F R Éd

Éric Robert

J Ér Ôme

V Étois

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

Jérôme Vétois
email: jerome.vetois@mcgill.ca

BLOWING-UP SOLUTIONS FOR SECOND-ORDER CRITICAL ELLIPTIC EQUATIONS: THE IMPACT OF THE SCALAR CURVATURE

Keywords: . MSC: 58J05, 35J20, 35J60, 53C21. compactness

 proved that a necessary condition for the existence of energy-bounded blowing-up solutions to perturbations of the equation

touches the Scalar curvature somewhere when n ≥ 4 (the condition is different for n = 6). In this paper, we prove that Druet's condition is also sufficient provided we add its natural differentiable version. For n ≥ 6, our arguments are local. For the low dimensions n ∈ {4, 5}, our proof requires the introduction of a suitable mass that is defined only where Druet's condition holds. This mass carries global information both on h 0 and (M, g).

Introduction and main results

Let (M, g) be a compact Riemannian manifold of dimension n ≥ 3 without boundary and h 0 ∈ C p (M ), 1 ≤ p ≤ ∞. We consider the equation [START_REF] Th | Equations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire[END_REF] ∆ g u + h 0 u = u 2 -1 , u > 0 in M, where ∆ g := -div g (∇) is the Laplace-Beltrami operator and 2 := 2n n-2 . We investigate the existence of families (h ) >0 ∈ C p (M ) and (u ) >0 ∈ C 2 (M ) satisfying [START_REF] Bianchi | A note on the Sobolev inequality[END_REF] ∆ g u + h u = u 2 -1 , u > 0 in M for all > 0, and such that h → h 0 in C p (M ) and max M u → ∞ as → 0. We say that (u ) blows up at some point ξ 0 ∈ M as → 0 if for all r > 0, lim →0 max Br(ξ0) u = +∞. Druet [START_REF] Druet | From one bubble to several bubbles: the low-dimensional case[END_REF][START_REF]La notion de stabilité pour des équations aux dérivées partielles elliptiques[END_REF] obtained the following necessary condition for blow-up:

Theorem 1.1 (Druet [START_REF] Druet | From one bubble to several bubbles: the low-dimensional case[END_REF][START_REF]La notion de stabilité pour des équations aux dérivées partielles elliptiques[END_REF]). Let (M, g) be a compact Riemannian manifold of dimension n ≥ 4. Let h 0 ∈ C 1 (M ) be such that ∆ g + h 0 is coercive. Assume that there exist families (h ) >0 ∈ C 1 (M ) and (u ) >0 ∈ C 2 (M ) satisfying (2) and such that h → h 0 strongly in C 1 (M ) and u u 0 weakly in L 2 (M ). Assume that (u ) blows-up. Then there exists ξ 0 ∈ M such that (u ) blows up at ξ 0 and

(3) (h 0 -c n Scal g ) (ξ 0 ) = 0 if n = 6 and (h 0 -c n Scal g -2u 0 ) (ξ 0 ) = 0 if n = 6.

Furthermore, if n ∈ {4, 5}, then u 0 ≡ 0.

Here c n := n-2 4(n-1) and Scal g is the Scalar curvature of (M, g). This result does not hold in dimension n = 3. Indeed, Hebey-Wei [START_REF] Hebey | Resonant states for the static Klein-Gordon-Maxwell-Proca system[END_REF] constructed examples of blowing-up solutions to (2) on the standard sphere (S 3 , g 0 ), which are bounded in L 2 (S 3 ) but do not satisfy [START_REF] Brendle | Blow-up phenomena for the Yamabe equation[END_REF]. This paper is concerned with the converse of Theorem 1.1 in dimensions n ≥ 4. For the sake of clarity, we state separately our results in the cases u 0 ≡ 0 in dimension n ≥ 4 (Theorem 1.2) and u 0 > 0 in dimension n ≥ 6 (Theorem 1.3): Theorem 1.2. Let (M, g) be a compact Riemannian manifold of dimension n ≥ 4. Let h 0 ∈ C p (M ), 1 ≤ p ≤ ∞, be such that ∆ g + h 0 is coercive. Assume that there exists a point ξ 0 ∈ M such that

(4) (h 0 -c n Scal g ) (ξ 0 ) = |∇ (h 0 -c n Scal g ) (ξ 0 )| = 0.
Then there exist families (h ) >0 ∈ C p (M ) and (u ) >0 ∈ C 2 (M ) satisfying (2) and such that h → h 0 strongly in C p (M ), u 0 weakly in L 2 (M ) and (u ) blows up at ξ 0 .

For convenience, for every h 0 , u 0 ∈ C 0 (M ), we define [START_REF] Chen | Infinitely many solutions for the Schrödinger equations in R n with critical growth[END_REF] ϕ h0 := h 0 -c n Scal g and ϕ h0,u0 := h 0 -c n Scal g if n = 6 h 0 -2u 0 -c n Scal g if n = 6.

Theorem 1.3. Let (M, g) be a compact Riemannian manifold of dimension n ≥ 6.

Let h 0 ∈ C p (M ), 1 ≤ p ≤ ∞, be such that ∆ g + h 0 is coercive. Assume that there exist a solution u 0 ∈ C 2 (M ) of (1) and a point ξ 0 ∈ M such that (6) ϕ h0,u0 (ξ 0 ) = |∇ϕ h0,u0 (ξ 0 )| = 0.

Then there exist families (h ) >0 ∈ C p (M ) and (u ) >0 ∈ C 2 (M ) satisfying (2) and such that h → h 0 strongly in C p (M ), u u 0 weakly in L 2 (M ) and (u ) blows up at ξ 0 .

Compared with Theorem 1.1, we have assumed here that condition [START_REF] Brendle | Blow-up phenomena for the Yamabe equation[END_REF] is also satisfied at order 1. However, this stronger condition is actually expected to be necessary for the existence of blowing-up solutions (see Theorem 14.1 in the last section of this paper and the discussion in Druet [START_REF]La notion de stabilité pour des équations aux dérivées partielles elliptiques[END_REF]Section 2.5]). Note that we do not make any nondegeneracy assumptions, neither on the solution u 0 , nor on the critical point ξ 0 .

We refer to Section 2 for examples of functions h 0 and u 0 satisfying the assumptions of Theorem 1.3. Recently, Premoselli-Thizy [START_REF] Premoselli | Bubbling above the threshold of the scalar curvature in dimensions four and five[END_REF] obtained a beautiful example of blowing-up solutions showing that in dimension n ∈ {4, 5}, condition (4) may not be satisfied at all blow-up points.

When h 0 ≡ c n Scal g , that is when [START_REF] Th | Equations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire[END_REF] is the Yamabe equation, several examples of blowing-up solutions have been obtained. In the perturbative case, that is when h ≡ c n Scal g , examples of blowing-up solutions have been obtained by Druet-Hebey [START_REF] Druet | Blow-up examples for second order elliptic PDEs of critical Sobolev growth[END_REF], Esposito-Pistoia-Vétois [START_REF] Esposito | The effect of linear perturbations on the Yamabe problem[END_REF], Morabito-Pistoia-Vaira [START_REF] Morabito | Towering phenomena for the Yamabe equation on symmetric manifolds[END_REF], Pistoia-Vaira [START_REF] Pistoia | Clustering phenomena for linear perturbation of the Yamabe equation, Partial Differential Equations Arising from Physics and Geometry[END_REF] and Robert-Vétois [START_REF]Examples of non-isolated blow-up for perturbations of the scalar curvature equation[END_REF]. In the nonpertubative case h ≡ c n Scal g , we refer to 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] regarding the non-compactness of Yamabe metrics. When solutions blow-up not only pointwise but also in energy, the function ϕ h0 may not vanish (see Chen-Wei-Yan [START_REF] Chen | Infinitely many solutions for the Schrödinger equations in R n with critical growth[END_REF] for n ≥ 5 and Vétois-Wang [START_REF] Vétois | Infinitely many solutions for cubic nonlinear Schrödinger equations in dimension four[END_REF] for n = 4).

When there does not exist any blowing-up solutions to the equations (2), then equation ( 1) is stable. We refer to the survey of Druet [START_REF]La notion de stabilité pour des équations aux dérivées partielles elliptiques[END_REF] and the book of Hebey [START_REF] Hebey | Compactness and stability for nonlinear elliptic equations[END_REF] for exhaustive studies of the various concepts of stability. Stability also arises in the Lin-Ni-Takagi problem (see for instance del Pino-Musso-Roman-Wei [START_REF] Del Pino | Interior bubbling solutions for the critical Lin-Ni-Takagi problem in dimension 3[END_REF] for a recent reference on this topic). In Geometry, stability is linked to the problem of Theorem 1. [START_REF] Brendle | Blow-up phenomena for the Yamabe equation. II[END_REF]. Let (M, g) be a compact Riemannian manifold of dimension n ≥ 4. Let h 0 ∈ C p (M ), p ≥ 2, be such that ∆ g +h 0 is coercive. Assume that there exists a point ξ 0 ∈ M satisfying [START_REF] Brendle | Blow-up phenomena for the Yamabe equation. II[END_REF]. Assume in addition that ξ 0 is a nondegenerate critical point of h 0 -c n Scal g and

(7) K h0 (ξ 0 ) :=    m h0 (ξ 0 ) if n = 4, 5 ∆ g (h 0 -c n Scal g ) (ξ 0 ) + c n 6 | Weyl g (ξ 0 )| 2 g if n ≥ 6    = 0,
where m h0 (ξ 0 ) is the mass of ∆ g +h 0 at the point ξ 0 (see Proposition-Definition 8.1), and Weyl g is the Weyl curvature tensor of the manifold. We fix a function f ∈ C p (M ) such that f (ξ 0 ) × K h0 (ξ 0 ) > 0. Then for small > 0, there exists u ∈ C 2 (M ) satisfying

(8) ∆ g u + (h 0 + f )u = u 2 -1 in M, u > 0,
and such that u 0 weakly in L 2 (M ) and (u ) blows up with one bubble at ξ 0 .

The definition of K h0 (ξ 0 ) outlines the major difference between high-and lowdimensions that was mentioned above: for n ≥ 6, it is a local quantity, but for n ∈ {4, 5}, it carries global information (see Section 8 for more discussions).

Next we deal with the case where u 0 > 0 in dimension n ≥ 6:

Theorem 1.5. Let (M, g) be a compact Riemannian manifold of dimension n ≥ 6. Let h 0 ∈ C p (M ), p ≥ 2, be such that ∆ g + h 0 is coercive. Assume that there exist a nondegenerate solution u 0 ∈ C 2 (M ) to equation (1) and ξ 0 ∈ M satisfying [START_REF] Del Pino | Interior bubbling solutions for the critical Lin-Ni-Takagi problem in dimension 3[END_REF]. Assume in addition that ξ 0 is a nondegenerate critical point of ϕ h0,u0 and (9)

K h0,u0 (ξ 0 ) :=                  ∆ g ϕ h0,u0 (ξ 0 ) + c 6 6 | Weyl g (ξ 0 )| 2 g if n = 6 u 0 (ξ 0 ) if 7 ≤ n ≤ 9 672u 0 (ξ 0 ) + ∆ g ϕ h0,u0 (ξ 0 ) + c 10 6 | Weyl g (ξ 0 )| 2 g if n = 10 ∆ g ϕ h0,u0 (ξ 0 ) + c n 6 | Weyl g (ξ 0 )| 2 g if n ≥ 11                  = 0.
We fix a function f ∈ C p (M ) such that

(10) K h0,u0 (ξ 0 ) × f + 2(∆ g + h 0 -2u 0 ) -1 (f u 0 ) (ξ 0 ) if n = 6 f (ξ 0 ) if n > 6 > 0.
Then for small > 0, there exists u ∈ C 2 (M ) satisfying [START_REF]Compactness for Yamabe metrics in low dimensions[END_REF] and such that u u 0 weakly in L 2 (M ) and (u ) blows up with one bubble at ξ 0 .

The paper is organized as follows. In Section 2, we discuss the question of existence of functions h 0 and u 0 satisfying the assumptions of Theorem 1.3. In Section 3, we introduce our notations and discuss the general setting of the problem. In Section 4, we establish a general C 1 -estimate on the energy functional, which holds in all dimensions. In Sections 5, 6 and 7, we then compute a C 1 -asymptotic expansion of the energy functional in the case where n ≥ 6, which we divide in the following subcases: [n ≥ 6 and u 0 ≡ 0] in Section 5, [n ≥ 7 and u 0 > 0] in Section 6 and [n = 6 and u 0 > 0] in Section 7. In Section 8, we discuss the specific setting of dimensions n ∈ {4, 5} and we define the mass of ∆ g + h 0 in this case. In Section 9, we then deal with the C 1 -asymptotic expansion of the energy functional when n ∈ {4, 5}. In Sections 10, 11, 12 and 13, we complete the proofs of Theorems 1.4, 1.5, 1.2 and 1.3, respectively. Finally, in Section 14, we deal with the necessity of condition (4) on the gradient 2. Existence results for h 0 and u 0 This short section deals with two results which provide conditions for the existence of functions h 0 and u 0 satisfying the assumptions of Theorem 1.3 with prescribed ϕ h0,u0 and ξ 0 . The first result is a straightforward consequence of classical works on the Yamabe equation: Theorem 2.1. (Aubin [START_REF] Th | Equations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire[END_REF], Schoen [START_REF] Schoen | Conformal deformation of a Riemannian metric to constant scalar curvature[END_REF], Trudinger [START_REF] Trudinger | Remarks concerning the conformal deformation of Riemannian structures on compact manifolds[END_REF]) Assume that n ≥ 3. Then there exists 0 ≥ 0 depending only on n and (M, g) such that 0 > 0 if (M, g) is not conformally diffeomorphic to the standard sphere, 0 = 0 otherwise, and for every ϕ 0 ∈ C 1 (M ) such that ϕ 0 ≤ 0 and λ 1 (∆ g + h 0 ) > 0, where h 0 := ϕ 0 + c n Scal g , there exists a solution u 0 ∈ C 2 (M ) of the equation (1). In particular, if n = 6 and ϕ 0 (ξ 0 ) = |∇ϕ 0 (ξ 0 )| = 0 at some point ξ 0 ∈ M , then h 0 satisfies [START_REF] Del Pino | Interior bubbling solutions for the critical Lin-Ni-Takagi problem in dimension 3[END_REF].

It remains to deal with the case where n = 6. In this case, we obtain the following:

Proposition 2.1. Assume that n = 6. Let ϕ 0 ∈ C p (M ), 1 ≤ p ≤ ∞, be such that (11) λ 1 (∆ g + ϕ 0 + c n Scal g ) < 0.
Then there exists h 0 ∈ C p (M ) such that the equation (1) admits a solution u 0 ∈ C 2 (M ) satisfying h 0 -c n Scal g -2u 0 ≡ ϕ 0 . In particular, if ϕ 0 (ξ 0 ) = |∇ϕ 0 (ξ 0 )| = 0 at some point ξ 0 ∈ M , then (h 0 , u 0 ) satisfy [START_REF] Del Pino | Interior bubbling solutions for the critical Lin-Ni-Takagi problem in dimension 3[END_REF].

Proof of Proposition 2.1. Note that 2 -1 = 2 when n = 6. In this case, we can rewrite the equation ( 1) as

(12) ∆ g u + (h 0 -2u)u = -u 2 , u > 0 in M.
Using [START_REF] Druet | Blow-up theory for elliptic PDEs in Riemannian geometry[END_REF] together with a standard variational method, we obtain that there exists a solution u 0 ∈ C p+1 (M ) ⊂ C 2 (M ) of the equation ( 12) with h 0 := ϕ 0 + c n Scal g +2u 0 ∈ C p (M ). This ends the proof of Proposition 2.1.

Notations and general setting

We follow the notations and definitions of Robert-Vétois [START_REF] Robert | A general theorem for the construction of blowing-up solutions to some elliptic nonlinear equations via Lyapunov-Schmidt's reduction[END_REF].

3.1. Euclidean setting. We define

(13) U 1,0 (x) := n(n -2) 1 + |x| 2 n-2 2
for all x ∈ R n , so that U 1,0 is a positive solution to the equation

∆ Eucl U = U 2 -1 in R n ,
where Eucl stands for the Euclidean metric. For every δ > 0 and ξ ∈ R n , we define

(14) U δ,ξ (x) := δ -n-2 2 U δ -1 (x -ξ) = n(n -2)δ δ 2 + |x -ξ| 2 n-2 2
for all x ∈ R n .

We define (15)

Z 0 := (∂ δ U δ,ξ ) |(1,0) and Z i := (∂ ξi U δ,ξ ) |(1,0) for all i = 1, . . . , n.
As one checks, ( 16)

Z 0 = - n -2 2 U -(x, ∇U ) = n(n -2) n-2 2 n -2 2 |x| 2 -1 (1 + |x| 2 ) n 2 and (17) Z i = -∂ xi U = n(n -2) n-2 2 (n -2) x i (1 + |x| 2 ) n 2
for all i = 1, . . . , n.

We denote

p = (p 0 , p 1 , . . . , p n ) := (δ, ξ) ∈ (0, ∞) × R n . Straightforward computa- tions yield (18) ∂ pi U δ,ξ = δ -1 (Z i ) δ,ξ := δ -1 δ -n-2 2 Z i δ -1 (x -ξ) for all i = 0, . . . , n, (19)
∂ δ U δ,ξ = n(n -2) n-2 2 n -2 2 δ n-2 2 -1 |x -ξ| 2 -δ 2 (δ 2 + |x -ξ| 2 ) n/2 and (20) ∂ ξi U δ,ξ = n(n -2) n-2 2 (n -2)δ n-2 2 (x -ξ) i (δ 2 + |x -ξ| 2 ) n/2 for all i = 1, . . . , n.
It follows from Rey [START_REF] Rey | The role of the Green's function in a nonlinear elliptic equation involving the critical Sobolev exponent[END_REF] (see also Bianchi-Egnell [START_REF] Bianchi | A note on the Sobolev inequality[END_REF]) that

{φ ∈ D 2 1 (R n ) : ∆ Eucl φ = (2 -1)U 2 -2 φ in R n } = Span{Z i } i=0,...,n , where D 2 1 (R n ) is the completion of C ∞ c (R n ) for u → ∇u 2 . 3.2.
Riemannian setting. We fix N > n -2 to be chosen large later. It follows from Lee-Parker [START_REF] Lee | The Yamabe problem[END_REF] that there exists a function Λ ∈ C ∞ (M × M ) such that, defining Λ ξ := Λ(ξ, •), we have [START_REF] Marques | A priori estimates for the Yamabe problem in the non-locally conformally flat case[END_REF] Λ ξ > 0, Λ ξ (ξ) = 1 and ∇Λ ξ (ξ) = 0 for all ξ ∈ M and, defining the metric g ξ := Λ 2 -2 ξ g conformal to g, we have

(22) Scal g ξ (ξ) = 0, ∇ Scal g ξ (ξ) = 0, ∆ g Scal g ξ (ξ) = 1 6 | Weyl g (ξ)| 2
g and ( 23)

dv g ξ (x) = (1 + O(|x| N )) dx via the chart exp g ξ
ξ around 0, where dx is the Euclidean volume element, dv g ξ is the Riemannian volume element of (M, g ξ ) and exp g ξ ξ is the exponential chart at ξ with respect to the metric g ξ . The compactness of M yields the existence of r 0 > 0 such that the injectivity radius of the metric g ξ satisfies i g ξ (M ) ≥ 3r 0 for all ξ ∈ M . We let χ ∈ C ∞ (R) be such that χ(t) = 1 for all t ≤ r 0 , χ(t) = 0 for all t ≥ 2r 0 and 0 ≤ χ ≤ 1. For every δ > 0 and ξ ∈ M , we then define the bubble as

U δ,ξ (x) : = χ(d g ξ (x, ξ))Λ ξ (x)δ -n-2 2 U 1,0 (δ -1 (exp g ξ ξ ) -1 (x)) (24) = χ(d g ξ (x, ξ))Λ ξ (x) δ n(n -2) δ 2 + d g ξ (x, ξ) 2 n-2 2
, where d g ξ (x, ξ) is the geodesic distance between x and ξ with respect to the metric g ξ . Since there will never be ambiguity, to avoid unnecessary heavy notations, we will keep the notations U δ,ξ as ( 14) when p = (δ, ξ) ∈ (0, ∞) × R n , and as [START_REF] Pistoia | Clustering phenomena for linear perturbation of the Yamabe equation, Partial Differential Equations Arising from Physics and Geometry[END_REF] when p = (δ, ξ) ∈ (0, ∞) × M . Finally, for every p = (δ, ξ) ∈ (0, ∞ × M , we define

K δ,ξ := Span{(Z i ) δ,ξ } i=0,...,n , where (Z i ) δ,ξ (x) := χ(d g ξ (x, ξ))Λ ξ (x)δ -n-2 2 Z i (δ -1 (exp g ξ ξ ) -1 (x)
) for all x ∈ M and i = 0, . . . , n.

General reduction theorem.

For every 1 ≤ q ≤ ∞, we let • q be the usual norm of L q (M ). For every h ∈ C 0 (M ), we define

J h (u) := 1 2 M |∇u| 2 g + hu 2 dv g - 1 2 M u 2 + dv g for all u ∈ H 2 1 (M ),
where u + := max(u, 0). The space H 2 1 (M ) is endowed with the bilinear form

•, • h , where u, v h := M (∇u∇v + huv) dv g for all u, v ∈ H 2 1 (M ). If ∆ g + h 0 is coercive and h -h 0 ∞ is small enough, then •, • h is positive definite and (H 2 1 (M ), •, h ) is a Hilbert space. We then have that J h ∈ C 1 (H 2 1 (M )) and J h (u)[φ] = M (∇u∇φ + huφ) dv g - M u 2 -1 + φ dv g = u, φ h - M u 2 -1 + φ dv g for all u, φ ∈ H 2 1 (M ). We let (δ, ξ) → B h,δ,ξ = B h (δ, ξ) be a function in C 1 ((0, ∞)× M, H 2 1 (M )
) such that for every δ > 0, there exists (δ) > 0 independent of h and ξ such that

(25) B h,δ,ξ H 2 1 + δ ∂ p B h,δ,ξ H 2 1 < (δ) for all p = (δ, ξ) ∈ (0, ∞) × M
and (δ) → 0 as δ → 0. The function B h,δ,ξ will be fixed later. We also let ũ0 ∈ C 2 (M ). We define

W h,ũ0,δ,ξ := ũ0 + U δ,ξ + B h,δ,ξ .
We fix a point ξ 0 ∈ M and a function h 0 ∈ C 0 (M ) such that ∆ g + h 0 is coercive. We let u 0 ∈ C 2 (M ) be a solution of the equation

∆ g u 0 + h 0 u 0 = u 2 -1 0 , u 0 ≥ 0 in M.
It follows from the strong maximum principle that either u 0 ≡ 0 or u 0 > 0. We assume that u 0 is nondegenerate, that is, for every φ ∈ H 2 1 (M ),

∆ g φ + h 0 φ = (2 -1)u 2 -2 0 φ ⇐⇒ φ ≡ 0.
It then follows from Robert-Vétois [START_REF] Robert | A general theorem for the construction of blowing-up solutions to some elliptic nonlinear equations via Lyapunov-Schmidt's reduction[END_REF] that there exist 0 > 0,

U 0 ⊂ M a small open neighborhood of ξ 0 and Φ h,ũ0 ∈ C 1 ((0, 0 ) × U 0 , H 2 1 (M )) such that, when h -h 0 ∞ < 0 and ũ0 -u 0 C 2 < 0 , we have (26) Π K ⊥ δ,ξ (W h,ũ0,δ,ξ + Φ h,ũ0,δ,ξ -(∆ g + h) -1 ((W h,ũ0,δ,ξ + Φ h,ũ0,δ,ξ ) 2 -1 + )) = 0 and (27) Φ h,ũ0,δ,ξ H 2 1 ≤ C W h,ũ0,δ,ξ -(∆ g +h) -1 ((W h,ũ0,δ,ξ ) 2 -1 + ) H 2 1 ≤ C R δ,ξ 2n n+2
for all (δ, ξ) ∈ (0, 0 ) × U 0 , where C > 0 does not depend on (h, ũ0 , δ, ξ), Φ h,ũ0,δ,ξ := Φ h,ũ0 (δ, ξ), Π K ⊥ δ,ξ is the orthogonal projection of H 2 1 (M ) onto K ⊥ δ,ξ (here, the orthogonality is taken with respect to •, • h ) and

(28) R δ,ξ := (∆ g + h)W h,ũ0,δ,ξ -(W h,ũ0,δ,ξ ) 2 -1 + .
Furthermore, for every (δ 0 , ξ 0 ) ∈ (0, 0 ) × U 0 , we have

(29) J h (W h,ũ0,δ0,ξ0 + Φ h,ũ0,δ0,ξ0 ) = 0 ⇐⇒ (δ 0 , ξ 0 ) is a critical point of (δ, ξ) → J h (W h,ũ0,δ,ξ + Φ h,ũ0,δ,ξ ).
It follows from Robert-Vétois [START_REF] Robert | A general theorem for the construction of blowing-up solutions to some elliptic nonlinear equations via Lyapunov-Schmidt's reduction[END_REF] that

(30) J h (W h,ũ0,δ,ξ + Φ h,ũ0,δ,ξ ) = J h (W h,ũ0,δ,ξ ) + O( Φ h,ũ0,δ,ξ 2 H 2 1 )
uniformly with respect to (δ, ξ) ∈ (0, 0 ) × U 0 and (h, ũ0 ) such that h -h 0 ∞ < 0 and ũ0 -u 0 C 2 < 0 .

Conventions:

• To avoid unnecessarily heavy notations, we will often drop the indices (h, , ũ0 , δ, ξ), so that U := U δ,ξ , B := B h,δ,ξ , W := W h,ũ0,δ,ξ , Φ := Φ h,ũ0,δ,ξ , etc. The differentiation with respect to the variable (δ, ξ) will always be denoted by ∂ p , and the differentiation with respect to x ∈ M (or R n ) by ∂ x . For example,

∂ xi ∂ pj W =        ∂ 2 W h,ũ0,δ,ξ (x) ∂x i ∂δ if j = 0 ∂ 2 W h,ũ0,δ,ξ (x) ∂x i ∂ξ j if j ≥ 1.
• For every ξ ∈ U 0 , we identify the tangent space T ξ M with R n . Indeed, assuming that the neighborhood U 0 is small enough, it follows from the Gram-Schmidt orthonormalization procedure that there exists an orthonormal frame with respect to the metric g ξ , which is smooth with respect to the point ξ. Such a frame provides a smooth family of linear isometries (ψ ξ ) ξ∈U0 , ψ ξ : R n → T ξ M , which allow to identify T ξ M with R n . In particular, in this paper, the chart exp g ξ ξ will denote the composition of the usual exponential chart with the isometry ψ ξ .

• Throughout the paper, C will denote a positive constant such that -C depends on n, (M, g), ξ 0 ∈ M , the functions h 0 , u 0 ∈ C 2 (M ) and a constant A > 0 such that h 0 C 2 < A and λ 1 (∆ g + h 0 ) > 1/A. In the case where u 0 > 0, we also assume that u 0 C 2 < A and u 0 > 1/A. -C does not depend on x ∈ M (or x ∈ R n , depending on the context), ξ in the neighborhood U 0 , δ > 0 small and h ∈ C 2 (M ) such that h C 2 < A and λ 1 (∆ g + h) > 1/A. In the case where u 0 > 0, C is also independent of ũ0 ∈ C 2 (M ) such that ũ0 C 2 < A and ũ0 > 1/A. The value of C might change from line to line, and even in the same line.

• For every f, g ∈ R, the notations f = O(g) and f = o(g) will stand for |f | ≤ C |g| and |f | ≤ C (h, δ, ξ) |g|, respectively, where (h, δ, ξ) → 0 as h → h 0 in C 2 (M ), δ → 0 and ξ → ξ 0 .

C 1 -estimates for the energy functional

For every δ > 0 and ξ ∈ U 0 , we define

(31) Ũδ,ξ (x) := δ n(n -2) δ 2 + d g ξ (x, ξ) 2 n-2 2
for all x ∈ M.

Our first result is the differentiable version of (30).

Proposition 4.1. In addition to the assumptions of Section 3, we assume that

(32) |B h,δ,ξ (x)| + δ|∂ p B h,δ,ξ (x)| ≤ C(U δ,ξ (x) + δ Ũδ,ξ (x)) for all x ∈ M.
We then have

(33) ∂ p J h (W + Φ) = ∂ p J h (W ) + O(δ -1 Φ H 2 1 ( R 2n n+2 + δ ∂ p R 2n n+2 + Φ H 2 1 )) + O(1 n≥7 δ -1 Φ 2 -1 H 2 1 ),
where R = R δ,ξ is as in [START_REF] Schoen | Conformal deformation of a Riemannian metric to constant scalar curvature[END_REF].

Proof of Proposition 4.1. It follows from [START_REF] Robert | A general theorem for the construction of blowing-up solutions to some elliptic nonlinear equations via Lyapunov-Schmidt's reduction[END_REF] that there exist real numbers λ j := λ j (δ, ξ) such that

W + Φ -(∆ g + h) -1 (W + Φ) 2 -1 + = n j=0 λ j Z j .
This can be written as (34)

J h (W + Φ) = n j=0 λ j Z j , • h .
We fix i ∈ {0, . . . , n}. We obtain

∂ pi J h (W + Φ) = J h (W + Φ)[∂ pi W + ∂ pi Φ] (35) = J h (W )[∂ pi W ] + (J h (W + Φ) -J h (W )) [∂ pi W ] + J h (W + Φ)[∂ pi Φ] = J h (W )[∂ pi W ] + (J h (W + Φ) -J h (W )) [∂ pi W ] + n j=0 λ j Z j , ∂ pi Φ h = ∂ pi J h (W ) + (J h (W + Φ) -J h (W )) [∂ pi W ] - n j=0 λ j ∂ pi Z j , Φ h ,
where, for the last line, we have used that (Z i ) δ,ξ , Φ h,ũ0,δ,ξ h = 0 for all (δ, ξ) since Φ h,ũ0,δ,ξ ∈ K ⊥ δ,ξ . We estimate separately the two last terms in the right-hand side of (35). As regards the first of these two term, we have

(J h (W + Φ) -J h (W )) [∂ pi W ] (36) = M (∇Φ∇∂ pi W + hΦ∂ pi W ) - M ((W + Φ) 2 -1 + -W 2 -1 + )∂ pi W dv g = M Φ((∆ g + h)∂ pi W -(2 -1)W 2 -1 + ∂ pi W ) dv g - M ((W + Φ) 2 -1 + -W 2 -1 + -(2 -1)W 2 -1 + Φ)∂ pi W dv g .
With the definition (28), Hölder's and Sobolev's inequalities, we obtain

(37) M Φ((∆ g + h)∂ pi W -(2 -1)W 2 -1 + ∂ pi W ) dv g = M Φ∂ pi R dv g = O( Φ 2 ∂ pi R 2n n+2 ) = O( Φ H 2 1 ∂ pi R 2n n+2
). In the sequel, we will need the following lemma: Lemma 4.1. We have

(38) U δ,ξ (x) + δ|∂ p U δ,ξ (x)| ≤ C Ũδ,ξ (x)
for all (δ, ξ) ∈ (0, 0 ) × U 0 and x ∈ M .

Proof of Lemma 38. Most of the proof is easy computations. The only delicate point is to prove that |∂ ξ d g ξ (x, ξ) 2 | ≤ Cd g ξ (x, ξ) for all x ∈ M and ξ ∈ U 0 . We define F (x, ξ) := d g ξ (x, ξ) 2 and G(ξ, y) := exp g ξ ξ (y). Proving the desired inequality amounts to proving that (∂ ξ F (x, ξ)) |ξ=x = 0 for all x ∈ M . Note that F (G(ξ, y), ξ) = |y| 2 for small y ∈ R n . Differentiating this equality with respect to ξ yields a relation between ∂ x F and ∂ ξ F , and the requested inequality follows.

End of proof of Proposition 4.1. Using Lemma 4.1, the assumption (32) on B h,δ,ξ , and that ∂ pi ũ0 = 0, we obtain

M ((W + Φ) 2 -1 + -W 2 -1 + -(2 -1)W 2 -2 + Φ)∂ pi W dv g ≤ Cδ -1 M |(W + Φ) 2 -1 + -W 2 -1 + -(2 -1)W 2 -2 + Φ| Ũ dv g .
We split the integral in two. First

|W |≤2|Φ| |(W + Φ) 2 -1 + -W 2 -1 + -(2 -1)W 2 -2 + Φ| Ũ dv g ≤ C M |Φ| 2 -1 Ũ dv g ≤ C Φ 2 -1 2 Ũ 2 ≤ C Φ 2 -1 H 2 1 .
As regards the other part, looking carefully at the signs of the different terms, we obtain

|Φ|≤|W |/2 |(W + Φ) 2 -1 + -W 2 -1 + -(2 -1)W 2 -2 + Φ| Ũ dv g = |Φ|≤|W |/2 |W | 2 -1 1 + Φ W 2 -1 -1 -(2 -1) Φ W Ũ dv g ≤ C |Φ|≤|W |/2 |W | 2 -1 Φ W 2 Ũ dv g = C |Φ|≤|W |/2 |W | 2 -3 Φ 2 Ũ dv g .
In case n ≤ 6, that is 2 ≥ 3, we obtain

|Φ|≤|W |/2 |W | 2 -3 |Φ| 2 Ũ dv g ≤ M Ũ 2 -2 |Φ| 2 dv g ≤ C Ũ 2 -2 2 Φ 2 2 ≤ C Φ 2 H 2 1 .
In case n ≥ 7, that is 2 < 3, arguing as above, we obtain

|Φ|≤|W |/2 |W | 2 -3 Φ 2 Ũ dv g ≤ C M |Φ| 2 -1 Ũ dv g ≤ C Φ 2 -1 H 2 1 .
Plugging these estimates together yields (39)

M ((W + Φ) 2 -1 + -W 2 -1 + -(2 -1)W 2 -2 + Φ)∂ pi W dv g ≤ Cδ -1 ( Φ 2 H 2 1 + 1 n≥7 Φ 2 -1 H 2 1
).

As regards the last term in the right-hand side of (35), arguing as in the proof of Lemma 4.1, we obtain ∂ pi Z j H 2 1 ≤ C/δ for all i, j = 0, . . . , n. Therefore, we obtain (40)

n j=0 λ j ∂ pi Z j , Φ h ≤ Cδ -1 Λ Φ H 2 1 , where Λ := n j=0 |λ j |.
It follows from (34) that

J h (W + Φ)[Z i ] = n j=0 λ j Z i , Z j h
for all i = 0, . . . , n. Since Z i , Z j h → 0 if i = j and → 1 if i = j as δ → 0 and uniformly with respect to ξ ∈ U 0 , we obtain

Λ ≤ C n i=0 |J h (W + Φ)[Z i ]| .
For every i = 0, . . . , n, using that Φ, Z i h = 0 and W 2 + Z i 2 ≤ C, we obtain 36), (37), ( 39), ( 40) and ( 41) into (35) yields (33). This proves Proposition 4.1.

|J h (W + Φ)[Z i ]| ≤ |J h (W )[Z i ]| + Φ, Z i h - M ((W + Φ) 2 -1 + -W 2 -1 + )Z i dv g ≤ M RZ i dv g + C M (|W | 2 -2 |Φ| + |Φ| 2 -1 )|Z i | dv g ≤ C R 2n n+2 + C( Φ 2 + Φ 2 -1 2 ) ≤ C R 2n n+2 + C Φ 2 . Therefore, (41) Λ ≤ C R 2n n+2 + C Φ 2 . Plugging (

5.

Energy and remainder estimates: the case n ≥ 6 and u 0 ≡ ũ0 ≡ 0

In this section, we consider the case where n ≥ 6 and u 0 ≡ ũ0 ≡ 0. In this case, we set B h,δ,ξ ≡ 0. Then W h,ũ0,δ,ξ = W δ,ξ ≡ U δ,ξ and the assumptions of Proposition 4.1 are satisfied. We prove the following estimates for R = R δ,ξ : Proposition 5.1. Assume that n ≥ 6 and u 0 ≡ ũ0 ≡ 0. Then

(42) R 2n n+2 + δ ∂ p R 2n n+2 ≤ C              δ 2 + D h,ξ δ 2 (ln(1/δ)) 2/3 if n = 6 δ n-2 2 + D h,ξ δ 2 if 7 ≤ n ≤ 9 δ 4 (ln(1/δ)) 3/5 + D h,ξ δ 2 if n = 10 δ 4 + D h,ξ δ 2 if n ≥ 11,
where

(43) D h,ξ := h -h 0 ∞ + d g (ξ, ξ 0 ) 2 . Proof of Proposition 5.1. Let L g := ∆ g + c n Scal g be the conformal Laplacian. For a metric g = w 4/(n-2) g conformal to g (w ∈ C ∞ (M ) is positive), the conformal invariance law gives that (44) L g φ = w -(2 -1) L g (wφ) for all φ ∈ C ∞ (M ).
Therefore, we have

R = (∆ g + h)U -U 2 -1 = L g U -U 2 -1 + ϕ h U = Λ 2 -1 ξ (L g ξ (Λ -1 ξ U ) -(Λ -1 ξ U ) 2 -1 ) + ϕ h U = Λ 2 -1 ξ (∆ g ξ (Λ -1 ξ U ) -(Λ -1 ξ U ) 2 -1
) + ĥξ U, where ϕ h is as in ( 5) and ( 45)

ĥξ := ϕ h + c n Λ 2 -2 ξ Scal g ξ .
Via the exponential chart, using the radial symmetry of U δ,0 : R n → R, we obtain that around 0, (46)

∆ g ξ (Λ -1 ξ U ) -(Λ -1 ξ U ) 2 -1 = ∆ Eucl U δ,0 + ∂ r |g ξ | |g ξ | ∂ r U δ,0 -U 2 -1 δ,0 = ∂ r |g ξ | |g ξ | ∂ r U δ,0 .
It then follows from ( 23) that

(47) R(x) = ĥξ (x)U (x) + δ n-2 2 Θ δ,ξ (x), where |Θ δ,ξ (x)| + |∂ p Θ δ,ξ (x)| ≤ C
for all (δ, ξ) ∈ (0, ∞) × U 0 and x ∈ M . Note that these estimates are a consequence of (46) when x is close to ξ, and they are straightforward when x is far from ξ.

Using Lemma 4.1, we then obtain

(48) |R(x)| + δ|∂ δ R(x)| ≤ Cδ n-2 2 + C| ĥξ (x)| Ũδ,ξ (x)
and ( 49)

δ|∂ ξ R(x)| ≤ Cδ n-2 2 + C| hξ (x)| Ũδ,ξ (x) + Cδ|∂ p hξ (x)| Ũδ,ξ (x).
Since ( 6) and ( 22) hold, we have

(50) | ĥξ (x)| ≤ CD h,ξ + Cd g ξ (x, ξ) 2 and |∂ ξ ĥξ (x)| ≤ Cd g ξ (x, ξ).
It is a straightforward computation that for every α > 0 and p ≥ 1, we have (51)

d g ξ (•, ξ) α Ũδ,ξ p ≤ C        δ n-2 2 if n > (n -2 -α)p δ n-2 2 (ln(1/δ)) 1/p if n = (n -2 -α)p δ n p +α-n-2 2 if n < (n -2 -α)p.
Plugging together (48), ( 49), ( 50) and (51), long but painless computations yield (42). This ends the proof of Proposition 5.1.

Since n ≥ 6, that is 2 -1 ≤ 2, we have Φ 2

H 2 1 = O( Φ 2 -1 H 2 1
). Plugging together (30), ( 27), ( 33) and (42), we obtain (52)

J h (W + Φ) = J h (W ) + O        δ 4 + D 2 h,ξ δ 4 (ln(1/δ)) 4/3 if n = 6 δ n-2 + D 2 h,ξ δ 4 if 7 ≤ n ≤ 9 δ 8 (ln(1/δ)) 6/5 + D 2 h,ξ δ 4 if n = 10 δ 8 + D 2 h,ξ δ 4 if n ≥ 11        and (53) ∂ pi J h (W + Φ) = ∂ pi J h (W ) + O δ -1              δ 4 + D 2 h,ξ δ 4 (ln(1/δ)) 4/3 if n = 6 (δ n-2 2 + D h,ξ δ 2 ) 2 -1 if 7 ≤ n ≤ 9 (δ 4 (ln(1/δ)) 3/5 + D h,ξ δ 2 ) 2 -1 if n = 10 (δ 4 + D h,ξ δ 2 ) 2 -1 if n ≥ 11
for all i = 0, . . . , n. We now estimate J h (W + Φ):

Proposition 5.2. Assume that n ≥ 6 and u 0 ≡ ũ0 ≡ 0. Then

(54) J h (W + Φ) = 1 n R n U 2 1,0 dx + 1 2 ϕ h (ξ)δ 2 R n U 2 1,0 dx - 1 4n      24 2 ω 5 K h0 (ξ 0 )δ 4 ln(1/δ) + O(δ 4 (o(ln(1/δ) + D 2 h,ξ (ln(1/δ)) 4/3 )) if n = 6 K h0 (ξ 0 )δ 4 R n |x| 2 U 2 1,0 dx + o(δ 4 ) if n ≥ 7 as δ → 0, ξ → ξ 0 and h → h 0 in C 2 (M )
, where K h0 (ξ 0 ) is as in [START_REF] Druet | From one bubble to several bubbles: the low-dimensional case[END_REF].

Proof of Proposition 5.2. Integrating by parts, we obtain

J h (U ) = 1 2 M [(∆ g + h)U ] U dv g - 1 2 M U 2 dv g (55) = 1 2 M [(∆ g + h)U -U 2 -1 ]U dv g + 1 n M U 2 dv g .
It follows from ( 47) that ( 56)

M (∆ g U + hU -U 2 -1 )U dv g = M ĥξ U 2 dv g + O(δ n-2 ).
Using the volume estimate [START_REF] Premoselli | Bubbling above the threshold of the scalar curvature in dimensions four and five[END_REF], we obtain

M U 2 dv g = M (Λ -1 ξ U ) 2 dv g ξ = Br 0 (0) U 2 δ,0 (1 + O(|x| N ) dx + O(δ n ) (57) = R n U 2 1,0 dx + O(δ n ).
Plugging (56) and ( 57) into (55), we obtain

J h (U ) = 1 2 M ĥξ U 2 dv g + 1 n R n U 2 1,0 dx + O(δ n-2 ).
With the change of metric, the definition of the bubble [START_REF] Pistoia | Clustering phenomena for linear perturbation of the Yamabe equation, Partial Differential Equations Arising from Physics and Geometry[END_REF] and the property of the volume [START_REF] Premoselli | Bubbling above the threshold of the scalar curvature in dimensions four and five[END_REF], we obtain

(58) M ĥξ U 2 dv g = Br 0 (ξ) ĥξ U 2 dv g + O(δ n-2 ) = Br 0 (0) A h,ξ U 2 δ,0 dx + O(δ n-2 ),
where

(59) A h,ξ (x) := ( ĥξ Λ 2-2 ξ )(exp g ξ ξ (x)
). Using the radial symmetry of U δ,0 and since h 0 ∈ C 2 (M ), we obtain

Br 0 (0) A h,ξ U 2 δ,0 dx = Br 0 (0) (A h,ξ (0) + ∂ xα A h,ξ (0)x α (60) + 1 2 ∂ xα ∂ x β A h,ξ (0)x α x β + O( h -h 0 C 2 |x| 2 + h0,ξ (x)|x| 2 ))U 2 δ,0 dx = A h,ξ (0) Br 0 (0) U 2 δ,0 dx - 1 2n ∆ Eucl A h,ξ (0) Br 0 (0) |x| 2 U 2 δ,0 dx + O Br 0 (0) ( h -h 0 C 2 + h0,ξ (x))|x| 2 U 2 δ,0 dx + O(δ n-2 ),
where h0,ξ (x) → 0 as x → 0, uniformly in ξ ∈ U 0 . With a change of variable and Lebesgue convergence theorem, we obtain (61)

Br 0 (0) U 2 δ,0 dx = δ 2 R n U 2 1,0 dx + O(δ n-2 ), (62)
Br 0 (0)

|x| 2 U 2 δ,0 dx =    24 2 ω 5 δ 4 ln(1/δ) + O(δ 4 ) if n = 6 δ 4 R n |x| 2 U 2 1,0 dx + O(δ 5 ) if n ≥ 7, and (63)
Br 0 (0) h0,ξ (x)|x| 2 U 2 δ,0 dx = o δ 4 ln(1/δ) if n = 6 δ 4 if n ≥ 7 .
Furthermore, we have A h,ξ (0) = ϕ h (ξ) and

∆ Eucl A h,ξ (0) = ∆ g ξ ( ĥξ Λ 2-2 ξ )(ξ) = L g ξ (ϕ h Λ 2-2 ξ )(ξ) + c n ∆ g ξ Scal g ξ (ξ) (64) = L g (ϕ h Λ 3-2 ξ )(ξ) + c n 6 | Weyl g (ξ)| 2 g = L g (ϕ h0 Λ 3-2 ξ )(ξ) + c n 6 | Weyl g (ξ)| 2 g + O( h -h 0 C 2 ) = K h0 (ξ 0 ) + O( h0 (ξ) + h -h 0 C 2 ),
where h0 (ξ) → 0 as ξ → ξ 0 . Therefore, plugging together these identities yields

(65) J h (U ) = 1 n R n U 2 1,0 dx + 1 2 ϕ h (ξ)δ 2 R n U 2 1,0 dx - 1 4n    24 2 ω 5 K h0 (ξ 0 )δ 4 ln(1/δ) + o δ 4 ln(1/δ) if n = 6 K h0 (ξ 0 )δ 4 R n |x| 2 U 2 1,0 dx + o(δ 4 ) if n ≥ 7.
Plugging together (52) and (65), we obtain (54). This ends the proof of Proposition 5.2.

We now estimate the derivatives of J h (W + Φ):

Proposition 5.3. Assume that n ≥ 6 and u 0 ≡ ũ0 ≡ 0. Then (66) ∂ δ J h (W + Φ) = ϕ h (ξ)δ R n U 2 1,0 dx - 1 n      24 2 ω 5 K h0 (ξ 0 )δ 3 ln(1/δ) + o δ 3 ln(1/δ) + O(D 2 h,ξ δ 3 (ln(1/δ)) 4/3 ) if n = 6 K h0 (ξ 0 )δ 3 R n |x| 2 U 2 1,0 dx + o(δ 3 ) + O(D 2 -1 h,ξ δ n+6 n-2 ) if n ≥ 7 and (67) ∂ ξi J h (W + Φ) = 1 2 ∂ ξi ϕ h (ξ)δ 2 R n U 2 1,0 dx + O   o(δ 3 ln(1/δ)) + O(D 2 h,ξ δ 3 (ln(1/δ)) 4/3 ) if n = 6 o(δ 3 ) + O(D 2 -1 h,ξ δ n+6 n-2 ) if n ≥ 7   for all i = 1, . . . , n, as δ → 0, ξ → ξ 0 and h → h 0 in C 2 (M ).
Proof of Proposition 5.3. We fix i ∈ {0, . . . , n}. Using (47) and (38) and arguing as in (58), we obtain

∂ pi J h (U ) = J h (U )[∂ pi U ] = M (∆ g U + hU -U 2 -1 )∂ pi U dv g (68) = M R∂ pi U dv g = M ĥξ U ∂ pi U dv g + O δ n-2 2 M |∂ pi U | dv g = M ĥξ U ∂ pi U dv g + O δ -1 δ n-2 = Br 0 (0) A h,ξ U δ,ξ (Λ -1 ξ ∂ pi U )(exp g ξ ξ (x)) dx + O δ -1 δ n-2
As in (60), we write

(69) A h,ξ (x) = A h,ξ (0) + ∂ xα A h,ξ (0)x α + 1 2 ∂ xj ∂ x k A h,ξ (0)x j x k + O( h0,ξ (x)|x| 2 + h -h 0 C 2 |x| 2 )
for all x ∈ B r0 (0), where h0,ξ (x) → 0 as x → 0, uniformly in ξ ∈ U 0 . With (38), ( 62) and (63), we obtain (70)

Br 0 (0) ( h0,ξ (x) + h -h 0 C 2 )|x| 2 U δ,0 (Λ -1 ξ ∂ pi U )(exp g ξ ξ (x)) dx ≤ Cδ -1 Br 0 (0) ( h0,ξ (x)+ h-h 0 C 2 )|x| 2 Ũ 2 δ,0 dx = o(δ -1 ) δ 4 ln(1/δ) if n = 6 δ 4 if n ≥ 7.
We write

Br 0 (0) A h,ξ U δ,0 (Λ -1 ξ ∂ pi U )(exp g ξ ξ (x)) dx = Br 0 (0) A h,ξ U δ,0 ∂ pi (Λ -1 ξ U )(exp g ξ ξ (x)) dx - Br 0 (0) A h,ξ U 2 δ,0 (Λ -1 ξ ∂ pi Λ -1 ξ )(exp g ξ ξ (x)) dx.
Since ∇Λ ξ (ξ) = 0, we obtain

Br 0 (0) A h,ξ U 2 δ,0 (Λ -1 ξ ∂ pi Λ -1 ξ )(exp g ξ ξ (x)) dx = O A δ,ξ (0) Br 0 (0) |x|U 2 δ,0 dx + O Br 0 (0) |x| 2 U 2 δ,0 dx .
With the definition (59) of A h,ξ and the assumption (6) on h 0 , it follows that

Br 0 (0) A h,ξ U 2 δ,0 (Λ -1 ξ ∂ pi Λ -1 ξ )(exp g ξ ξ (x)) dx = O δ -1 δ 4 D h,ξ + δ ln(1/δ) if n = 6 δ if n ≥ 7 .
This estimate, the Taylor expansion (69) and the estimate (70) yield (71)

Br 0 (0) A h,ξ Λ -1 ξ U δ,0 (Λ -1 ξ ∂ pi U )(exp g ξ ξ (x)) dx = A h,ξ (0) Br 0 (0) U δ,0 ∂ pi (Λ -1 ξ U )(exp g ξ ξ (x)) dx + ∂ xα A h,ξ (0)
Br 0 (0)

x α U δ,0 ∂ pi (Λ -1 ξ U )(exp g ξ ξ (x)) dx + 1 2 ∂ xj ∂ x k A h,ξ (0)
Br 0 (0)

x j x k U δ,0 ∂ pi (Λ -1 ξ U )(exp g ξ ξ (x)) dx + o(δ -1 ) δ 4 ln(1/δ) if n = 6 δ 4 if n ≥ 7.
We first deal with the case i = 0, that is

∂ pi = ∂ p0 = ∂ δ .
For every homogeneous polynomial Q on R n , it follows from ( 14) and ( 18) that

Br 0 (0) QU δ,0 ∂ δ (Λ -1 ξ U )(exp g ξ ξ (x)) dx = Br 0 (0) Qδ -1 δ -n-2 2 U 1,0 (x/δ) δ -n-2 2 Z 0 (x/δ) dx.
The explicit expressions ( 13) and ( 15) of U and Z 0 and their radial symmetry then yield

Br 0 (0) U δ,0 ∂ δ (Λ -1 ξ U )(exp g ξ ξ (x)) dx = δ -1 δ 2 R n U 1,0 Z 0 dx + O(δ -1 δ n-2 ) for n ≥ 6,
Br 0 (0)

x j U δ,0 ∂ δ (Λ -1 ξ U )(exp g ξ ξ (x)) dx = 0 for n ≥ 6,
and

Br 0 (0) x j x k U δ,0 ∂ δ (Λ -1 ξ U )(exp g ξ ξ (x)) dx = jk n δ -1 δ 4    c 6 ln(1/δ) + O(δ -1 δ 4 ) if n = 6 R n |x| 2 U 1,0 Z 0 dx + O(δ -1 δ n-2 ) if n ≥ 7,
where jk is the Kronecker symbol and c 6 > 0 is a constant that will be discussed later. Putting these estimates in (68), and (71), we obtain

∂ δ J h (U ) = A h,ξ (0)δ -1 δ 2 R n U 1,0 Z 0 dx - 1 2n δ -1 δ 4    c 6 ∆ Eucl A h,ξ (0) ln(1/δ) + o(ln(1/δ)) if n = 6 ∆ Eucl A h,ξ (0) R n |x| 2 U 1,0 Z 0 dx + o(1) if n ≥ 7.
For every δ > 0, we have

R n U 2 δ,0 dx = δ 2 R n U 2 1,0 dx for n ≥ 5 and R n |x| 2 U 2 δ,0 dx = δ 4 R n |x| 2 U 2 1,0 dx for n ≥ 7.
Differentiating these equalities with respect to δ at δ = 1, we obtain

R n U 1,0 Z 0 dx = R n U 2 1,0 for n ≥ 5 and R n |x| 2 U 1,0 Z 0 dx = 2 R n |x| 2 U 2 1,0 for n ≥ 7.
Therefore, with the computation (64) and the definition [START_REF] Druet | From one bubble to several bubbles: the low-dimensional case[END_REF], we obtain

(72) ∂ δ J h (U ) = ϕ h (ξ)δ -1 δ 2 R n U 2 1,0 dx - 1 n δ -1 δ 4    c 6 K h0 (ξ 0 ) ln(1/δ) + o(ln(1/δ)) if n = 6 K h0 (ξ 0 ) R n |x| 2 U 2 1,0 dx + o(1) if n ≥ 7.
Differentiating (65), we obtain c 6 /2 = 24 2 ω 5 . Therefore, with (53), we obtain (66). We now deal with the case where i ≥ 1, that is

∂ pi = ∂ ξi . We first claim that (73) [∂ ξi (Λ -1 ξ U δ,ξ )](ξ, exp g ξ ξ (x)) + [∂ xi (Λ -1 ξ U δ,ξ )](ξ, exp g ξ ξ (x)) = O δ n-2 2 |x| 3 (δ 2 + |x| 2 ) n/2
, where the differential for ξ is taken via the exponential chart. Before proving this claim, let us remark that it is trivial in the Euclidean context. Indeed, for every ξ, x ∈ R n and δ > 0, with the notation (14), we have

∂ ξi U δ,ξ (x) = ∂ ξi (δ -n-2 2 U (δ -1 (x -ξ))) = -∂ xi U δ,ξ (x).
We now prove the claim (73). We fix ξ ∈ U 0 . We define the path ξ(t) := exp g ξ ξ (t e i ) for small t ∈ R, where e i is the i-th vector in the canonical basis of R n . With [START_REF] Trudinger | Remarks concerning the conformal deformation of Riemannian structures on compact manifolds[END_REF], we obtain

[∂ xi (Λ -1 ξ U δ,ξ )](ξ, exp g ξ ξ (x)) = d dt Ũδ,ξ (exp g ξ ξ (x + t e i )) |t=0 (74) = - n -2 2 δ n-2 2 (δ 2 + |x| 2 ) n/2 • 2x i and [∂ ξi (Λ -1 ξ U δ,ξ )](ξ, exp g ξ ξ (x)) = d dt Ũδ,ξ(t) (exp g ξ ξ (x)) |t=0 (75) = - n -2 2 δ n-2 2 (δ 2 + |x| 2 ) n/2 • d dt d 2 g ξ(t) (ξ(t), exp g ξ ξ (x)). It follows from Esposito-Pistoia-Vétois [12, Lemma A.2] that (76) d dt d 2 g ξ(t) (ξ(t), exp g ξ ξ (x)) + 2x i = O(|x| 3 ) as x → 0.
Putting together all these estimates yields (73). This proves the claim. With the definition ( 14), we obtain

Br 0 (0) U δ,0 δ n-2 2 |x| 3 (δ 2 + |x| 2 ) n/2 dx = O(δ 3 ) for n ≥ 6, Br 0 (0) |x|U δ,0 δ n-2 2 |x| 3 (δ 2 + |x| 2 ) n/2 dx = O δ 4 ln(1/δ) if n = 6 δ 4 if n ≥ 7

and

Br 0 (0)

|x| 2 U δ,0 δ n-2 2 |x| 3 (δ 2 + |x| 2 ) n/2 dx = O    δ 4 if n = 6 δ 5 ln(1/δ) if n = 7 δ 5 if n ≥ 8.    . Noting that [∂ xi (Λ -1 ξ U δ,ξ )](ξ, exp g ξ ξ (x)) = ∂ xi U δ,0
, we obtain by symmetry that

Br 0 (0) Λ -1 ξ U δ,0 ∂ xi (Λ -1 ξ U )(exp g ξ ξ (x)) dx = Br 0 (0) U δ,0 ∂ xi U δ,0 dx = 0
and similarly, Br 0 (0)

x j x k U δ,0 ∂ xi (Λ -1 ξ U )(exp g ξ ξ (x)) dx = 0.
Integrating by parts, straightforward estimates yield Br 0 (0)

x j U δ,0 ∂ xi (Λ -1 ξ U )(exp g ξ ξ (x)) dx = Br 0 (0) x j U δ,0 ∂ xi U δ,0 dx = 1 2 Br 0 (0) x j ∂ xi (U 2 δ,0 ) dx = - ij 2 Br 0 (0) U 2 δ,0 dx + 1 2 ∂Br 0 (0) x j ν i U 2 δ,0 dσ = - ij 2 δ 2 R n U 2 1,0 dx + O(δ n-2 ) for n ≥ 6,
where ν := ( ν 1 , . . . , ν n ) is the outward unit normal vector and dσ is the volume element of ∂B r0 (0). Since A h,ξ (0) = O(D h,ξ ), plugging these estimates together with (68) and (71), we obtain (77)

∂ ξi J h (U ) = 1 2 ∂ ξi ϕ h (ξ)δ 2 R n U 2 1,0 dx + o δ 3 ln(1/δ) if n = 6 δ 3 if n ≥ 7 .
With (53), we then obtain (67). This ends the proof of Proposition 5.3.

Theorem 1.4 for n ≥ 6 will be proved in Section 10.

6. Energy and remainder estimates: the case n ≥ 7 and u 0 , ũ0 > 0

In this section, we assume that u 0 , ũ0 > 0 and n ≥ 7, that is 2 -1 < 2. As in the previous case, we set B h,δ,ξ ≡ 0, so that W h,ũ0,δ,ξ = W ũ0,δ,ξ ≡ ũ0 + U δ,ξ and the assumptions of Proposition 4.1 are satisfied. We prove the following estimates for R = R δ,ξ : Proposition 6.1. Assume that n ≥ 7 and u 0 , ũ0 > 0. Then (78)

R 2n n+2 ≤ C ∆ g ũ0 + hũ 0 -ũ2 -1 0 ∞ + C(D h,ξ + δ 2 + δ n- 6
2 )δ 2 and ∂ p R 2n n+2 ≤ Cδ, where D h,ξ is as in (43).

Proof of Proposition 6.1. We have

(79) R = (∆ g ũ0 + hũ 0 -ũ2 -1 0 ) + R 0 -((ũ 0 + U ) 2 -1 -ũ2 -1 0 -U 2 -1 ),
where R 0 := ∆ g U + hU -U 2 -1 . Concerning the derivatives, given i ∈ {0, . . . , n}, we have

∂ pi R = ∆ g ∂ pi U + h∂ pi U -(2 -1)(ũ 0 + U ) 2 -2 ∂ pi U (80) = ∂ pi R 0 -(2 -1)((ũ 0 + U ) 2 -2 -U 2 -2 )∂ pi U.
A straightforward estimate yields

|(ũ 0 + U ) 2 -1 -ũ2 -1 0 -U 2 -1 | ≤ C1 U ≤ũ0 ũ2 -2 0 U + C1 ũ0≤U ũ0 U 2 -2 .
With the expression [START_REF] Pistoia | Clustering phenomena for linear perturbation of the Yamabe equation, Partial Differential Equations Arising from Physics and Geometry[END_REF], we obtain

{U (x) ≤ ũ0 (x) ⇒ d g ξ (x, ξ) ≥ c 1 √ δ} and {U (x) ≥ ũ0 (x) ⇒ d g ξ (x, ξ) ≤ c 2 √ δ}
for all x ∈ M , where c 1 , c 2 > 0 depend only on n, (M, g) and A > 0 such that 1/A < ũ0 < A Therefore, with r := d g ξ (x, ξ),

|(ũ 0 + U ) 2 -1 -ũ2 -1 0 -U 2 -1 | ≤ C1 r≥c1 √ δ U + C1 r≤c2 √ δ U 2 -2 . Since U ≤ Cδ n-2 2 (δ 2 + r 2 ) 1-n/2 , we then obtain (81) (ũ 0 + U ) 2 -1 -ũ2 -1 0 -U 2 -1 2n n+2 ≤ Cδ n+2 4
for n ≥ 7.

Since 0 < 2 -2 < 1, we have

|(ũ 0 + U ) 2 -2 -U 2 -2 | ≤ C.
Therefore, with [START_REF] Trudinger | Remarks concerning the conformal deformation of Riemannian structures on compact manifolds[END_REF] and (38), we obtain

(82) ((ũ 0 + U ) 2 -2 -U 2 -2 )∂ pi U 2n n+2 ≤ Cδ -1 U 2n n+2 ≤ Cδ -1 δ 2 for n ≥ 7.
Merging the estimates (42), (79), (80), (81) and (82), we obtain (78). This ends the proof of Proposition 6.1.

Plugging (78) and (78) together with [START_REF]On the number of constant scalar curvature metrics in a conformal class[END_REF], ( 27) and (33), we obtain

(83) J h (W + Φ) = J h (W ) + O( ∆ g ũ0 + hũ 0 -ũ2 -1 0 2 ∞ + D 2 h,ξ δ 4 + δ 8 + δ n-2 ) and (84) ∂ pi J h (W + Φ) = ∂ pi J h (W ) + O( ∆ g ũ0 + hũ 0 -ũ2 -1 0 2n n+2 δ + ∆ g ũ0 +hũ 0 -ũ2 -1 0 2 -1 ∞ δ -1 +(D h,ξ +δ 2 +δ n-6 2 ) 2 -1 δ n+6 n-2 +D h,ξ δ 3 +δ 5 +δ n/2 )
for all i = 0, . . . , n. We now estimate J h (W + Φ): Proposition 6.2. Assume that n ≥ 7 and u 0 , ũ0 > 0. Then

(85) J h (W + Φ) = J h (ũ 0 ) + 1 n R n U 2 1,0 dx + 1 2 ϕ h (ξ)δ 2 R n U 2 1,0 dx - 1 4n K h0 (ξ 0 )δ 4 R n |x| 2 U 2 1,0 dx + o(δ 4 ) -u 0 (ξ 0 )δ n-2 2 R n U 2 -1 1,0 dx +O( ∆ g ũ0 +hũ 0 -ũ2 -1 0 2 ∞ +δ n-2 2 ( ∆ g ũ0 +hũ 0 -ũ2 -1 0 ∞ + ũ0 -u 0 ∞ +o(1))) as δ → 0, ξ → ξ 0 and h → h 0 in C 2 (M ).
Proof of Proposition 6.2. We first write

J h (ũ 0 + U ) = J h (ũ 0 ) + J h (U ) - M ũ0 U 2 -1 dv g + M (∆ g ũ0 + hũ 0 -ũ2 -1 0 )U dv g - 1 2 M ((ũ 0 + U ) 2 -ũ2 0 -U 2 -2 ũ2 -1 0 U -2 ũ0 U 2 -1 ) dv g . We fix 0 < θ < 2 n-2 < 2 -2.
There exists C > 0 such that

|(ũ 0 + U ) 2 -ũ2 0 -U 2 -2 ũ2 -1 0 U -2 ũ0 U 2 -1 | ≤ C1 ũ0≤U ũ1+θ 0 U 2 -1-θ + C1 U ≤ũ0 ũ2 -1-θ 0 U 1+θ .
Using the definition [START_REF] Pistoia | Clustering phenomena for linear perturbation of the Yamabe equation, Partial Differential Equations Arising from Physics and Geometry[END_REF] and arguing as in the proof of (81), we obtain

M ((ũ 0 + U ) 2 -ũ2 0 -U 2 -2 ũ2 -1 0 U -2 ũ0 U 2 -1 ) dv g ≤ Cδ n-2 2 + n-2 2 θ .
Furthermore, we obtain

M (∆ g ũ0 + hũ 0 -ũ2 -1 0 )U dv g ≤ C ∆ g ũ0 + hũ 0 -ũ2 -1 0 ) ∞ M U dv g ≤ C ∆ g ũ0 + hũ 0 -ũ2 -1 0 ) ∞ δ n-2 2 .
Using [START_REF] Pistoia | Clustering phenomena for linear perturbation of the Yamabe equation, Partial Differential Equations Arising from Physics and Geometry[END_REF], that Λ ξ (x) = 1 + O(d g (x, ξ) 2 ) for all x ∈ M and that U δ,0 is radially symmetrical, we obtain

M ũ0 U 2 -1 dv g = Br 0 (0)
ũ0 (exp

g ξ ξ (x))(1 + O(|x| 2 ))U 2 -1 δ,0 dx + O(δ n-2 2 (2 -1) ) = Br 0 (0) (ũ 0 (ξ) + x α ∂ xα ũ0 (exp g ξ ξ (ξ)) + O(|x| 2 ))U 2 -1 δ,0 dx + O(δ n+2 2 )
= ũ0 (ξ)

Br 0 (0) U 2 -1 δ,0 dx + O Br 0 (0) |x| 2 U 2 -1 δ,0 dx + O(δ n+2 2 ) = ũ0 (ξ)δ n-2 2 B r 0 /δ (0) U 2 -1 1,0 dx + O δ n+2 2 B r 0 /δ (0) |x| 2 U 2 -1 1,0 dx + O(δ n+2 2 ). Since U 1,0 ≤ C(1 + |x| 2 ) 1-n/2
, we obtain

B r 0 /δ (0) U 2 -1 1,0 dx = R n U 2 -1 1,0 dx + O(δ 2 )
and

B r 0 /δ (0) |x| 2 U 2 -1 1,0 dx = O (ln(1/δ)) for n ≥ 7.
Therefore, plugging all these estimates together yields

M ũ0 U 2 -1 dv g = ũ0 (ξ)δ n-2 2 R n U 2 -1 1,0 dx + O(δ n+2 2 ln(1/δ)).
Consequently, we obtain that for every 0 < θ < 2 n-2 ,

J h (ũ 0 + U ) = J h (ũ 0 ) + J h (U ) -ũ0 (ξ)δ n-2 2 R n U 2 -1 1,0 dx + M (∆ g ũ0 + hũ 0 -ũ2 -1 0 )U dv g + O(δ n-2 2 + n-2 2 θ ).
Now, with the expansion (65), we obtain that for n ≥ 7, (86

) J h (ũ 0 + U ) = J h (ũ 0 ) + 1 n R n U 2 1,0 dx + 1 2 ϕ h (ξ)δ 2 R n U 2 1,0 dx - 1 4n K h0 (ξ 0 )δ 4 R n |x| 2 U 2 1,0 dx + o(δ 4 ) -u 0 (ξ 0 )δ n-2 2 R n U 2 -1 1,0 dx + O(δ n-2 2 ( ∆ g ũ0 + hũ 0 -ũ2 -1 0 ∞ + ũ0 -u 0 ∞ + d g (ξ, ξ 0 ) + δ n-2 2 θ )).
Plugging together (83) and (86), we then obtain (85). This ends the proof of Proposition 6.2.

We now estimate the derivatives of J h (W + Φ):

Proposition 6.3. Assume that n ≥ 7 and u 0 , ũ0 > 0. Then

(87) ∂ δ J h (W + Φ) = ϕ h (ξ)δ R n U 2 1,0 dx - 1 n K h0 (ξ 0 )δ 3 R n |x| 2 U 2 1,0 dx + o(δ 3 ) - n -2 2 u 0 (ξ 0 )δ n-4 2 R n U 2 -1 1,0 dx + O(δ n-4 2 ( ũ0 -u 0 ∞ + o(1)) + ∆ g ũ0 + hũ 0 -ũ2 -1 0 ∞ δ + ∆ g ũ0 + hũ 0 -ũ2 -1 0 2 -1 ∞ δ -1 + D 2 -1 h,ξ δ n+6 n-2 )
and

(88) ∂ ξi J h (W + Φ) = 1 2 ∂ ξi ϕ h (ξ)δ 2 R n U 2 1,0 dx + o(δ 3 ) + O(δ n-4 2 ( ũ0 -u 0 ∞ + o(1)) + ∆ g ũ0 + hũ 0 -ũ2 -1 0 ∞ δ + ∆ g ũ0 + hũ 0 -ũ2 -1 0 2 -1 ∞ δ -1 + D 2 -1 h,ξ δ n+6 n-2 )
for all i = 1, . . . , n, as δ → 0, ξ → ξ 0 and h → h 0 in C 2 (M ).

Proof of Proposition 6.3. We fix i ∈ {0, . . . , n}. We have

∂ pi J h (ũ 0 + U ) = M (∆ g ũ0 + hũ 0 -ũ2 -1 0 )∂ pi U dv g -(2 -1) M ũ0 U 2 -2 ∂ pi U dv g + ∂ pi J h (U ) - M ((ũ 0 + U ) 2 -1 -U 2 -1 -(2 -1)ũ 0 U 2 -2 )∂ pi U dv g .
There exists C > 0 such that

|(ũ 0 + U ) 2 -1 -ũ2 -1 0 -U 2 -1 -(2 -1)ũ 0 U 2 -2 | ≤ C1 ũ0≤U ũ2 -1 0 + C1 U ≤ũ0 U 2 -1 .
Since |∂ pi U | ≤ C Ũ /δ (see (38)), arguing as in the proof of (81), we obtain

M ((ũ 0 + U ) 2 -1 -U 2 -1 -(2 -1)ũ 0 U 2 -2 )∂ pi U dv g ≤ Cδ n-2 2 .
Furthermore, we obtain

M (∆ g ũ0 + hũ 0 -ũ2 -1 0 )∂ pi U dv g ≤ C ∆ g ũ0 + hũ 0 -ũ2 -1 0 ) ∞ δ -1 M Ũ dv g ≤ C ∆ g ũ0 + hũ 0 -ũ2 -1 0 ) ∞ δ -1 δ n-2 2 .
Independently, using again (38), straightforward computations yield

M ũ0 U 2 -2 ∂ pi U dv g = M (u 0 (ξ 0 ) + O( ũ0 -u 0 ∞ + d g (., ξ 0 )) U 2 -2 ∂ pi U dv g = u 0 (ξ 0 ) M U 2 -2 ∂ pi U dv g + O δ -1 M ( ũ0 -u 0 ∞ + d g (ξ, ξ 0 ) + d g (., ξ)) Ũ 2 -1 dv g = u 0 (ξ 0 ) M U 2 -2 ∂ pi U dv g + O(δ -1 δ n-2 2 ( ũ0 -u 0 ∞ + d g (ξ, ξ 0 ) + δ)).
Arguing as in the proof of (71), we obtain

M U 2 -2 ∂ pi U dv g = Br 0 (0) (Λ ξ U ) 2 -2 ∂ pi (Λ -1 ξ U )(exp g ξ ξ (x)) dx + O δ -1 Br 0 (0) |x| Ũ 2 -1 dx = Br 0 (0) (Λ ξ U ) 2 -2 ∂ pi (Λ -1 ξ U )(exp g ξ ξ (x)) dx + O(δ n- 2
2 ).

We first deal with the case where i = 0, that is [START_REF] Li | Compactness of solutions to the Yamabe problem[END_REF], we obtain

∂ pi = ∂ p0 = ∂ δ . With
Br 0 (0) (Λ ξ U ) 2 -2 ∂ δ (Λ -1 ξ U )(exp g ξ ξ (x)) dx = Br 0 (0) U 2 -2 δ,0 ∂ δ U δ,0 dx = Br 0 (0) U 2 -2 δ,0 ∂ δ U δ,0 dx = δ -1 Br 0 (0) (δ -n-2 2 U 1,0 (δ -1 x)) 2 -2 δ -n-2 2 Z 0 (δ -1 x) dx = δ -1 δ n-2 2 B r 0 /δ (0) U 2 -2 1,0 Z 0 dx.
Since Z 0 ≤ CU 1,0 , an asymptotic estimate yields

Br 0 (0) (Λ ξ U ) 2 -2 ∂ δ (Λ -1 ξ U )(exp g ξ ξ (x)) dx = δ -1 δ n-2 2 R n U 2 -2 1,0 Z 0 dx + O(δ n 2 ).
Note that for every δ > 0, we have

R n U 2 -1 δ,0 dx = δ n-2 2 R n U 2 -1 1,0 dx.
Differentiating this equality with respect to δ at 1, we obtain

(2 -1) R n U 2 -2 1,0 Z 0 dx = n -2 2 R n U 2 -1 1,0 dx.
Therefore, we obtain

(2 -1) M U 2 -2 ∂ δ U dv g = n -2 2 δ -1 δ n-2 2 R n U 2 -1 1,0 dx + O(δ n- 2
2 ).

We now deal with the case i ≥ 1, that is ∂ pi = ∂ ξi . It follows from ( 75) and ( 76) that

Br 0 (0) (Λ ξ U ) 2 -2 ∂ ξi (Λ -1 ξ U )(exp g ξ ξ (x)) dx = Br 0 (0) U 2 -2 δ,0 - n -2 2 δ n-2 2 (δ 2 + |x| 2 ) n/2 -2x i + O(|x| 3 ) dx = O Br 0 (0) U 2 -2 δ,0 U δ,0 δ 2 + |x| 2 |x| 3 dx = O Br 0 (0) |x|U 2 -1 δ,0 dx = O(δ n- 2
2 ).

Putting these results together yields

∂ ξi J h (ũ 0 + U ) = ∂ ξi J h (U ) - n -2 2 i,0 u 0 (ξ 0 )δ -1 δ n-2 2 R n U 2 -1 1,0 dx + O(δ -1 δ n-2 2 ( ∆ g ũ0 + hũ 0 -ũ2 -1 0 ∞ + ũ0 -u 0 ∞ + d g (ξ, ξ 0 ) + δ))
for all i = 0, . . . , n. Using the estimates (72) and (77) for the derivatives of J h (U δ,ξ ), we obtain

∂ δ J h (ũ 0 + U ) = ϕ h (ξ)δ -1 δ 2 R n U 2 1,0 dx -4K h0 (ξ 0 )δ 3 R n |x| 2 U 2 1,0 dx - n -2 2 u 0 (ξ 0 )δ n-4 2 R n U 2 -1 1,0 dx + o(δ 3 ) + O(δ n-4 2 ( ∆ g ũ0 + hũ 0 -ũ2 -1 0 ∞ + ũ0 -u 0 ∞ + d g (ξ, ξ 0 ) + δ)) and ∂ ξi J h (ũ 0 + U ) = 1 2 ∂ ξi ϕ h (ξ)δ 2 R n U 2 1,0 dx + o(δ 4 ) + O(δ n-4 2 ( ∆ g ũ0 + hũ 0 -ũ2 -1 0 ∞ + ũ0 -u 0 ∞ + d g (ξ, ξ 0 ) + δ)).
With (84), we then obtain (87) and (88). This ends the proof of Proposition 6.3.

Theorem 1.5 for n ≥ 7 will be proved in Section 11.

7. Energy and remainder estimates: the case n = 6 and u 0 , ũ0 > 0

In this section, we assume that u 0 , ũ0 > 0 and n = 6, that is 2 -1 = 2. Here again, we set B h,δ,ξ ≡ 0, so that W h,ũ0,δ,ξ = W ũ0,δ,ξ ≡ ũ0 +U δ,ξ and the assumptions of Proposition 4.1 are satisfied. The remark underlying this section is that

∆ g (u 0 + U ) + h(u 0 + U ) -(u 0 + U ) 2 = ∆ g U + (h -2u 0 )U -U 2 .
Therefore, to obtain a good approximation of the blowing-up solution, we will subtract a perturbation of 2u 0 to the potential. We first estimate R = R δ,ξ : Proposition 7.1. Assume that n = 6 and u 0 , ũ0 > 0. Then

(89) R 3/2 + δ ∂ p R 3/2 ≤ C ∆ g ũ0 + hũ 0 -ũ2 0 ∞ + Cδ 2 (1 + D h,ξ (ln(1/δ)) 2/3 ),
where

(90) D h,ξ := h -h0 ∞ + d g (ξ, ξ 0 ) 2 .
Proof of Proposition 7.1. Since 2 -1 = 2, we have

R = ∆ g (ũ 0 + U ) + h(ũ 0 + U ) -(ũ 0 + U ) 2 = ∆ g ũ0 + hũ 0 -ũ2 0 + ∆ g U + (h -2ũ 0 )U -U 2 and ∂ pi R = ∂ pi ∆ g U + (h -2ũ 0 )U -U 2
for all i = 0, . . . , n. For convenience, we write h := h -2ũ 0 and h0 := h 0 -2u 0 .

The estimate (89) then follows from (42). This ends the proof of Proposition 7.1.

We now estimate the derivatives of J h (W + Φ):

Proposition 7.2. Assume that n = 6 and u 0 , ũ0 > 0. Then

(91) J h (W + Φ) = J h (ũ 0 ) + 1 n R n U 2 1,0 dx + 1 2 ϕ h,ũ0 (ξ)δ 2 R n U 2 1,0 dx -24ω 5 K h0,u0 (ξ 0 )δ 4 ln(1/δ) + O( ∆ g ũ0 + hũ 0 -ũ2 0 2 ∞ + ∆ g ũ0 + hũ 0 -ũ2 0 ∞ δ 2 ) + O(δ 4 ln(1/δ)(o(1) + D 2 h,ξ (ln(1/δ)) 1/3 )), ( 92
) ∂ δ J h (W + Φ) = ϕ h,ũ0 (ξ)δ R n U 2 1,0 dx -96ω 5 K h0,u0 (ξ 0 )δ 3 ln(1/δ) + O( ∆ g ũ0 + hũ 0 -ũ2 0 ∞ δ + ∆ g ũ0 + hũ 0 -ũ2 0 2 ∞ δ -1 ) + O(δ 3 ln(1/δ)(o(1) + D 2 h,ξ (ln(1/δ)) 1/3 ))
and

(93) ∂ ξi J h (W + Φ) = 1 2 ∂ ξi ϕ h,ũ0 (ξ)δ 2 R n U 2 1,0 dx + O( ∆ g ũ0 + hũ 0 -ũ2 0 ∞ δ + ∆ g ũ0 + hũ 0 -ũ2 0 2 ∞ δ -1 ) + O(δ 3 ln(1/δ)(o(1) + D 2 h,ξ (ln(1/δ)) 1/3 ))
for all i = 1, . . . , n, as δ → 0, ξ → ξ 0 and h → h 0 in C 2 (M ), where ϕ h,ũ0 , K h0,u0 (ξ 0 ) and D h,ξ are as in (5), ( 9) and (90).

Proof of Proposition 7.2. As one checks, since n = 6 and 2 = 3, we have

J h (ũ 0 + U ) = J h (ũ 0 ) + Jh(U ) + M ∆ g ũ0 + hũ 0 -ũ2 0 U dv g and ∂ pi J h (ũ 0 + U ) = ∂ pi Jh(U ) + M ∆ g ũ0 + hũ 0 -ũ2 0 ∂ pi U dv g
for all i = 0, . . . , n. Using the definition [START_REF] Pistoia | Clustering phenomena for linear perturbation of the Yamabe equation, Partial Differential Equations Arising from Physics and Geometry[END_REF] and since

|∂ pi U | ≤ C Ũ /δ, we obtain M (∆ g ũ0 + hũ 0 -ũ2 0 )U dv g ≤ C ∆ g ũ0 + hũ 0 -ũ2 0 ) ∞ M U dv g ≤ C ∆ g ũ0 + hũ 0 -ũ2 0 ) ∞ δ 2 .
and

M (∆ g ũ0 + hũ 0 -ũ2 0 )∂ pi U dv g ≤ C ∆ g ũ0 + hũ 0 -ũ2 0 ) ∞ δ -1 M Ũ dv g ≤ C ∆ g ũ0 + hũ 0 -ũ2 0 ) ∞ δ -1 δ 2 .
Putting these estimates together with ( 6), ( 30), (33), ( 89), (65), ( 72) and (77), we obtain (91), ( 92) and (93). This ends the proof of Proposition 7.2. Theorem 1.5 for n = 6 will be proved in Section 11.

8. Setting and definition of the mass in dimensions n = 3, 4, 5

In this section, we assume that n ≤ 5. Our first lemma is a simple computation: Lemma 8.1. There exist two functions (ξ, x) → f i (ξ, x), i = 1, 2, defined and smooth on M × M such that for every function f : R n → R that is radially symmetrical, we have

(∆ g + h)(χ(r)Λ ξ (x)f (r)) = Λ ξ (x) 2 -1 χ∆ Eucl (f (r)) + f 1 (ξ, x)f (r) + f 2 (ξ, x)f (r) + ĥξ χ(x)Λ ξ (x)f (r)
for all x ∈ M \ {ξ}, where r := d g ξ (x, ξ) and ĥξ is as in (45). Furthermore, f i (ξ, x) = 0 when d g (x, ξ) ≥ r 0 and there exists C N > 0 such that

|f 1 (ξ, x)(x)| ≤ C N d g (x, ξ) N -1 and |f 2 (ξ, x)| ≤ C N d g (x, ξ) N -2 for all x, ξ ∈ M.
The proof of Lemma 8.1 follows the computations in (47). We leave the details to the reader.

We define

Γ ξ (x) := χ(d g ξ (x, ξ))Λ ξ (x) (n -2)ω n-1 d g ξ (x, ξ) n-2
for all x ∈ M \ {ξ}. It follows from Lemma 8.1 and the definition ( 14) that

(94) ∆ g U δ,ξ + hU δ,ξ = U 2 -1 δ,ξ + F δ (ξ, x)δ n-2 2 + ĥξ U δ,ξ and
(∆ g + h)Γ ξ = δ ξ + F 0 (ξ, x) k n + ĥξ Γ ξ , where k n := (n -2)ω n-1 n(n -2) n-2 2 ,
δ ξ is the Dirac mass at ξ and (t, ξ, x)

→ F t (ξ, x) is of class C p on [0, ∞) × M × M ,
with p being as large as we want provided we choose N large enough. This includes t = 0 and, therefore, (95) lim t→0

F t = F 0 in C p (M × M ).
For every t ≥ 0, we define β h,t,ξ ∈ H 2 1 (M ) as the unique solution to

(∆ g + h)β h,t,ξ = - F t (ξ, x) k n + ĥξ χ(d g ξ (ξ, x))Λ ξ (x) (n -2)ω n-1 (t 2 + d g ξ (ξ, x) 2 ) n-2 2 (96) = - F t (ξ, x) k n -ĥξ    U t,ξ k n t n-2 2 if t > 0 Γ ξ if t = 0.
Since N > n -2 and n ≤ 5, the right-hand-side is uniformly bounded in L q (M ) for some q > 2n n+2 , independently of t ≥ 0, ξ ∈ U 0 and h ∈ C 2 (M ) satisfying h ∞ < A and λ 1 (∆ g + h) > 1/A. Therefore, β h,t,ξ is well defined and we have (97) β h,t,ξ -β h,0,ξ H 2 1 = o(1) as t → 0 uniformly with respect to ξ and h. Furthermore, we have β h,t,ξ ∈ C 2 (M ) when t > 0. As one checks, with these definitions, we obtain that

G h,ξ := Γ ξ + β h,0,ξ
is the Green's function of the operator ∆ g + h at the point ξ. We now define the mass of ∆ g + h at the point ξ:

Proposition-Definition 8.1. Assume that 3 ≤ n ≤ 5 and N > n -2. Let h ∈ C 2 (M ) be such that ∆ g + h is coercive. In the case where n ∈ {4, 5}, assume in addition that there exists ξ ∈ M such that ϕ h (ξ) = |∇ϕ h (ξ)| = 0, where ϕ h is as in [START_REF] Chen | Infinitely many solutions for the Schrödinger equations in R n with critical growth[END_REF]. Then β h,0,ξ ∈ C 0 (M ). Furthermore, the number β h,0,ξ (ξ) does not depend on the choice of N > n -2 and g ξ satisfying (21) and [START_REF] Premoselli | Bubbling above the threshold of the scalar curvature in dimensions four and five[END_REF]. We then define the mass of ∆ g + h at the point ξ as m h (ξ) := β h,0,ξ (ξ).

Proof of Proposition-Definition 8.1. As one checks, when n = 3, we have

ĥξ (x)Γ ξ (x) = O(d g (x, ξ) -1 )
and when n ∈ {4, 5} and

ϕ h (ξ) = |∇ϕ h (ξ)| = 0, we have ĥξ (x)Γ ξ (x) = O(d g (x, ξ) 4-n ).
Furthermore, we have

F 0 (ξ, x) = O(d g (x, ξ) N -n ).
When N > n -2, this implies that β h,0,ξ ∈ C 0 (M ). The fact that the number β h,0,ξ (ξ) does not depend on the choice of N and g ξ then follows from the uniqueness of conformal normal coordinates up to the action of O(n) and the choice of the metric's one-jet at the point ξ (see Lee-Parker [START_REF] Lee | The Yamabe problem[END_REF]). This ends the proof of Proposition-Definition 8.1.

We now prove a differentiation result that will allow us to obtain Theorem 1.2:

Proposition 8.1. Assume that 3 ≤ n ≤ 5. Let h ∈ C 2 (M ) be such that ∆ g + h is coercive.
In the case where n ∈ {4, 5}, assume that there exists

ξ ∈ M such that ϕ h (ξ) = |∇ϕ h (ξ)| = 0. Let H ∈ C 2 (M ) be such that H(ξ) = |∇H(ξ)| = 0.
Then m h+ H (ξ) is well defined for small ∈ R and differentiable with respect to . Furthermore,

∂ (m h+ H (ξ)) |0 = - M HG 2 h,ξ dv g .
Proof of Proposition 8.1. In order to differentiate the mass with respect to the potential function h, it is convenient to write

G h,ξ = G cn Scalg,ξ + βh,ξ , where βh,ξ ∈ H 2 1 (M ) is the solution to (98) (∆ g + h) βh,ξ = -ϕ h G cn Scalg,ξ .
Under the assumptions of the proposition, we have βh,ξ ∈ C 0 (M ) and

βh,ξ (ξ) = - M ϕ h G cn Scalg,ξ G h,ξ dv g .
Furthermore, as one checks, we have

(99) m h (ξ) = m cn Scalg (ξ) -βh,ξ (ξ).
It follows from standard elliptic theory that βh+ H,ξ is differentiable with respect to . Differentiating (98) then yields

(∆ g + h)∂ ( βh+ H,ξ ) |0 + H βh,ξ = -HG cn Scalg,ξ , which gives (∆ g + h)∂ ( βh+ H,ξ ) |0 = -HG h,ξ .
Therefore,

∂ ( βh+ H,ξ (x)) |0 = - M G h,x HG h,ξ dv g .
It then follows from (99) that

∂ (m h+ H (ξ)) |0 = - M HG 2 h,ξ dv g .
This ends the proof of Proposition 8.1.

9.

Energy and remainder estimates in dimensions n = 3, 4, 5

In this section, we assume that n ≤ 5 and u 0 ≡ ũ0 ≡ 0. When n ∈ {4, 5}, we assume in addition that the condition (4) is satisfied. We define

(100) W h,ũ0,δ,ξ = W h,δ,ξ := U δ,ξ + B h,δ,ξ , where B h,δ,ξ := k n δ n-2 2 β h,δ,ξ .
In order to use the C 1 -estimates of Proposition 4.1, our first step is to obtain estimates for β h,δ,ξ and its derivatives in H 2 1 (M ): Proposition 9.1. For 3 ≤ n ≤ 5, let B h,δ,ξ be as in (100). Then (25) holds.

Proof of Proposition 9.1. It follows from (97) that

β h,δ,ξ H 2 1 ≤ C. Differentiating (96) with respect to ξ i , i = 1, . . . , n, we obtain (∆ g + h)(∂ ξi β h,δ,ξ ) = - 1 k n ∂ ξi F δ (ξ, •) + ∂ ξi ĥξ U δ,ξ δ n-2 2 + ĥξ ∂ ξi U δ,ξ δ n-2 2
.

It follows from (95) that

∂ ξi F δ (ξ, •) ∞ ≤ C
With the definition (45) of ĥξ , we obtain

∂ ξi ĥξ = ∂ ξi (c n Scal g ξ Λ 2-2 ξ ) = O(d g (•, ξ)).
Therefore, with [START_REF] Hebey | Compactness and stability for nonlinear elliptic equations[END_REF], we obtain

∂ ξi ĥξ U δ,ξ δ n-2 2 ≤ C d g (x, ξ) (δ 2 + d g (x, ξ) 2 ) n-2 2
.

With ( 73) and (74), we obtain

|δ -n-2 2 ∂ ξi U δ,ξ | ≤ C 1 (δ 2 + d g (x, ξ) 2 ) n-2 2 + C d g (x, ξ) (δ 2 + d g (x, ξ) 2 ) n/2 .
The definition (45) of ĥξ and the assumption ϕ h0 (ξ

0 ) = |∇ϕ h0 (ξ 0 )| = 0 yield (101) ĥξ (x) = O(d g (x, ξ) 2 + D h,ξ ),
where D h,ξ is as in (43). Putting together these inequalities yields

(102) |(∆ g +h)(∂ ξi β h,δ,ξ )| ≤ C+C d g (x, ξ) (δ 2 + d g (x, ξ) 2 ) n-2 2 +CD h,ξ δ 2 + d g (x, ξ) (δ 2 + d g (x, ξ) 2 ) n/2 .
It then follows from standard elliptic theory and straightforward computations that

∂ ξi β h,δ,ξ H 2 1 ≤ C        1 if n = 3 (ln(1/δ)) 4/3 if n = 4 δ -1/2 if n = 5.
Similarly, differentiating with respect to δ, we obtain

|(∆ g + h)(∂ δ β h,δ,ξ )| = - 1 k n ∂ δ F δ (ξ, •) + ĥξ ∂ δ (δ -n-2 2 U δ,ξ ) (103) ≤ C + C δ(d g (x, ξ) 2 + D h,ξ ) (δ 2 + d g (x, ξ) 2 ) n/2
and, therefore, elliptic estimates and straightforward computations yield

∂ δ β h,δ,ξ 2n n+2 ≤ C + C δ (δ 2 + d g (x, ξ) 2 ) n/2 H 2 1 ≤ C 1 if n = 3 δ 2-n/2 if n = 4, 5.
With the definition (100), all these estimates yield [START_REF] Rey | The role of the Green's function in a nonlinear elliptic equation involving the critical Sobolev exponent[END_REF]. This ends the proof of Proposition 9.1.

The sequel of the analysis requires a pointwise control for β h,δ,ξ and its derivatives. This is the objective of the following proposition: Proposition 9.2. We have

(104) |β h,δ,ξ (x)| ≤ C        1 if n = 3 1 + | ln δ 2 + d g (x, ξ) 2 | if n = 4 δ 2 + d g (x, ξ) 2 -1/2 if n = 5, ( 105
) |∂ δ β h,δ,ξ (x)| ≤ C + CD h,ξ δ ln(1/δ) δ 2 + d g (x, ξ) 2 -n-2 2
and (106)

|∂ ξi β h,δ,ξ (x)| ≤ C + C        D h,ξ ln(δ 2 + d g (x, ξ) 2 ) if n = 3 D h,ξ δ 2 + d g (x, ξ) 2 -1/2 if n = 4 ln(δ 2 + d g (x, ξ) 2 ) + D h,ξ (δ 2 + d g (x, ξ) 2 ) -1 if n = 5
for all i = 1, . . . , n, where D h,ξ is as in (43).

Proof of Proposition 9.2. These estimates will be consequences of Green's representation formula and Giraud's Lemma. More precisely, it follows from (96) that (107)

β h,δ,ξ (x) = - M G h,x (y) F δ (ξ, y) k n + ĥξ χ(d g ξ (y, ξ))Λ ξ (y) (n -2)ω n-1 (δ 2 + d g ξ (y, ξ) 2 ) n-2 2 dv g (y)
for all x ∈ M . With (95) and the standard estimates of the Green's function 0 < G h,x (y) ≤ Cd g (x, y) 2-n for all x, y ∈ M , x = y, we obtain

(108) |β h,δ,ξ (x)| ≤ C + C M d g (x, y) 2-n (δ 2 + d g (y, ξ) 2 ) n-2 2
dv g (y).

Recall Giraud's Lemma (see [START_REF] Druet | Blow-up theory for elliptic PDEs in Riemannian geometry[END_REF] for the present statement): For every α, β such that 0 < α, β < n and x, z ∈ M , x = z, we have

M d g (x, y) α-n d g (y, z) β-n dv g (z) ≤ C      d g (x, z) α+β-n if α + β < n 1 + | ln d g (x, z)| if α + β = n 1 if α + β > n.
Therefore, (108) yields (104) when

d g (x, ξ) ≥ δ. When d g (x, ξ) ≤ δ, (108) yields |β h,δ,ξ (x)| ≤ C + C M d g (x, y) 2-n (δ 2 + d g (y, x) 2 ) n-2 2 dv g (y),
which in this case also yields (104). To prove (106), we use (102) and the same method as for (104). The inequality (105) is a little more delicate. With (103) and Green's identity, we obtain

|∂ δ β h,δ,ξ (x)| = M G h,x (y) (∆ g + h) ∂ δ β h,δ,ξ (y) dv g (y) ≤ C + C M d g (x, y) 2-n δ(d g (y, ξ) 2 + D h,ξ ) (δ 2 + d g (y, ξ) 2 ) n/2 dv g (y).
We then obtain

|∂ δ β h,δ,ξ (x)| ≤ C + Cδ M d g (x, y) 2-n d g (y, ξ) 2-n dv g (y) + CδD h,ξ M d g (x, y) 2-n (δ 2 + d g (y, ξ) 2 ) n/2 dv g (y).
We estimate the first two terms in the right-hand side by using Giraud's lemma as in the proof of (104). We split the integral of the third term as

M d g (x, y) 2-n (δ 2 + d g (y, ξ) 2 ) n/2 dv g (y) = {dg(x,y)<dg(x,ξ)/2} + {dg(x,y)≥dg(x,ξ)/2} . Since d g (y, ξ) > d g (x, ξ)/2 when d g (x, y) < d g (x, ξ)/2, we have {dg(x,y)<dg(x,ξ)/2} d g (x, y) 2-n (δ 2 + d g (y, ξ) 2 ) n/2 dv g (y) ≤ Cd g (x, ξ) -n {dg(x,y)<dg(x,ξ)/2} d g (x, y) 2-n dv g (y) ≤ Cd g (x, ξ) 2-n .
As regards the second part of the integral, we have {dg(x,y)≥dg(x,ξ)/2}

d g (x, y) 2-n (δ 2 + d g (y, ξ) 2 ) n/2 dv g (y) ≤ Cd g (x, ξ) 2-n M (δ 2 + d g (y, ξ) 2 ) -n/2 dv g (y) ≤ Cd g (x, ξ) 2-n ln(1/δ).
This yields (105) when d g (x, ξ) > δ. Finally, we treat the case d g (x, ξ) ≤ δ in the same way as (106). This ends the proof of Proposition 9.2.

It is a direct consequence of Proposition 9.2 that ( 25) is satisfied. Therefore Proposition 4.1 applies. It follows from ( 27), ( 30) and ( 33) that (109)

J h (W + Φ) = J h (W ) + O( R 2 2n n+2
) and, since n ≤ 5, (110)

∂ p J h (W + Φ) = ∂ p J h (W ) + O(δ -1 R 2n n+2 ( R 2n n+2 + δ ∂ p R 2n n+2
)), where R = R δ,ξ is as in [START_REF] Schoen | Conformal deformation of a Riemannian metric to constant scalar curvature[END_REF]. We prove the following estimates for R: Proposition 9.3. We have

(111) R 2n n+2 + δ ∂ p R 2n n+2 ≤ C      δ if n = 3 δ 2 ln(1/δ) if n = 4 D h,ξ δ 2 ln(1/δ) + δ 2 if n = 5.
Proof of Proposition 9.3. Note that since n < 6, we have 2 > 3. The definitions (96), ( 100) and (100) combined with (94) yield

R = (∆ g + h)U + (∆ g + h)B -(U + B) 2 -1 + = U 2 -1 -(U + B) 2 -1 + (112) = -(2 -1)U 2 -2 B + O(U 2 -3 B 2 + |B| 2 -1 ),
where we have used that U ≥ 0. Therefore,

R 2n n+2 ≤ C U 2 -2 B 2n n+2 + |B| 2 -1 2n n+2 . Since B = k n δ n-2
2 β, the pointwise estimate (104), the estimate U ≤ C Ũ and the estimates (51) yield

R 2n n+2 ≤ C      δ if n = 3 δ 2 ln(1/δ) if n = 4 δ 2 if n = 5.
We now deal with the gradient term. We fix i ∈ {0, . . . , n}. We have

∂ pi R = ∂ pi (U 2 -1 -(U + B) 2 -1 + ) = -(2 -1)((U + B) 2 -2 + (∂ pi U + ∂ pi B) -U 2 -2 ∂ pi U ) = -(2 -1)(((U + B) 2 -2 + -U 2 -2 )∂ pi U + (U + B) 2 -2 + ∂ pi B).
Using that 2 > 3 together with ( 32) and (38), we obtain

δ|∂ pi R| ≤ C Ũ 2 -2 |B| + C Ũ |B| 2 -2 + Cδ|∂ pi B| Ũ 2 -2 . Since B = k n δ n-2
2 β, using the estimates of β and its derivatives in Proposition 9.2 and the estimates (51), long but easy computations yield

δ ∂ pi R 2n n+2 ≤ C      δ if n = 3 δ 2 ln(1/δ) if n = 4 D h,ξ δ 2 ln(1/δ) + δ 2 if n = 5.
Therefore, we obtain (111). This ends the proof of Proposition 9.3.

With (111), the estimates (109) and (110) become

J h (W + Φ) = J h (W ) + O     δ 2 if n = 3 δ 4 (ln(1/δ)) 2 if n = 4 δ 4 + D 2 h,ξ δ 4 (ln(1/δ)) 2 if n = 5    
and

∂ pi J h (W + Φ) = ∂ pi J h (W ) + O    δ if n = 3 δ 3 (ln(1/δ)) 2 if n = 4 δ 3 + D 2 h,ξ δ 3 (ln(1/δ)) 2 if n = 5    .
We now estimate J h (W + Φ):

Proposition 9.4. We have

(113) J h (W + Φ) = 1 n R n U 2 1,0 dx + 1 2 ϕ h (ξ)          0 if n = 3 8ω n-1 δ 2 ln(1/δ) if n = 4 δ 2 R n U 2 1,0 dx if n = 5          - k 2 n 2 m h0 (ξ 0 )δ n-2 + o(δ n-2 ) as δ → 0, ξ → ξ 0 and h → h 0 in C 2 (M ).
Proof of Proposition 9.4. We have

J h (W ) = 1 2 M |∇W | 2 + hW 2 dv g - 1 2 M W 2 + dv g (114) = 1 2 M RW dv g + 1 2 - 1 2 M W 2 + dv g .
Using that U ≥ 0, we obtain 112) and (115) into (114), and using [START_REF] Vétois | Infinitely many solutions for cubic nonlinear Schrödinger equations in dimension four[END_REF] and (38), we obtain

(115) W 2 + = (U + B) 2 + = U 2 + 2 BU 2 -1 + O B 2 U 2 -2 + |B| 2 . Plugging (
J h (W ) = 1 n M U 2 dv g - 1 2 M BU 2 -1 dv g + O M ( Ũ 2 -2 B 2 + Ũ |B| 2 -1 + |B| 2 ) dv g . Since B = k n δ n-2
2 β, the pointwise estimate (104), the definition ( 14) and ( 57) yield

(116) J h (W ) = 1 n R n U 2 1,0 dx - 1 2 M BU 2 -1 dv g + O     δ 2 if n = 3 δ 4 (ln(1/δ)) 3 if n = 4 δ 4 if n = 5     .
The definitions (96) and (100) of β and B yield (117)

∆ g B + hB = U 2 -1 -(∆ g U + hU ) in M.
Therefore, we obtain

M BU 2 -1 dv g = M B(U 2 -1 -(∆ g U + hU )) dv g + M B(∆ g U + hU ) dv g = M |∇B| 2 + hB 2 dv g + M (∆ g B + hB)U dv g = M |∇B| 2 + hB 2 dv g -δ n-2 2 M F δ (ξ, •)U dv g - M ĥξ U 2 dv g . Since B = k n δ n-2
2 β, using (97) and (95) together with Lebesgue's convergence theorem, we obtain (118)

M BU 2 -1 dv g = δ n-2 k 2 n M |∇β h,0,ξ | 2 + hβ 2 h,0,ξ dv g - 1 k n M F 0 (ξ, •)Γ ξ dv g - M ĥξ U 2 dv g + o(δ n-2 ).
Since U (x) 2 ≤ Cδ n-2 d g (ξ, x) 4-2n , letting ξ → ξ 0 and h → h 0 in C 2 (M ), integration theory yields

M ĥξ U 2 dv g = δk 2 n M (h 0 ) ξ0 Γ 2 ξ0 dv g + o(δ) when n = 3.
We now assume that n ∈ {4, 5}. We write

M ĥξ U 2 dv g = ĥξ (ξ) M U 2 dv g + ∂ ξi ĥξ (ξ) M x i U 2 dv g + M ( ĥξ -ĥξ (ξ) -∂ ξi ĥξ (ξ)x i )U 2 dv g ,
where the coordinates are taken with respect to the exponential chart at ξ. As one checks, there exists C > 0 such that

| ĥξ -ĥξ (ξ) -∂ ξi ĥξ (ξ)x i |U 2 ≤ Cδ n-2 d g (ξ, x) 6-2n
for all x, ξ ∈ M , x = ξ. Since n < 6 and ξ remains in a neighborhood of ξ (so that the exponential chart remains nicely bounded), integration theory then yields

M ( ĥξ -ĥξ (ξ) -∂ ξi ĥξ (ξ)x i )U 2 dv g = δ n-2 k 2 n M ( ĥξ -ĥξ (ξ) -∂ ξi ĥξ (ξ)x i )Γ 2 ξ dv g + o(δ n-2 ).
Furthermore, letting ξ → ξ 0 , h → h 0 and using (4), we obtain (119)

M ( ĥξ -ĥξ (ξ) -∂ ξi ĥξ (ξ)x i )U 2 dv g = δ n-2 k 2 n M (h 0 ) ξ0 Γ 2 ξ0 dv g + o(δ n-2 ).
Using the explicit expression ( 14) of U together with the facts that Λ ξ (ξ) = 1, ∇Λ ξ (ξ) = 0 and |x|∂ xi U = O( Ũ ), we obtain

M ĥξ U ∂ xi U dv g = Br 0 (0)
ĥξ (exp

g ξ ξ (x))U δ,0 ∂ xi U δ,0 (1 + O(|x| 2 )) dx + O Br 0 (0)
| ĥξ (exp

g ξ ξ (x))||x| Ũ 2 δ,0 dx + O(δ n-2 )
With a Taylor expansion of ĥξ , using the radial symmetry of U δ,0 and the explicit expressions given in [START_REF] Li | Yamabe type equations on three-dimensional Riemannian manifolds[END_REF], we then obtain that there exists c 4 , c 5 > 0 such that

M ĥξ U ∂ xi U dv g = -∂ ξi ϕ h (ξ)      0 if n = 3 c 4 δ 2 ln(1/δ) if n = 4 c 5 δ 2 if n = 5      + O(δ n-2 )
and then

∂ ξi J h (W ) = ∂ ξi ϕ h (ξ)      0 if n = 3 c 4 δ 2 ln(1/δ) if n = 4 c 5 δ 2 if n = 5      + O    δ if n = 3 δ 2 + D h,ξ δ 2 ln(1/δ) if n = 4 δ 3 + D h,ξ δ 2 if n = 5.    .
We now consider the case where i = 0, so that ∂ pi = ∂ p0 = ∂ δ . In this case, we have

M ∂ δ U (∆ g + h)B dv g = - M ĥξ U ∂ δ U dv g -δ n-2 2 M F ∂ δ U dv g = - M ( ĥξ (ξ) + x i ∂ ξi ĥξ (ξ))U ∂ δ U dv g - M (δ n-2 2 F + ( ĥξ -ĥξ (ξ) -x i ∂ ξi ĥξ (ξ))U )∂ δ U dv g ,
where the coordinates are taken with respect to the exponential chart at ξ. With (18), ( 16) and [START_REF]Compactness of solutions to the Yamabe problem. III[END_REF], arguing as in the proof of (119), we obtain

δ n-2 2 M (F + ( ĥξ -ĥξ (ξ) -x i ∂ ξi ĥξ (ξ))δ -n-2 2 U )∂ δ U dv g = n -2 2 k 2 n δ -1 δ n-2 M F ξ,0 k n + ĥξ0 Γ h ξ0 Γ h ξ0 dv g + o(δ -1 δ n-2 ).
Using (61) and arguing as in the estimate of (60), we obtain that there exist c 4 , c 5 > 0 such that

M ( ĥξ (ξ) + x i ∂ ξi ĥξ (ξ))U ∂ δ U dv g = ĥξ (ξ) δ B0(r0) U δ,0 Z δ,0 dx + o(δ -1 δ n-2 ) = ĥξ (ξ) δ      0 if n = 3 c 4 δ 2 ln(1/δ) if n = 4 c 5 δ 2 if n = 5      + o(δ -1 δ n-2 ).
Putting these estimates together yields

-(2 -1) M U 2 -2 B∂ pi U dv g = ĥξ (ξ) δ      0 if n = 3 c 4 δ 2 ln(1/δ) if n = 4 c 5 δ 2 if n = 5      - n -2 2 k 2 n I h0,ξ0 δ -1 δ n-2 + o(δ -1 δ n-2 ),
where I h0,ξ0 is as in (120). Since I h0,ξ0 = m h0 (ξ 0 ) (see ( 121)), we obtain (122) and (123) up to the value of the constants. These values then follow from Proposition 9.4 together with the above estimates. This ends the proof of Proposition 9.4.

Theorem 1.4 for n ∈ {4, 5} will be proved in Section 10.

Proof of Theorem 1.4

We let h 0 , f ∈ C p (M ), p ≥ 2, and ξ 0 ∈ M satisfy the assumptions of Theorem 1.4. For small > 0 and τ ∈ R n , we define (126) h := h 0 + f and ξ (τ ) := exp

g ξ 0 ξ0 ( √ τ ).
We fix R > 0 and 0 < a < b to be chosen later.

10.1. Proof of Theorem 1.4 for n ≥ 6. In this case, we let (δ ) >0 > 0 be such that δ → 0 as → 0. We define (127) δ (t) := δ t and F (t, τ

) := J h (U δ (t),ξ (τ ) + Φ h ,0,δ (t),ξ (τ ) )
for all τ ∈ R n such that |τ | < R and t > 0 such that a < t < b. Using the assumption ϕ h0 (ξ 0 ) = |∇ϕ h0 (ξ 0 )| = 0, we obtain

ϕ h (ξ (τ )) = 1 2 ∇ 2 ϕ h0 (ξ 0 )[τ, τ ] + f (ξ 0 ) + o( ) and ∇ϕ h (ξ (τ )) = ∇ 2 ϕ h0 (ξ 0 )[τ, •] √ + o( √ )
as → 0 uniformly with respect to |τ | < R. We distinguish two cases:

Case n ≥ 7. In this case, we set δ := √ . It follows from (54) that

(128) lim →0 F (t, τ ) -1 n R n U 2 1,0 dx 2 = E 0 (t, ξ) in C 0 loc ((0, ∞) × R n ),
where

E 0 (t, τ ) := C n 1 2 ∇ 2 ϕ h0 (ξ 0 )[τ, τ ] + f (ξ 0 ) t 2 -D n K h0 (ξ 0 )t 4 , with (129)
C n := 1 2 R n U 2 1,0 dx and D n := 1 4n R n |x| 2 U 2 1,0 dx.
Furthermore, we have

∂ t F (t, τ ) = √ ∂ δ J h (U δ (t),ξ (τ ) + Φ δ (t),ξ (τ ) )
and

∂ τi F (t, τ ) = √ ∂ ξi J h (U δ (t),ξ (τ ) + Φ δ (t),ξ (τ ) ) .
Therefore, it follows from (66) and (67) that the limit in (128) actually holds in

C 1 loc ((0, ∞) × R n ).
Assuming that f (ξ 0 ) × K h0 (ξ 0 ) > 0, we can define

t 0 := C n f (ξ 0 ) 2D n K h0 (ξ 0 )
.

As one checks, (t 0 , 0) is a critical point of E 0 . In addition, the Hessian matrix at the critical point (t 0 , 0) is

∇ 2 E 0 (t 0 , 0) = -8t 2 0 D n K h0 (ξ 0 ) 0 0 t 2 0 C n ∇ 2 ϕ h0 (ξ 0 )
. Therefore, if ξ 0 is a nondegenerate critical point of ϕ h0 , then (t 0 , 0) is a nondegenerate critical point of E 0 . With the convergence in C 1 loc ((0, ∞) × R n ), we then obtain that there exists a critical point (t , τ ) of F such that (t , τ ) → (t 0 , 0) as → 0. It then follows from (29) that u := U δ (t ),ξ (τ ) + Φ h ,0,δ (t ),ξ (τ ) is a solution to [START_REF]Compactness for Yamabe metrics in low dimensions[END_REF]. As one checks, u 0 weakly in L 2 (M ) and (u ) blows up with one bubble at ξ 0 . This proves Theorem 1.4 for n ≥ 7.

Case n = 6. In this case, we let δ > 0 be such that (130) δ 2 ln(1/δ ) = .

As one checks, δ → 0 as → 0. As in the previous case, we obtain

lim →0 F (t, τ ) -1 n R n U 2 1,0 dx δ 2 = E 0 (t, ξ) in C 1 loc ((0, ∞) × R n ),
where

E 0 (t, τ ) := C 6 1 2 ∇ 2 ϕ h0 (ξ 0 )[τ, τ ] + f (ξ 0 ) t 2 -24 2 ω 5 K h0 (ξ 0 )t 4
for all t > 0 and τ ∈ R n . As in the previous case, E 0 has a nondegenerate critical point ( t0 , 0), which yields the existence of a critical point of F and, therefore, a blowing-up solution to [START_REF]Compactness for Yamabe metrics in low dimensions[END_REF] satisfying the desired conditions. This proves Theorem 1.4 for n = 6.

10.2. Proof of Theorem 1.4 for n ∈ {4, 5}. When n ∈ {4, 5}, we define

F (t, τ ) := J h (U δ (t),ξ (τ ) + B h ,δ (t),ξ (τ ) + Φ h ,0,δ (t),ξ (τ ) ),
where δ (t) will be chosen differently depending on the dimension.

Case n = 5. In this case, we set δ (t) := t . It follows from (113) that lim

→0 F (t, τ ) -1 n R n U 2 1,0 dx 3 = E 0 (t, ξ) in C 0 loc ((0, ∞) × R n ),
where

E 0 (t, τ ) := C 5 1 2 ∇ 2 ϕ h0 (ξ 0 )(τ, τ ) + f (ξ 0 ) t 2 - k 2 5 2 m h0 (ξ 0 )t 3 .
It follows from the C 1 -estimates of Proposition 9.5 that the convergence holds in

C 1 loc ((0, ∞) × R n ).
Assuming that f (ξ 0 ) × m h0 (ξ 0 ) > 0, we then define

t 0 := 4C 5 f (ξ 0 ) (n -2)k 2 5 m h0 (ξ 0 )
.

As in the previous cases, we obtain that (t 0 , 0) is a nondegenerate critical point of E 0 , which yields the existence of a critical point for F and, therefore, a blowing-up solution to [START_REF]Compactness for Yamabe metrics in low dimensions[END_REF] satisfying the desired conditions. This proves Theorem 1.4 for n = 5.

Case n = 4. In this case, we set δ (t) := e -t/ . It follows from the C 1 -estimates of Proposition 9.5 that lim

→0 -δ (t) -2 ∂ t F (t, τ ), δ (t) -2 ∂ τ F (t, τ ) = (ψ 0 (t, τ ), ψ 1 (t, τ )) in C 0 loc ((0, ∞) × R n ),
where

ψ 0 (t, τ ) := C 4 1 2 ∇ 2 ϕ h0 (ξ 0 )(τ, τ ) + f (ξ 0 ) t - n -2 2 k 2 n m h0 (ξ 0 )
and

ψ 1 (t, τ ) := 1 2 C 4 ∇ 2 ϕ h0 (ξ 0 )[τ, •]t.
As one checks, since ξ 0 is a nondegenerate critical point of ϕ h0 , the function ψ has a unique zero point in (0, ∞) × R n which is of the form (t 0 , 0) for some t 0 > 0. Furthermore, the nondegeneracy implies that the Jacobian determinant of ψ at (t 0 , 0) is nonzero and, therefore, the degree of ψ at 0 is well-defined and nonzero. The invariance of the degree under uniform convergence then yields the existence of a critical point (t , τ ) of F such that (t , τ ) → (t 0 , 0) as → 0. It then follows from (29) that

u := U δ (t),ξ (τ ) + B h ,δ (t),ξ (τ ) + Φ h ,0,δ (t),ξ (τ )
is a critical point of J h that blows up at ξ 0 and converges weakly to 0 in L 2 (M ). This proves Theorem 1.4 for n = 4.

11. Proof of Theorem 1.5

We let h 0 , f ∈ C p (M ), p ≥ 2, u 0 ∈ C 2 (M ) and ξ 0 ∈ M satisfy the assumptions of Theorem 1.5. We let h be as in [START_REF]Compactness for Yamabe metrics in low dimensions[END_REF]. We let ξ (τ ) and δ (t) be as in ( 126) and (127). Since u 0 is nondegenerate, the implicit function theorem yields the existence of 0 ∈ (0, 0 ) and (u 0,

) 0< < 0 ∈ C 2 (M ) such that (131) ∆ g u 0, + h u 0, = u 2 -1 0,
, u 0, > 0 in M and (u 0, ) is smooth with respect to , which implies in particular that

u 0, -u 0 C 2 ≤ C .
We fix 0 < a < b and R > 0 to be chosen later. We define

F (t, τ ) := J h (u 0, + U δ (t),ξ (τ ) + Φ h ,u0, ,δ (τ ),ξ (τ ) )
for all τ ∈ R n such that |τ | < R and t > 0 such that a < t < b. With (85), we obtain that for n ≥ 7,

F (t, τ ) = J h (u 0, ) + 1 n R n U 2 1,0 dx + C n 1 2 ∇ 2 ϕ h0 (ξ 0 )(τ, τ ) + f (ξ 0 ) t 2 δ 2 + o( δ 2 ) -D n K h0 (ξ 0 )t 4 δ 4 + o(δ 4 ) -B n u 0 (ξ 0 )t n-2 2 δ n-2 2 + o(δ n-2 2 )
as → 0 uniformly with respect to a < t < b and |τ | < R, where C n and D n are as in (129) and

B n := R n U 2 -1 1,0 dx.
We distinguish three cases:

Case 7 ≤ n ≤ 10, that is n ≥ 7 and n-2 2 ≤ 4. In this case, we set δ :=

2 n-6 , so that δ 2 = δ n-2 2 . We then obtain (132) lim →0 F (t, τ ) -A δ 2 = E 0 (t, τ )
uniformly with respect to a < t < b and |τ | < R, where (133)

A := J h (u 0, ) + 1 n R n U 2 1,0 dx and E 0 (t, τ ) := C n 1 2 ∇ 2 ϕ h0 (ξ 0 )(τ, τ ) + f (ξ 0 ) t 2 -(B n u 0 (ξ 0 ) + 1 n=10 D n K h0 (ξ 0 )) t n-2 2 .
Moreover, the estimates (87) and (88) yield the convergence (132) in C 1 loc ((0, ∞) × R n ). Straightforward changes of variable yield

B 10 D 10 = 40 R 10 U 3/2 1,0 dx R 10 |x| 2 U 3/2 1,0 dx = 40 ∞ 0 r 9 dr (1+r 2 ) 6 ∞ 0 r 11 dr (1+r 2 ) 8 = 40 ∞ 0 s 4 ds (1+s) 6 ∞ 0 s 5 ds (1+s) 8 .
Integrating by parts, we then obtain The assumption K h0,u0 (ξ 0 ) = 0 then gives B n u 0 (ξ 0 ) + 1 n=10 D n K h0 (ξ 0 ) = 0 with same sign as f (ξ 0 ). As in the proof of Theorem 1.4 for n ≥ 7, we obtain that E 0 has a unique critical point in (0, ∞) × R n , say (t 0 , 0), and this critical point is nondegenerate. Mimicking again the proof of Theorem 1.4 for n ≥ 7, we obtain the existence of a critical point (t , τ ) of F such that (t , τ ) → (t 0 , 0) as → 0. It then follows that

B 10 D 10 = 40 × 4 5 × 3 4 × 2 3 × 1 2 ∞ 0 ds (1+s) 2 5 7 × 4 6 × 3 5 × 2 4 × 1 3 ∞ 0 ds (1+s) 3 = 40 × 6 × 7 ∞ 0 ds (1+s)
u := u 0, + U δ (t ),ξ (τ ) + Φ h ,u0, ,δ (t ),ξ (τ )
is a solution to [START_REF]Compactness for Yamabe metrics in low dimensions[END_REF]. As one checks, u 0 weakly in L 2 (M ) and (u ) blows up with one bubble at ξ 0 . This proves Theorem 1.5 for 7 ≤ n ≤ 10.

Case 4 < n-2
4 , that is n ≥ 11. In this case, we set δ := √ , so that

δ 2 = δ 4 and δ n-2 2
= o(δ 4 ) as → 0.

We then obtain

lim →0 F (t, τ ) -A δ 2 = E 0 (t, τ ) in C 0 loc ((0, ∞) × R n ),
where A is as in ( 133) and

E 0 (t, τ ) := C n 1 2 ∇ 2 ϕ h0 (ξ 0 )(τ, τ ) + f (ξ 0 ) t 2 -D n K h0 (ξ 0 )t 4 .
As in the previous case, we obtain that the convergence holds in C 1 loc ((0, ∞) × R n ) and E 0 has a nondegenerate critical point in (0, ∞) × R n , which yields the existence of a blowing-up solution (u ) to [START_REF]Compactness for Yamabe metrics in low dimensions[END_REF] satisfying the desired conditions. This proves Theorem 1.5 for n ≥ 11.

Case n=6. Note that in this case, we have 2 -1 = 2. Differentiating (131) with respect to at 0, we obtain

(∆ g + h 0 -2u 0 )(∂ u 0, ) |0 + f u 0 = 0 in M.
Using that u 0 is nondegenerate, we then obtain

(∂ u 0, ) |0 = -(∆ g + h 0 -2u 0 ) -1 (f u 0 ). It follows that ϕ h ,u = h -2u 0, -c n Scal g = ϕ h0,u0 + f + o( ) as → 0, where f := f + 2(∆ g + h 0 -2u 0 ) -1 (f u 0 ).
We let δ > 0 be as in (130). With (91), we then obtain

lim →0 F (t, τ ) -A δ 2 = E 0 (t, τ ) in C 0 loc ((0, ∞) × R n ),
where A is as in (133) and

E 0 (t, τ ) := C 6 1 2 ∇ 2 ϕ h0 (ξ 0 )(τ, τ ) + f (ξ 0 ) t 2 -24 2 ω 5 K h0,u0 (ξ 0 )t 4 .
As in the previous case, using (92) and (93), we obtain that the convergence holds in C 1 loc ((0, ∞) × R n ). Furthermore, using [START_REF] Druet | Blow-up examples for second order elliptic PDEs of critical Sobolev growth[END_REF], we obtain that E 0 has a nondegenerate critical point in (0, ∞) × R n and, therefore, that there exists a blowing-up solution to (8) satisfying the desired conditions. This proves Theorem 1.5 for n = 6.

12. Proof of Theorem 1.2

We let h 0 ∈ C p (M ), 1 ≤ p ≤ ∞, and ξ 0 ∈ M be such that ∆ g + h 0 is coercive and the condition (4) is satisfied. In the case where p = 1, a standard regularization argument give the existence of ( ĥ ) >0 ∈ C 2 (M ) such that ĥ → h 0 in C 1 (M ) as → 0. In the case where p ≥ 2, we set ĥ = h 0 . We then define h := ĥ + f , where f (x) := χ(x)((h 0 -ĥ )(ξ 0 ) + ∇(h 0 -ĥ )(ξ 0 ), x + λ |x| 2 ), where λ > 0, χ is a smooth cutoff function around 0 and the coordinates are taken with respect to the exponential chart at ξ 0 . As one checks, for some suitable λ → 0, we then have that h

→ h 0 in C p (M ), ϕ h (ξ 0 ) = ϕ h0 (ξ 0 ) = 0, |∇ϕ h (ξ 0 )| = |∇ϕ h0 (ξ 0 )| = 0 and for small > 0, ξ 0 is a nondegenerate critical point of ϕ h .
Assume first that n ∈ {4, 5}. Then the mass of h is defined at ξ 0 . As is easily seen, there exists ψ : (0, 1) → (0, 1) such that ψ( ) → 0 as → 0 and either {m hψ( ) (ξ 0 ) > 0 for all ∈ (0, 1)}, {m hψ( ) (ξ 0 ) < 0 for all ∈ (0, 1)} or {m hψ( ) (ξ 0 ) = 0 for all ∈ (0, 1)}. If m h ψ( ) (ξ 0 ) = 0 for all ∈ (0, 1), then it follows from Proposition 8.1 that if we choose µ > 0 small enough, then we obtain mȟ (ξ 0 ) < 0 for small > 0 with ȟ = hψ( ) + µ χ| • | 2 . Therefore, in all cases, we can assume that m h (ξ 0 ) = 0 for small > 0, with a sign independent of . Assume now that n ≥ 6. With a similar argument, we can assume that, for small > 0, K h (ξ 0 ) = 0 with a sign independent of , where K h (ξ 0 ) is as in [START_REF] Druet | From one bubble to several bubbles: the low-dimensional case[END_REF].

In all cases, we can now fix f 0 ∈ C ∞ (M ) such that f 0 (ξ 0 ) × K h (ξ 0 ) > 0 for small > 0. It then follows from Theorem 1.4 that there exist α > 0 and a family (ũ ,α ) 0<α<α of solutions to the equation

∆ g ũ ,α + ( h + αf 0 )ũ ,α = ũ2 -1
,α , ũ ,α > 0 in M such that ũ ,α 0 weakly in L 2 (M ) and (ũ ,α ) α blows up with one bubble at ξ 0 as α → 0. Therefore, we obtain that for every > 0, there exists α > 0 such that

0 < α < min( , α ), ũ ,α 2 < , M |ũ ,α | 2 dv g - R n U 2 1,0 dx < and M \B (ξ0) |ũ ,α | 2 dv g < .
We then define u := ũ ,α , so that

∆ g u + h u = u 2 -1 in M, where h := h + α f 0 = h 0 + f + α f 0 .
As one checks, u 0 weakly in L 2 (M ) and (u ) blows up with one bubble at ξ 0 as → 0. This proves Theorem 1.2.

Proof of Theorem 1.3

We let h 0 ∈ C p (M ), 1 ≤ p ≤ ∞, u 0 ∈ C 2 (M ) and ξ 0 ∈ M be such that ∆ g + h 0 is coercive, u 0 is a solution of (1) and the condition (6) is satisfied. We begin with proving the following: Lemma 13.1. There exists a neighborhood Ω 0 of ξ 0 and families ( h

) >0 ∈ C p (M ) and (ũ ) >0 ∈ C 2 (M ) such that h → h 0 in C p (M ), ũ → u 0 in C 2 (M ) as → 0, h ≡ h 0 and ũ ≡ u 0 in Ω 0 and ũ is a nondegenerate solution of (134) ∆ g ũ + h ũ = ũ2 -1 , ũ > 0 in M for all > 0.
Proof of Lemma 13.1. For all v ∈ C p+2 (M ) such that v > -u 0 , we define

u(v) := u 0 + v and h(v) := u(v) 2 -2 - u 2 -1 0 -h 0 u 0 + ∆ g v u(v) = u(v) 2 -2 - ∆ g u(v) u(v) , so that (135) ∆ g u(v) + h(v)u(v) = u(v) 2 -1 in M.
By elliptic regularity, we have

u 0 ∈ C p+1 (M ). Since moreover h 0 ∈ C p (M ) and v ∈ C p+2 (M ), we obtain that u(v) ∈ C p+1 (M ) and h(v) ∈ C p (M ). Furthermore, we have that h(v) → h 0 in C p (M ) and u(v) → u 0 in C 2 (M ) as v → 0 in C p+2 (M ).
As is easily seen, to prove the lemma, it suffices to show that there exists a neighborhood Ω 0 of ξ 0 and a family (v ) >0 ∈ C p+2 (M ) such that v → 0 in C p+2 (M ) as → 0, v ≡ 0 in Ω 0 and u(v ) is a nondegenerate solution of (135). Assume by contradiction that this is not true, that is for every neighborhood Ω of ξ 0 , there exists a small neighborhood

V Ω of 0 in C p+2 (M ) such that for every v ∈ V Ω , if v ≡ 0 in Ω, then u(v) is degenerate i.e. there exists φ(v) ∈ K v \ {0}, where
K v := {φ ∈ H 2 1 (M ) : ∆ g φ + h(v)φ = (2 -1)u(v) 2 -2 φ in M }.
By renormalizing, we can assume that φ(v) ∈ S Kv := {φ ∈ K v : φ H 2 1 = 1}. Since h(tv), u(tv) → h 0 , u 0 in C 0 (M ) as t → 0, it then follows that there exists φ v ∈ K 0 and (t k ) k∈N > 0 such that t k → 0 and φ(t k v) φ v weakly in H 2 1 (M ). By compactness of the embedding H 2 1 (M ) → L 2 (M ), we obtain that φ(t k v) → φ v strongly in L 2 (M ). By standard elliptic theory that we apply to the linear equation satisfied by φ(t k v), we then obtain that φ(t k v) → φ v strongly in H 2 1 (M ), so that in particular φ v ∈ S K0 . We then define

ψ k (v) := φ(t k v) -φ v t k .
It is easy to check that ψ k (v) satisfies the equation

(136) ∆ g ψ k (v) + h 0 ψ k (v) = (2 -1)u 2 -2 0 ψ k (v) + f k (v)φ(t k v) in M,
where

f k (v) := 1 t k ((2 -1)(u(t k v) 2 -2 -u 2 -2 0 ) + h 0 -h(t k v)) = 1 t k ((2 -2)(u(t k v) 2 -2 -u 2 -2 0 ) + t k u 0 ∆ g v -v∆ g u 0 u 0 u(t k v) ).
A straightforward Taylor expansion gives

(137) f k (v) = (2 -2) 2 u 2 -3 0 v + u -1 0 ∆ g v -u -2 0 v∆ g u 0 + o(1) = u -1 0 L 0 (v) + o(1), as k → ∞, uniformly in v ∈ V Ω , where (138) L 0 (v) := ∆ g v + h 0 v -(1 -(2 -2) 2 )u 2 -2 0 v. It follows that Π K ⊥ 0 (ψ k (v)) H 2 1 ≤ C f k (v)φ(t k v) 2n n+2 ≤ C φ(t k v) 2n n+2 ≤ C φ(t k v) H 2 1 ≤ C,
where Π K ⊥ 0 is the orthogonal projection of H 2 1 onto K ⊥ 0 and the letter C stands for positive constants independent of k ∈ N and v ∈

V Ω . Since (Π K ⊥ 0 (ψ k (v))) k is bounded in H 2 1 (M )
, up to a subsequence, we may assume that there exists

ψ v ∈ K ⊥ 0 such that Π K ⊥ 0 (ψ k (v)) ψ v weakly in H 2 1 (
M ). Passing to the limit in (136) and using (137), we then obtain that ψ v satisfies the equation

∆ g ψ v + h 0 ψ v = (2 -1)u 2 -2 0 ψ v + u -1 0 L 0 (v)φ v in M.
In particular, since φ v ∈ K 0 , multiplying this equation by φ v and integrating by parts yields

(139) M u -1 0 L 0 (v)φ 2 v dv g = 0.
We now construct v contradicting (139). For every α > 0, we choose Ω := B α (ξ 0 ) and we consider the neighborhood V Bα(ξ0) of 0 in C p+2 (M ). We let r α ∈ (0, α) be such that B 0 (r α ) ⊂ V Bα(ξ0) and χ ∈ C ∞ (R) be such that χ(t) = 0 for t ≤ 1 and χ(t) = 1 for t ≥ 2. We define v α (x) := e -1/α r α χ(d g (x, ξ 0 )/α)u 0 (x) for all x ∈ M and α > 0.

As one checks, for small α > 0,

v α ≡ 0 in B α (ξ 0 ) and v α ∈ B 0 (r α ) ⊂ V Bα(ξ0) . (140)
Therefore, u(v α ) is degenerate and the above analysis applies. Since φ vα H 2 1 = 1, φ vα ∈ K 0 ⊂ C 2 (M ) and K 0 is of finite dimension, up to a subsequence, we can assume that there exists φ 0 ∈ K 0 such that (141) lim

α→0 φ vα = φ 0 = 0 in C 2 (M ).
Since L 0 is self-adjoint, it follows from (139) that M v α L 0 (u -1 0 φ 2 vα ) dv g = 0 for all > 0.

Since e 1/α r -1 α v α → u 0 in L 2 (M ) as α → 0, passing to the limit in this equation and using (140) and (141), we obtain M u 0 L 0 (u -1 0 φ 2 0 ) dv g = 0.

Integrating again by parts and noting that L 0 (u 0 ) = (2 -2) 2 u 2 -1 0 , we then obtain

0 = M u -1 0 φ 2 0 L 0 (u 0 ) dv g = (2 -2) 2 M u 2 -2 0 φ 2 0 dv g ,
which is a contradiction since u 0 > 0 and φ 0 ≡ 0. This ends the proof of Lemma 13.1.

We can now end the proof of Theorem 1.3. We let Ω 0 , ( h ) >0 and (ũ ) >0 be given by Lemma 13.1. Since h ≡ h 0 and ũ ≡ u 0 in Ω 0 , we obtain that ϕ h ,ũ ≡ ϕ h0,u0 in Ω 0 and, therefore, ϕ h ,ũ (ξ 0 ) = |∇ϕ h ,ũ (ξ 0 )| = 0. For every > 0, we can then mimick the first part of the proof of Theorem 1.2 to construct a family ( h ,α ) α>0 ∈ C max(p,2) (M ) such that h ,α → h in C p (M ) as α → 0, ϕ h ,α ,ũ (ξ 0 ) = 0, ξ 0 is a nondegenerate critical point of ϕ h ,α ,ũ and K h ,α ,ũ (ξ 0 ) = 0. We now distinguish two cases:

Case n ≥ 7. Note that in this case, we have ϕ h ,α ,ũ = ϕ h . Since ũ is nondegenerate and h ,α → h in C 1 (M ) as α → ∞, the implicit function theorem gives that for small α > 0, there exists a nondegenerate solution ũ ,α ∈ C 2 (M ) to the equation ∆ g ũ ,α + h ,α ũ ,α = ũ2 -1 ,α , ũ ,α > 0 in M such that ũ ,α → ũ in C 2 (M ) as α → 0. Applying Theorem 1.5, we then obtain that there exist β ,α > 0, ( h ,α,β ) 0<β<β ,α ∈ C max(p,2) (M ) and (ũ ,α,β ) 0<β<β ,α ∈ C 2 (M ) satisfying ∆ g ũ ,α,β + h ,α,β ũ ,α,β = ũ2 -1

,α,β in M, ũ ,α,β > 0 for all 0 < β < β ,α and such that h ,α,β → h ,α in C max(p,2) (M ), ũ ,α,β ũ ,α weakly in L 2 (M ) and (ũ ,α,β ) β blows up with one bubble at ξ 0 as β → 0. Therefore, we obtain that for every > 0, there exists α ∈ (0, ) and β > 0 such that h ,α -h C p < , ũ ,α -ũ C 2 < , 0 < β < min( , β ,α ), ũ ,α ,β -u 0 2 < , |ũ ,α ,β -ũ ,α | 2 dv g < .

We then define u := ũ ,α ,β , so that ∆ g u + h u α = u 2 -1 in M, where h := h ,α ,β .

As one checks, h → h 0 in C p (M ), u u 0 weakly in L 2 (M ) and (u ) blows up with one bubble at ξ 0 as → 0. This proves Theorem 1.3 for n ≥ 7.

Case n = 6. In this case, we have ϕ h ,α ,ũ = ϕ h ,α -2ũ . Furthermore, noting that 2 -1 = 2 when n = 6, we can rewrite the equation (134) as

∆ g ũ + ( h -2ũ )ũ = -ũ 2 in M.
Since h ,α -2ũ → h -2ũ in C 0 (M ) as α → 0, a standard minimization method gives that for small α > 0, there exists a unique nondegenerate solution ũ ,α to the equation ∆ g ũ ,α + ( h ,α -2ũ )ũ ,α = -ũ 2 ,α , ũ ,α > 0 in M. As is easily seen, this equation can be rewritten as (142) ∆ g ũ ,α + h ,α ũ ,α = ũ2

,α , ũ ,α > 0 in M, where h ,α := h ,α -2ũ + 2ũ ,α .

Since h ,α → h in C p (M ) as α → 0, we obtain that h ,α → h in C p (M ) and ũ ,α → ũ in C p+1 (M ) as α → 0. Furthermore, since ũ is nondegenerate, we have that ũ ,α is nondegenerate for small α > 0. Similarly, since K h ,α ,ũ (ξ 0 ) = 0, we obtain that K h ,α ,ũ ,α (ξ 0 ) = 0 for small α > 0. Furthermore, we have ϕ h ,α ,ũ ,α = h ,α -2ũ ,α -c n Scal g = h ,α -2ũ -c n Scal g = ϕ h ,α ,ũ .

In view of the properties satisfied by h ,α , it follows that ϕ h ,α ,ũ ,α (ξ 0 ) = 0 and ξ 0 is a nondegenerate critical point of ϕ h ,α ,ũ ,α . Applying Theorem 1.5, we then obtain that there exist β ,α > 0, ( h ,α,β ) 0<β<β ,α ∈ C max(p,2) (M ) and (ũ ,α,β ) 0<β<β ,α ∈ C 2 (M ) satisfying ∆ g ũ ,α,β + h ,α,β ũ ,α,β = ũ2 -1 ,α,β in M, ũ ,α,β > 0 for all 0 < β < β ,α and such that h ,α,β → h ,α in C max(p,2) (M ), ũ ,α,β ũ ,α weakly in L 2 (M ) and (ũ ,α,β ) β blows up with one bubble at ξ 0 as β → 0. Finally, as in the previous case, we obtain the existence of α > 0 and β > 0 such that u := ũ ,α ,β satisfies the desired conditions. This proves Theorem 1.3 for n = 6.

14. Necessity of the condition on the gradient Theorem 14.1. Let (M, g) be a compact Riemannian manifold of dimension n ≥ 4. Let h 0 ∈ C 1 (M ) be such that ∆ g + h 0 is coercive. Assume that there exist families (h ) >0 ∈ C p (M ) and (u ) >0 ∈ C 2 (M ) satisfying (2) and such that h → h 0 strongly in C 1 (M ). Assume that (M, g) is locally conformally flat. If (u ) blows up with one bubble at some point ξ 0 ∈ M and u 0 weakly as → 0, then (4) holds true.

Proof of Theorem 14.1. Let ϕ h0 be as in [START_REF] Chen | Infinitely many solutions for the Schrödinger equations in R n with critical growth[END_REF]. The identity ϕ h0 (ξ 0 ) = 0 is a consequence of the results of Druet [START_REF] Druet | From one bubble to several bubbles: the low-dimensional case[END_REF][START_REF]La notion de stabilité pour des équations aux dérivées partielles elliptiques[END_REF]. Since (M, g) is locally conformally flat, there exists Λ ∈ C ∞ (M ) positive such that ĝ := Λ ∆ ĝ û + ĥ û = û2 -1 , û > 0 in M.

As one checks, on (M, ĝ), û blows-up at ξ 0 in the sense that û = U δ ,ξ + o(1) as → 0 in H 2 1 (M ), where U δ ,ξ is as in [START_REF] Pistoia | Clustering phenomena for linear perturbation of the Yamabe equation, Partial Differential Equations Arising from Physics and Geometry[END_REF] (with respect to the metric ĝ) and (δ , ξ ) → (0, ξ 0 ) as → 0. It then follows from Druet-Hebey-Robert [START_REF] Druet | Blow-up theory for elliptic PDEs in Riemannian geometry[END_REF] that there exist C, 0 > 0 such that for every ∈ (0, 0 ), for all x ∈ R n , where χ is a cutoff function on a small ball centered at ξ 0 , we have (145) lim

→0 U = U 1,0 = n(n -2) 1 + | • | 2 n-2 2 in C 2 loc (R n ).
Without loss of generality, via a chart, we may assume that ĝ is the Euclidean metric on B 2ν (ξ 0 ) for some ν > 0. We fix i ∈ {1, .. for all x ∈ M \ B ν/4 (ξ 0 ) and ∈ (0, 0 ). It then follows from (143) and standard elliptic theory that there exists

C 1 > 0 such that |∇û (x)| ≤ C 1 δ n-2 2
for all x ∈ M \ B ν/2 (ξ 0 ) and ∈ (0, 0 ). Plugging these inequalities into (146) yields ( 147)

Bν (ξ )
∂ xi ĥ û2 dx = O(δ n-2 ) as → 0.

On the other hand, with a change of variable, we obtain Combining this identity with (147), we obtain that ∂ xi (ϕ h0 Λ 2-2 )(ξ 0 ) = 0 when n ≥ 5. Since Λ > 0 and ϕ h0 (ξ 0 ) = 0, it follows that ∂ xi ϕ h0 (ξ 0 ) = 0 when n ≥ 5.

We now assume that n = 4. With (144), we obtain In all cases, we thus obtain that ∇ϕ h0 (ξ 0 ) = 0. This ends the proof of Theorem 14.1.

2

 2

M

  |ũ ,α ,β -ũ ,α | 2 dv g -

4 n- 2 g

 42 is flat around ξ 0 . Define û := Λ -1 u and ĥ := (h -c n Scal g ) Λ 2-2 + c n Scal ĝ .The conformal law (44) yields(143)

2 + 2 n-2 2 ≤ 2 n-2 2 2 2

 222222 d ĝ (x, ξ ) û (x) ≤ C δ δ 2 + d ĝ (x, ξ )for all x ∈ M and, definingU (x) := δ nχ(x)û (ξ + δ x)

Bν (ξ ) ∂ xi ĥ û2 dx = δ 2 B

 2 ν/δ (0) (∂ xi ĥ )(ξ + δ x)U (x) 2 dx.The control (144) gives U ≤ CU 1,0 . Therefore, when n ≥ 5, Lebesgue's dominated convergence Theorem and (145) yieldBν (ξ )∂ xi ĥ û2 dx = δ 2 ∂ xi ĥ (ξ ) dx + o(1) as → 0.

Bν  δ 2 Bν 2 +

 22 (ξ ) |x -ξ |û 2 dx = O(δ 2 ).Therefore, with (147), we obtain(148) ∂ xi ĥ (ξ ) = O |x -ξ | 2 n-2 dx ≥ Cδ 2 ln(1/δ ).It follows from (148) and (149) that ∂ xi ĥ (ξ ) = o(1) as → 0 and so again ∂ xi ϕ h0 (ξ 0 ) = 0 when n = 4.

  ., n}. Differentiating the Pohozaev identity for û on B ν (ξ ) (see for instance Ghoussoub-Robert [13, Proposition 7])where dσ is the volume element on ∂B ν (ξ ). It follows from (144) that there exists C(ν) > 0 such that û (x) ≤ C(ν)δ

	and integrating by parts, we obtain
	(146)	1 2 Bν (ξ )	∂ xi ĥ û2 dx
		=	∂Bν (ξ )	x i |x|	|∇û | 2 + ĥ û2 2	-	û2 2	-	x |x|	, ∇û ∂ xi û dσ(x),
						n-2
						2

Via the exponential chart, using the radial symmetry of U , we obtain

Br 0 (0)

since n < 6. It then follows from (61), (62), (63) and the above estimates that M ĥξ U 2 dv g = ĥξ (ξ)

Combining this estimate with (118), we obtain

where (120) I h0,ξ0 := M |∇β h0,0,ξ0 | 2 + h 0 β 2 h0,0,ξ0 dv g -

Integrating by parts and using the definition (96), we obtain

We now use (107) at the point ξ 0 , which makes sense since β h0,0,ξ0 is continuous on M . We then obtain (121)

Putting these results together yields (113), which proves Proposition 9.4.

We now estimate the derivatives of J h (W + Φ):

Proposition 9.5. We have

and

Proof of Proposition 9.5. We fix i ∈ {0, . . . , n}. With (112), ( 38) and ( 32), we obtain

The estimates (106) and ( 105) and the definition

where i0 is the Kronecker symbol. Since W = U + B, we then obtain

Differentiating (117), we obtain

Multiplying by B and integrating by parts, we then obtain (124)

We begin with estimating the left-hand-side of (124). Using that

With (96) and the pointwise estimates ( 106) and (105), we obtain

.

We now deal with the second term in the right-hand-side of (124). We first consider the case where i ≥ 1, so that ∂ pi = ∂ ξi . In this case, it follows from (73) that ∂ ξi U = -∂ xi U + O( Ũ ). Then, using (96), we obtain


Thomas Maitre

Lorenzo Guglielmetti

Defining optimal fluoroquinolone exposure against Mycobacterium tuberculosis : contribution of murine studies

Compared to in vitro models, the main advantage for using an animal model is its ability to model both innate and adaptive immune response that participate in vivo to the treatment efficacy, particularly against mycobacterial infections. Among animal models, the murine model has been used for more than sixty years and its capability to predict activity in human has been well established. When comparing results provided by murine studies to those provided by HFS, we have noted that the levofloxacin target AUC 0-24h /MIC ratio measured in mice was similar, around 150 [START_REF] Maitre | Are moxifloxacin and levofloxacin equally effective to treat XDR tuberculosis?[END_REF][START_REF] Deshpande | Levofloxacin Pharmacokinetics/Pharmacodynamics, Dosing, Susceptibility Breakpoints, and Artificial Intelligence in the Treatment of Multidrug-resistant Tuberculosis[END_REF]. Nonetheless, the target AUC 0-24h /MIC ratio for moxifloxacin was measured in mice by two independent research teams at around 100 [START_REF] Maitre | Are moxifloxacin and levofloxacin equally effective to treat XDR tuberculosis?[END_REF][START_REF] Shandil | Moxifloxacin, ofloxacin, sparfloxacin, and ciprofloxacin against Mycobacterium tuberculosis: evaluation of in vitro and pharmacodynamic indices that best predict in vivo efficacy[END_REF] i.e. two times higher than the one measured in the HFS study cited by the authors [START_REF] Gumbo | Selection of a moxifloxacin dose that suppresses drug resistance in Mycobacterium tuberculosis, by use of an in vitro pharmacodynamic infection model and mathematical modeling[END_REF]. If we take in consideration this higher ratio, the percentage of patients included in the study by Forsman et al. that reached the target exposure would be even lower, around 20-30%.

The authors suggest that, against strains harbouring the gyrA A90V mutation, generating a moxifloxacin MIC of 2 mg/L [START_REF] Maitre | Are moxifloxacin and levofloxacin equally effective to treat XDR tuberculosis?[END_REF], high doses of moxifloxacin from 600 to 800 mg daily should be sufficient to reach the target exposure to kill M. tuberculosis.

We agree with Forsman et al. that residual fluoroquinolone activity against the lowlevel resistant mutant A90V is a crucial issue, since this mutation represents the second most frequent DNA gyrase mutation, which is harboured by approximately 20% of fluoroquinolone-resistant strains [START_REF] Avalos | Frequency and Geographic Distribution of gyrA and gyrB Mutations Associated with Fluoroquinolone Resistance in Clinical Mycobacterium Tuberculosis Isolates: A Systematic Review[END_REF]. However, it appears doubtful that doubling the dose of moxifloxacin can restore the activity against strains with 4-to 8times higher MIC. Our group has measured the activity of moxifloxacin against A90V mutants in two different murine strains: these studies showed that high doses of moxifloxacin, equivalent to 800 mg/day in humans, generated bacteriostatic activity only [START_REF] Maitre | Are moxifloxacin and levofloxacin equally effective to treat XDR tuberculosis?[END_REF][START_REF] Poissy | Should moxifloxacin be used for the treatment of extensively drugresistant tuberculosis? An answer from a murine model[END_REF]. When combined with other second-line drugs, high doses of moxifloxacin increased the activity of the regimen but did not reach the level of the same regimen against a susceptible strain [START_REF] Fillion | Impact of fluoroquinolone resistance on bactericidal and sterilizing activity of a moxifloxacincontaining regimen in murine tuberculosis[END_REF]. Thus, adding moxifloxacin 800 mg/day to a drug regimen against M. tuberculosis strains harbouring gyrA A90V mutation may likely only have limited effectiveness. This benefit has to be balanced with toxicity, in particular QT interval prolongation, which is also reported in association with the use of new drugs such as bedaquiline or delamanid. In addition to the clinical trial cited by the authors, observational evidence has also shown that, in bedaquiline-based regimens for multidrug-resistant tuberculosis, high-dose moxifloxacin is the main risk factor for QT prolongation [START_REF] Guglielmetti | Long-term outcome and safety of prolonged bedaquiline treatment for multidrug-resistant tuberculosis[END_REF].

In conclusion, we agree with the authors that moxifloxacin 400 mg/day and levofloxacin 500 mg/day are suboptimal doses with regard to anti-tuberculous activity. This finding is even more relevant in light of the possible inclusion of fluoroquinolones in shorter regimens for drug-susceptible tuberculosis, where fluoroquinolone exposure could be further reduced by the co-administration of rifamycins [START_REF] Naidoo | Effect of rifampicin and efavirenz on moxifloxacin concentrations when co-administered in patients with drug-susceptible TB[END_REF]. Possible approaches may include routine therapeutic drug monitoring and/or the use of higher fluoroquinolone doses. However, the benefit-risk of the latter strategy should be carefully evaluated.

3 1

 3 Defining optimal fluoroquinolone exposure against Mycobacterium tuberculosis: contribution of murine studies Thomas Maitre1 , Lorenzo Guglielmetti1,2 , Nicolas Veziris 1,2,Sorbonne Université, INSERM, U1135, Centre d'Immunologie et des Maladies Infectieuses, Cimi-Paris, équipe 13, Paris, France 2 APHP, Groupe Hospitalier Universitaire Sorbonne Université, Hôpital Pitié-Salpêtrière, Centre National de Référence des Mycobactéries et de la Résistance des Mycobactéries aux Antituberculeux, Paris, France 3 APHP, Groupe Hospitalier Universitaire Sorbonne Université, Hôpital Saint-Antoine, Département de Bactériologie, Paris, France We read with interest the research letter published by Forsman et al. in the European Respiratory Journal[START_REF] Forsman | Suboptimal moxifloxacin and levofloxacin drug exposure during treatment of patients with multidrug-resistant tuberculosis: results from a prospective study in China[END_REF]. The authors determined the proportion of multidrug-resistant tuberculosis patients treated with moxifloxacin or levofloxacin who attained an optimal drug exposure for these drugs. The target exposure corresponded to the area under the curve (AUC 0-24h )/ minimal inhibitory concentration (MIC) ratio generating the optimal bactericidal activity. The authors based their calculations on the AUC 0-24h /MIC ratio measured in a preclinical model called Hollow Fiber System (HFS), an in vitro model used to assess anti-tuberculosis activity of candidate drugs. Based on this model, they reported that the target ratio was reached in only half of patients receiving moxifloxacin, and in none of those receiving levofloxacin.Like other research groups, we have previously conducted studies in murine models that brought conclusions different from those of the HFS. No single preclinical model accurately reflects the very specific PK/PD conditions observed in human postprimary tuberculosis: in fact, each model has its own advantages and disadvantages.