesic normal chart, we get that

when ε → 0. Taylor's expansion at ξ yields on the one hand

when ε → 0, where

Plugging together (52), (54), and (53) yields

Step

when ε → 0, where K n is as in (46), and β n is as in (48). We have used here that

∆ Eucl U dx and integrated by parts.

Step 4: Estimate of I 3,ε,t,ξ .

We have that

As is easily checked, Taylor expansions of F (u 0 -W δε(t),ξ ) yield

Bounding u 0 and W δε(t),ξ pointwisely roughly from above in (58) and ( 59) and plugging this in (57) yields (60)

Step 5: End of proof of Lemma 5.2. The asymptotic expansion (47) follows from (49), ( 50), ( 55), ( 56) and (60).

In Lemma 5.3 below, we show that the first order terms in the asymptotic expansion of J ε (t, ξ), defined in (44), are the same as for J ε u 0 -W δε(t),ξ . Lemma 5.3. Assume that either {3 ≤ n ≤ 9} or {(M, g) is locally conformally flat and h ≡ c n Scal g }. Then

as ε → 0, uniformly with respect to t in compact subsets of R >0 and with respect to the point ξ in M .

Proof. All the estimates in this proof are uniform with respect to t on compact subsets of R >0 , with respect to the point ξ in M , and with respect to ε in (0, ε 0 ) for some fixed positive real number ε 0 . We get that

when ε → 0. Proposition 4.1 and Lemma 7.1 yield

if n ≥ 7, h ≡ c n Scal g , (M, g) loc. conformally flat when ε → 0. Finally, (61) follows from (62), and (63).

Step 3.2: Estimate of I 3,ε,t,ξ in the general case. In general, we get that (81)

Step 3.3: Estimates of the terms in (80) and (81).

Since χ ξ ≡ 1 on B ξ (r 0 /2) and χ ξ ≡ 0 on M \ B ξ (r 0 ), we get that

where the g ij 's are the coordinate of the metric g = g ξ and Γ γ αβ 's are the Christoffel symbols of the metric g in the normal chart exp ξ . Cartan's expansion of the metric yields |g ij (x) -δ ij | ≤ C|x| 2 and |Γ k ij (x)| ≤ C|x| around 0, and therefore when ε → 0.

Step 4: End of proof of (76). Finally, (76) follows from (77), ( 78), ( 79), (88), and (89).