Laurent Bakri
email: laurent.bakri@gmail.com.

Jean-Baptiste Casteras
email: baptiste.casteras@univ-brest.fr

Non-stability of Paneitz-Branson type equations in arbitrary dimensions

Keywords: Paneitz-Branson type equations, blow up solutions, Liapunov-Schmidt reduction procedure Mathematics Subject Classification (2010) : 35J30, 35J60, 35B33, 35B35

Let (M, g) be a compact riemannian manifold of dimension n ≥ 5. We consider a Paneitz-Branson type equation with general coefficients

where

and ε is a small positive parameter. Assuming that there exists a positive nondegenerate solution of (E) when ε = 0 and under suitable conditions, we construct solutions u ε of type (u 0 -BBl ε ) to (E) which blow up at one point of the manifold when ε tends to 0 for all dimensions n ≥ 5.

Introduction and statements of the results

The existence of conformally covariant operators and the study of their associated curvature invariants have attracted a lot of attention these last decades. In 1983, Paneitz in [START_REF] Stephen | A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds (summary)[END_REF] introduced a fourth order operator, conformally covariant, on 4-dimensional manifold. A few years later, Branson [START_REF] Thomas | Group representations arising from Lorentz conformal geometry[END_REF] generalized this operator to all n-dimensional manifolds for n ≥ 5. In analogy with the conformal laplacian, a curvature called the Q-curvature has been associated to this operator (see [START_REF] Branson | Explicit functional determinants in four dimensions[END_REF]). It turns out that this curvature appears in a lot of geometric and physic problems. For instance, it is worth pointing out that, in dimension 4, the integrand of the Chern-Gauss-Bonnet formula for the Euler characteristic is the Q-curvature (up to the conformally invariant Weyl's tensor). The Q-curvature also appears in the study of fully nonlinear equations involving the symmetric functions of the Schouten tensor and in the zeta function determinant. The Q-curvature is also linked to the scattering theory of conformally compact manifolds whose study was initiated by Fefferman and Graham. We refer to the articles of Branson and Gover [START_REF] Branson | Origins, applications and generalisations of the Q-curvature[END_REF], Chang [START_REF] Chang | Non-linear elliptic equations in conformal geometry[END_REF], [START_REF] Chang | Conformal invariants and partial differential equations[END_REF], Chang and Yang [START_REF] Chang | Non-linear partial differential equations in conformal geometry[END_REF], and Gursky [START_REF] Gursky | Conformal invariants and nonlinear elliptic equations[END_REF] (and the references therein) for more details and very interesting material on the geometric and physic aspects associated to the notion of Q-curvature.

In what follows we let (M, g) be a compact riemannian manifold of dimension n ≥ 5. We will be interested in solutions u ∈ C 4,θ (M ), θ ∈ (0, 1), of the following equation

P g u := ∆ 2 g u -div g (A g du) + hu = |u| 2 * -2 u, (1.1)
where A g is a smooth symmetric (2, 0)-tensor, h ∈ C ∞ (M ) and 2 * = 2n n -4 .

Following the terminology introduced in [START_REF] Djadli | Paneitz-type operators and applications[END_REF], the operator P g has been referred to as a Paneitz-Branson type operator with general coefficients. When A g is given by

A g = A paneitz := (n -2) 2 + 4 2(n -1)(n -2) R g g - 4 n -2 Ric g , (1.2)
where R g (resp. Ric g ) stands for the scalar curvature (resp. Ricci curvature)

with respect to the metric g, and h = n -4 2 Q g where Q g is the Q-curvature with respect to the metric g which is defined by

Q g = 1 2(n -1) ∆ g R g + n 3 -4n 2 + 16n -16 8(n -1) 2 (n -2) 2 R 2 g - 2 (n -2) 2 |Ric g | 2 g
, then P g is the so-called Paneitz-Branson operator and equation (1.1) is referred to as the Paneitz-Branson equation. It is well known that the Paneitz operator is conformally invariant, i.e. if g = ϕ 4 n-4 g then, for all u ∈ C ∞ (M ), we have P n g (uϕ) = ϕ n+4 n-4 P n g (u).

We also point out that if (M, g) is Einstein (Ric g = λg, λ ∈ R), then the Paneitz-Branson operator takes the form

P g u = ∆ 2 g u + b∆ g u + cu, (1.3)
where b = n 2 -2n -4 2(n -1) λ and c = n(n -4)(n 2 -4) 16(n -1) 2 λ. More generally, when b and c are two real numbers, the operator P g defined in (1.3) is referred to as a Paneitz-Branson type operator with constant coefficients. Existence, compactness and stability of solutions to (1.1) when P g is a Paneitz-Branson type operator with constant coefficients, have been widely investigated this last decade (see for example [START_REF] Felli | Fourth order equations of critical Sobolev growth. Energy function and solutions of bounded energy in the conformally flat case[END_REF][START_REF] Hebey | Coercivity and Struwe's compactness for Paneitz type operators with constant coefficients[END_REF][START_REF] Hebey | Asymptotic analysis for fourth order Paneitz equations with critical growth[END_REF][START_REF] Qing | Compactness for conformal metrics with constant Q curvature on locally conformally flat manifolds[END_REF][START_REF] Wei | Non-compactness of the prescribed Qcurvature problem in large dimensions[END_REF] and the references therein). However, less is known for solutions of (1.1) in the case where P g is a Paneitz-Branson type operator with general coefficients. Esposito and Robert [START_REF] Esposito | Mountain pass critical points for Paneitz-Branson operators[END_REF] proved the existence of a non trivial solution to (1.1) under the hypothesis that n ≥ 8 and min M T r g (A g -A paneitz ) < 0. In [START_REF] Sandeep | A compactness type result for Paneitz-Branson operators with critical nonlinearity[END_REF], Sandeep studied the stability of equation (1.1) in the following sense : he considered sequences of positive solutions (u α ) α of

∆ 2 g u α -div g (A α du α ) + a α u α = u 2 * -1 α , u α ∈ C 4,θ ,
where A α are smooth (2, 0) symmetric tensors and a α are smooth functions. Sandeep proved that if A α converges in C 1 (M ) to a smooth symmetric tensor A g , a α converges in C 0 (M ) to a smooth positive function a and u α converges weakly in H 2 (M ) to a function u 0 , then u 0 is nontrivial provided that A g -A paneitz is either positive or negative definite (generalizing a result of [START_REF] Hebey | Compactness and global estimates for a fourth order equation of critical Sobolev growth arising from conformal geometry[END_REF]).

Recently, Pistoia and Vaira [START_REF] Pistoia | On the stability for Paneitz-type equations[END_REF] studied the stability of (1.1) when it is the Paneitz-Branson equation, namely they considered the following equation

∆ 2 g u -div g ((A paneitz + εB)du) + Q g u = |u| 2 * -2 u, (1.4)
where ε is a small positive parameter and B is a smooth symmetric (2, 0) tensor. They proved that if (M, g) is not conformally flat, n ≥ 9 and there exists ξ 0 ∈ M a C 1 stable critical point (see below for the definition) of the

function ξ → T r g B(ξ) |W eyl g (ξ)| g
, such that T r g B(ξ 0 ) > 0, then equation (1.4) is not stable, i.e. there exists ε 0 > 0 such that, for any ε ∈ (0, ε 0 ), equation (1.4) admits a solution u ε such that u ε (ξ 0 ) -→ ε→0 + +∞. The aim of this paper is to investigate the stability in the sense of Deng-Pistoia of (1.1). We say that (1.1) is stable if, for any sequences of real positive numbers (ε α ) α such that ε α -→ α→∞ 0 and for any sequences of solutions

(u α ) α ∈ C 4,θ (M ), θ ∈ (0, 1), of ∆ 2 g u α -div g (A g du α ) + hu α = |u α | 2 * -2-εα u α , (1.5)
bounded in H 2 (M ), then up to a subsequence, u α converges in C 4 (M ) to some smooth function u solution of (1.1). Deng and Pistoia [START_REF] Deng | Blow-up solutions for Paneitz-Branson type equations with critical growth[END_REF] proved that (1.1) is not stable if a. n ≥ 7, T r g (A g -A paneitz ) is not constant and min

M T r g (A g -A paneitz ) > 0, b. or n ≥ 8 and ξ 0 ∈ M a C 1 stable critical point of T r g (A g -A paneitz ) such that T r g (A g -A paneitz )(ξ 0 ) > 0.
Our main result shows that under suitable assumptions, equation (1.1) is not stable for any n ≥ 5. In fact, inspired by the recent result of Robert and Vétois [START_REF] Robert | Sign-changing blow-up for scalar curvature type equations[END_REF] on scalar curvature type equations, we investigate the existence of families

(u ε ) ε ∈ C 4,θ (M ) of blow-up solutions to (1.5) of type (u 0 -BBl ε ).
Following the terminology of Robert and Vétois, we say that a blow-up sequence

(u ε ) ε is of type (u 0 -BBl ε ) if there exists u 0 ∈ C 4,θ (M ) and a bubble BBl ε (x) = [n(n -4)(n 2 -4)] n-4 8 µ ε µ ε + d g (x, x ε ) 2 n-4 2
, where x, x ε ∈ M and

µ ε ∈ R + is such that µ ε -→ ε→0 0, such that u ε = u 0 -BBl ε + o(1),
where o(1) -→ ε→0 0. Before stating more precisely the results, we would like to recall that a solution of (1.5) is called nondegenerate if the kernel of the linearization of the equation is trivial (see (2.3)). Let φ ∈ C 1 (M ), we also recall that a critical point ξ 0 of φ is said C 1 stable if there exists an open neighborhood Ω of ξ 0 such that, for any point ξ ∈ Ω, there holds ∇ g φ(ξ) = 0 if and only if ξ = ξ 0 and such that the Brower degree deg(∇ g φ, Ω, 0) = 0. We obtain :

Theorem 1.1. Let (M, g) be a compact riemannian manifold of dimension n, A g and h be such that P g is coercive. Let u 0 ∈ C 4,θ , θ ∈ (0, 1), be a positive nondegenerate solution of (1.1). Assume in addition that one of the following condition holds:

a. 5 ≤ n < 7,
b. 8 ≤ n ≤ 13 and there exists ξ 0 ∈ M a C 1 stable critical point of

ϕ(ξ) = (n -1) (n -6)(n 2 -4) (T r g (A g -A paneitz ))(ξ) + 2 n u 0 (ξ)ω n-1 (n + 2)(n(n -4)(n 2 -4)) n-4 8 ω n 1 n=8 , ξ ∈ M, (1.6)
such that ϕ(ξ 0 ) > 0, c. n > 13 and min

M T r g (A g -A paneitz ) > 0,
then, for any ε > 0, there exists a solution u ε of type u 0 -BBl ε to (1.5). In particular, (1.5) is not stable.

Let us notice that in the geometric case i.e. when A g = A paneitz , the previous theorem only applies if 5 ≤ n ≤ 8. However, with a small modification of the proof, we can construct a solution of type u 0 -BBl ε to (1.5) when 5 ≤ n ≤ 11 and A g = A paneitz . More precisely, we prove the following result : Theorem 1.2. Let (M, g) be a compact riemannian manifold of dimension n, A g and h be such that P g is coercive. Let u 0 ∈ C 4,θ , θ ∈ (0, 1), be a positive nondegenerate solution of (1.1). Assume that A g = A paneitz . Then, for any 5 ≤ n ≤ 11 and any ε > 0, there exists a solution u ε of type u 0 -BBl ε to (1.5). In particular, (1.5) is not stable.

The proof of the theorems relies on a well known Lyapunov-Schmidt reduction procedure which permits to reduce the problem to a finite dimensional one for which we defined a reduced energy. The solutions to (1.5) will then be obtained as critical points of this reduced energy. We refer to [START_REF] Ambrosetti | Perturbation methods and semilinear elliptic problems on R n[END_REF] and the references therein for more information on the Lyapunov-Schmidt reduction procedure. We would like to emphasize that the proof of Theorem 1.1 is inspired by the previous work of Robert and Vétois [START_REF] Robert | Sign-changing blow-up for scalar curvature type equations[END_REF]. Thus we will keep their notations. We also want to point out that we use without proof computations done in Deng and Pistoia [START_REF] Deng | Blow-up solutions for Paneitz-Branson type equations with critical growth[END_REF] (for more details on these computations, see their paper). The plan of the paper is the following : in section 2 we introduce notations and perform the finite dimensional reduction. In section 3 we study the reduced problem and prove Theorem 1.1. The error estimate and the C 1 uniform asymptotic expansion of the reduced energy are done in the appendix.

In all the following, we will denote by ., . Pg , the scalar product, for u, v ∈ H2 (M ),

u, v Pg = M ∆ g u∆ g vdV + M A g (∇ g u, ∇ g v)dV + M huvdV,
where here and in the following dV stands for the volume element with respect to the metric g, and . Pg , for the associated norm which is then equivalent to the standard norm of H 2 (M ). We denote by i * :

L 2n n+4 (M ) → H 2 (M ) the adjoint operator of the embedding i : H 2 (M ) → L 2n n-4 (M ), i.e. for all ϕ ∈ L 2n n+4 (M ), the function u = i * (ϕ) ∈ H 2 (M ) is the unique solution of ∆ 2 g u -div g (A g du) + hu = ϕ.
Using this notation, we see that equation (1.5) can be rewritten as, for u ∈ H 2 (M ),

u = i * (f ε (u)),
where f ε (u) = |u| 2 * -2-ε u. Before proceeding we recall some basic facts. It is well known (see [START_REF] Lin | A classification of solutions of a conformally invariant fourth order equation in R n[END_REF]) that all solutions u ∈ H 2 (R n ) of the equation

∆ 2 eucl u = u 2 * -1 = u n+4 n-4 in R n
are given by

U δ,y (x) = δ 4-n 2 U ( x -y δ ), δ > 0, y ∈ R n
where

U (x) = [n(n -4)(n 2 -4)] n-4 8 1 1 + |x| 2 n-4 2 = α n 1 1 + |x| 2 n-4 2 . (2.1)
It is also well known (see [START_REF] Lu | On a Sobolev inequality with remainder terms[END_REF]) that all solutions v ∈ H 2 (R n ) of

∆ 2 eucl v = (2 * -1)U 2 * -2 v are linear combinations of V 0 (x) = α n n -4 2 |x| 2 -1 (1 + |x| 2 ) n-2 2 and V i (x) = α n (n -4) x i (1 + |x| 2 ) n-2 2 , i = 1, . . . , n. Let χ : R → R be a smooth cutoff function such that 0 ≤ χ ≤ 1, χ(x) = 1 if x ∈ [- r 0 2 , r 0 2
] and χ(x) = 0 if x ∈ R\(-r 0 , r 0 ). We define, for any real δ strictly positive, ξ ∈ M and x ∈ M ,

W δ,ξ (x) = χ(d g (x, ξ))δ 4-n
where d g (x, ξ) stands for the distance from x to ξ with respect to the metric g and exp ξ is the exponential map with respect to the metric g. We also define, for any real δ strictly positive, ξ ∈ M and x ∈ M ,

Z δ,ξ (x) = χ(d g (x, ξ))δ n-4 2 d(x, ξ) 2 -δ 2 (δ 2 + d(x, ξ) 2 ) n-2 2
, and, for ω

∈ T ξ M , Z δ,ξ,ω (x) = χ(d g (x, ξ))δ n-2 2 exp -1 ξ x, ω g (δ 2 + d(x, ξ) 2 ) n-2 2
.

We denote by Π δ,ξ respectively Π ⊥ δ,ξ the projection of H 2 (M ) onto

K δ,ξ = span {Z δ,ξ , (Z δ,ξ,e i ) i=1..n } respectively K ⊥ δ,ξ = φ ∈ H 2 (M )/ φ, Z δ,ξ Pg = 0 and φ, Z δ,ξ,ω Pg = 0, ∀ω ∈ T ξ M . (2.
2) We recall that a solution u 0 of (1.5) is nondegenerate if the linearization of the equation has trivial kernel, that is

K = ϕ ∈ C 4,θ (M )/P g ϕ = (2 * -1)|u 0 | 2 * -2 ϕ = {0} . (2.3)
We are looking for solution u to (1.5) of the form

u = u 0 -W δε(tε),ξε + φ δε(tε),ξε ,
where u 0 is a nondegenerate positive solution of (1.5), φ δε(tε),ξε ∈ K ⊥ δε(tε),ξε and

δ ε (t ε ) = √ t ε ε if n ≥ 8 (t ε ε) 2 n-4 if 5 ≤ n ≤ 8 , t ε > 0. (2.4)
It is easy to see that equation (1.5) is equivalent to the following system

Π δε(t),ξ (u 0 -W δε(t),ξ + φ δε(t),ξ -i * (f ε (u 0 -W δε(t),ξ + φ δε(t),ξ ))) = 0, (2.5)
and

Π ⊥ δε(t),ξ (u 0 -W δε(t),ξ + φ δε(t),ξ -i * (f ε (u 0 -W δε(t),ξ + φ δε(t),ξ ))) = 0. (2.6)
We begin by solving (2.6).

Proposition 2.1. Let u 0 ∈ C 4,θ (M ) be a nondegenerate positive solution of (1.5). Given two real numbers a < b, there exists a positive constant C a,b such that for ε small, for any t ∈ [a, b] and any ξ ∈ M , there exists a unique function φ δε(t),ξ ∈ K ⊥ δε(t),ξ which solves equation (2.6) and satisfies φ δε(t),ξ Pg ≤ C a,b ε| ln ε|.

(2.7)

Moreover, φ δε(t),ξ is continuously differentiable with respect to t and ξ.

In order to prove the previous proposition, we set, for ε small, for any positive real number δ and ξ ∈ M , the map L ε,δ,ξ :

K ⊥ δ,ε → K ⊥ δ,ε defined by, for φ ∈ K ⊥ δ,ε , L ε,δ,ξ (φ) = Π ⊥ δ,ξ (φ -i * (f ε (u 0 -W δ,ξ )φ)
). We will first prove that this map is inversible for δ and ε small.

∈ K ⊥ δ,ε , we have L εα,δε α (tα),ξα (φ) Pg ≥ C a,b φ Pg .
Proof. Assume by contradiction that there exist two sequences of positive real numbers (ε α ) α and (t α ) α such that ε α -→ α→+∞ 0 and a ≤ t α ≤ b, a sequence of points (ξ α )) α of M and a sequence of functions (φ α ) α such that

φ α ∈ K ⊥ δε α (tα),ξα , φ α Pg = 1 and L εα,δε α (tα),ξα (φ α ) Pg -→ α→∞ 0. (2.8)
To simplify notations, we set L α = L εα,δε α (tα),ξα , W α = W δε α (tα),ξα , Z 0,α = Z δε α (tα),ξα and Z i,α = Z δε α (tα),ξα,e i for i = 1, . . . , n where e i is the i-th vector in the canonical basis of R n . By definition of L α , there exist real numbers λ i,α , i = 0, . . . , n such that

φ α -i * (f εα (u 0 -W α )φ α ) -L α (φ α ) = n i=0 λ i,α Z i,α . (2.9)
Standard computations give

Z i,α , Z j,α Pg -→ α→∞ ∆ eucl V i 2 L 2 (R n ) δ ij , (2.10)
where δ ij stands for the Kronecker symbol. Therefore, taking the scalar product of (2.9) with Z i,α , using the previous limit and recalling that φ α and L α (φ α ) belong to K ⊥ δε α (tα),ξα , we deduce that

M f εα (u 0 -W α )φ α Z i,α dV = -λ i,α ∆ eucl V i 2 L 2 (R n ) + n i=0 |λ i,α | o(1), (2.11)
where, here and in the following, o(1) -→ α→+∞ 0. It is easy to see using the definition of W α and Z i,α and a change of variables that, for α large enough,

M f εα (u 0 -W α )φ α Z i,α dV = M f εα (W α )φ α Z i,α dV + o(1) (2.12) = (2 * -1 -ε α )δ εα (t α ) εα n+4 2 R n χ 2 * -2-εα α U 2 * -2-εα V i φα dV gα + o(1)
,

where χ α = χ(δ εα (t α )|x|), φα (x) = δ εα (t α ) n-4 2 χ α φ α (exp ξα (δ εα (t α )x)) and gα (x) = exp * ξα g(δ εα (t α )x). Since (φ α ) α is bounded in H 2 (M )
, passing to a subsequence if necessary, we can assume that ( φα ) α converges weakly to a function φ ∈ H 2 (R n ). Letting α → +∞ in (2.12), we deduce that

M f εα (u 0 -W α )φ α Z i,α dV -→ α→∞ (2 * -1) R n U 2 * -2 V i φdV g eucl = 0, (2.13) where we used that V i is solution of ∆ 2 eucl V i = n + 4 n -4 U 2 * -2 V i in R n and φ α ∈ K ⊥ δε α (tα)
,ξα to obtain the last equality. Therefore, from (2.11) and (2.13), we have

λ i,α = o(1) + o( n i=0 |λ i,α |).
From (2.9), this implies

φ α -i * (f εα (u 0 -W α )φ α ) -L α (φ α ) -→ α→∞ 0.
Since by assumption L εα,δε α (tα),ξα (φ α ) Pg -→ α→∞ 0, we finally obtain that

φ α -i * (f εα (u 0 -W α )φ α ) Pg -→ α→∞ 0.
(2.14)

Since (φ α ) α is bounded in H 2 (M )
, up to taking a subsequence, we can assume that φ α converges weakly in H 2 (M ) to a function φ ∈ H 2 (M ). Then, using (2.14), we get, for any ϕ ∈ H 2 (M ),

ϕ, φ α Pg - M f εα (u 0 -W α )ϕφ α dV = ϕ, φ α -i * (f εα (u 0 -W α )φ α ) Pg ≤ ϕ Pg φ α -i * (f εα (u 0 -W α )φ α ) Pg = o( ϕ Pg ). (2.15)
We deduce from this that φ is a weak solution of P g φ = (2 * -1)u 2 * -2 0 φ. Since u 0 is a nondegenerate solution of (1.5), we obtain that φ = 0. Therefore, φ α α→∞ 0 weakly in H 2 (M ). Now we will show that φα α→∞ 0 weakly in

H 2 (R n ).
Let φ be a smooth function with compact support in R n , we will use (2.15) with, for x ∈ M ,

ϕ(x) = χ(d g ξα (x, ξ α ))δ εα (t α ) 4-n 2 φ(δ εα (t α ) -1 exp -1 ξα (x)).
Thus, applying (2.15) to the previous ϕ and using a change of variable, we have,

R n ∆ gα φα ∆ gα φdV gα + δ εα (t α ) 2 R n A gα (∇ gα φα , ∇ gα φ)dV gα + δ εα (t α ) 4 R n h(exp ξα (δ εα (t α )x)) φα φdV gα (2.16) = δ εα (t α ) 4 R n f εα (u 0,α -W α (exp ξα (δ εα (t α )x))) φα ϕdV gα + o(1),
where u 0,α (.) = u 0 (exp ξα (δ εα (t α ).)). Now it is easy to see that, letting α → ∞ in (2.16),

R n ∆ eucl φ∆ eucl φdV g eucl = (2 * -1) R n U 2 * -2 φ φdV g eucl .
Thus φ is a weak solution of ∆ 2 eucl φ = n + 4 n -4 U 2 * -2 φ. So, from [START_REF] Lu | On a Sobolev inequality with remainder terms[END_REF], we know that there exists λ i ∈ R, i = 0, . . . , n, such that φ = n i=0 λ i V i . Since φ α ∈ K ⊥ δε α (tα),ξα , using the same argument as in (2.13), we deduce that φ ≡ 0. Using one more time (2.15) with ϕ = φ α , a change of variables and since φ α α→∞ 0 weakly in H 2 (M ) and φα α→∞ 0 weakly in H 2 (R n ), we get

φ α 2 Pg = (2 * -1 -ε α ) M |u 0 -W α | 2 * -2-εα φ 2 α dV + o(1) ≤ C M φ 2 α dV + C M |W α | 2 * -2-εα φ 2 α dV + o(1) ≤ C M φ 2 α dV + C M |U | 2 * -2-εα φ2 α dV gα + o(1) -→ α→∞ 0.
This yields to a contradiction with (2.8).

Proof of Proposition 2.1. It is easy to see that equation (2.6) is equivalent to

L ε,δε(t),ξ (φ) = N ε,δε(t),ξ (φ) + R ε,δε(t),ξ ,
where

N ε,δε(t),ξ (φ) = Π ⊥ δε(t),ξ (i * (f ε (u 0 -W δε(t),ξ + φ)) -f ε (u 0 -W δε(t),ξ ) -f ε (u 0 -W δε(t),ξ )φ),
and

R ε,δε(t),ξ = Π ⊥ δε(t),ξ (i * ε (f ε (u 0 -W δε(t),ξ )) -u 0 + W δε(t),ξ ).
Let T ε,δε(t),ξ : K ⊥ δε α (tα),ξα → K ⊥ δε α (tα),ξα be the application defined by

T ε,δε(t),ξ (φ) = L -1 ε,δε(t),ξ (N ε,δε(t),ξ (φ) + R ε,δε(t),ξ ), and B ε,δε(t),ξ (γ) = φ ∈ K ⊥ δε α (tα),ξα | φ Pg ≤ γ R ε,δε(t),ξ Pg
, where γ is a positive constant which will be chosen later in order to apply the fixed point theorem for T ε,δε(t),ξ restricted to B ε,δε(t),ξ (γ). Since, from Lemma 2.1, the map L ε,δε(t),ξ is inversible and has a continuous inverse, we have

T ε,δε(t),ξ (φ) Pg ≤ C( N ε,δε(t),ξ (φ) Pg + R ε,δε(t),ξ Pg ), (2.17)
and

T ε,δε(t),ξ (φ 1 ) -T ε,δε(t),ξ (φ 2 ) Pg ≤ C N ε,δε(t),ξ (φ 1 ) -N ε,δε(t),ξ (φ 2 ) Pg . (2.18) Since i * : L 2n n+4 (M ) → H 2 (M ) is continuous, we get N ε,δε(t),ξ (φ) Pg ≤ C f ε (u 0 -W δε(t),ξ + φ)) -f ε (u 0 -W δε(t),ξ ) -f ε (u 0 -W δε(t),ξ )φ L 2n n+4 ,
where, here and in the following, . L p = . L p (M ) , p ∈ R + . Using the mean value theorem, Hölder and Sobolev inequalities, we have, for τ ∈ (0, 1),

N ε,δε(t),ξ (φ) Pg ≤ C f ε (u 0 -W δε(t),ξ + τ φ) -f ε (u 0 -W δε(t),ξ ) (φ) L 2n n+4 ≤C f ε (u 0 -W δε(t),ξ + τ φ) -f ε (u 0 -W δε(t),ξ ) L n 4 φ L 2 * .
We will use here and through the paper the following easy consequences of Taylor's expansion [17, lemma 2.2], for all α > 0, β ∈ R,

||α + β| θ -α θ | ≤ C θ min |β| θ , α θ-1 |β| if 0 < θ ≤ 1, C θ (α θ-1 |β| + |β| θ ) if θ > 1, (2.19)
and

||α + β| θ (α + β) -α θ+1 -(1 + θ)α θ β| ≤ C θ min |β| θ+1 , α θ-1 |β| 2 if θ < 1, C θ max{|β| θ+1 , α θ-1 |β| 2 } if θ ≥ 1.
(2.20) Thus, we obtain

N ε,δε(t),ξ (φ) Pg ≤ C φ 2 * -1-ε Pg if n ≥ 12, C( u 0 -W 2 * -3-ε L 2 * φ 2 Pg + φ 2 * -1-ε Pg ) if 5 ≤ n < 12.
(2.21) From the mean value theorem, Hölder and Sobolev inequalities, and (2.19), we also get, for some τ ∈ (0, 1),

N ε,δε(t),ξ (φ 1 ) -N ε,δε(t),ξ (φ 2 ) Pg (2.22) ≤ C f ε (u 0 -W δε(t),ξ + φ 1 ) -f ε (u 0 -W δε(t),ξ + φ 2 ) -f ε (u 0 -W δε(t),ξ )(φ 1 -φ 2 ) L 2n n+4 ≤ C f ε (u 0 -W δε(t),ξ + τ φ 2 + (1 -τ )φ 1 ) -f ε (u 0 -W δε(t),ξ ) (φ 1 -φ 2 ) L 2n n+4 ≤ C f ε (u 0 -W δε(t),ξ + τ φ 2 + (1 -τ )φ 1 ) -f ε (u 0 -W δε(t),ξ ) L n 4 × φ 1 -φ 2 L 2 * ≤      C( φ 1 2 * -2-ε Pg + φ 2 2 * -2-ε Pg ) φ 1 -φ 2 Pg if n ≥ 12, C( u 0 -W δε(t),ξ L 2 * (M ) + φ 1 Pg + φ 2 Pg ) 2 * -3-ε ×( φ 1 Pg + φ 2 Pg ) φ 1 -φ 2 Pg if 5 ≤ n < 12
Since u 0 -W δε(t),ξ L 2 * = O(1), it follows from (2.17), (2.18), (2.21) and (2.22), that, for all φ, φ 1 , φ 2 ∈ B ε,δε(t),ξ (γ),

T ε,δε(t),ξ (φ) Pg ≤        C(γ 2 * -1-ε R ε,δε(t),ξ 2 * -1-ε Pg + R ε,δε(t),ξ Pg ) if n ≥ 12 C(γ 2 R ε,δε(t),ξ 2
Pg + γ 2 * -1-ε R ε,δε(t),ξ 2 * -1-ε Pg + R ε,δε(t),ξ Pg ) if 5 ≤ n < 12 and T ε,δε(t),ξ (φ 1 ) -T ε,δε(t),ξ (φ 2 ) Pg ≤ Cγ 2 * -2-ε R ε,δε(t),ξ 2 * -2-ε Pg φ 1 -φ 2 Pg ,
where C stands for positive constants not depending on γ, ε, ξ, t, φ, φ 1 and φ 2 . Thus from Lemma 5.1, if γ is fixed large enough, for ε small, for any t ∈ [a, b] and any ξ ∈ M , T ε,δε(t),ξ is a contraction mapping from B ε,δε(t),ξ (γ) onto B ε,δε(t),ξ (γ). Therefore, using the fixed point theorem, there exists a function φ δε(t),ξ ∈ K ⊥ δε(t),ξ which solves equation (2.6). Now, (2.7) follows from Lemma 5.1. The fact that φ δε(t),ξ is continuously differentiable with respect to t and ξ is standard.

3 The reduced problem.

For ε > 0 small enough, we defined the energy associated to (1.5) by, for u ∈ H 2 (M ),

J ε (u) = 1 2 M (∆ g u) 2 + 1 2 M A g (∇ g u, ∇ g u)dV + 1 2 M hu 2 dV - M F ε (u)dV,
where

F ε (u) = u 0 f ε (s)ds. We set I ε (t, ξ) = J ε (u 0 -W δε(t),ξ +φ δε(t),ξ ), t ∈ R * +
and ξ ∈ M where φ δε(t),ξ ∈ K ⊥ δε(t),ξ is the function defined in Proposition 2.1. In the next proposition, we give the expansion of I ε with respect to ε. Proposition 3.1. Let u 0 ∈ C 4,θ (M ), θ ∈ (0, 1) be a nondegenerate positive solution of (1.5). Then there exist constants c i (n, u 0 ), i = 2, 5 depending on n and u 0 and c i (n), i = 1, 3, 4, depending on n such that

I ε (t, ξ) = c 5 (n, u 0 ) + c 2 (n, u 0 )ε + c 3 (n)ε ln ε -c 4 (n)ε ln(t) + c 1 (n)ϕ(ξ)εt + o(ε)
(3.1) as ε → 0 C 0 uniformly with respect to t in compact subsets of R * + and with respect to ξ ∈ M and C 1 uniformly if 8 ≤ n ≤ 13. Moreover, we have that

c 4 (n) > 0, c 1 (n) = 2 n K -n 4 n and ϕ(ξ) = (n -1) (n -6)(n 2 -4) (T r g (A g -A paneitz )(ξ)1 n≥8 + 2 n u 0 (ξ)ω n-1 (n + 2)(n(n -4)(n 2 -4)) n-4 8 ω n 1 n≤8 ,
where ω n stands for the volume of S n and K n is the sharp constant for the

embedding of H 2 (R n ) into L 2 * (R n ) given by K -1 n = n(n -4)(n 2 -4)ω 4 n n 16 .
Proof. We begin by proving that

I ε (t, ξ) = J ε (u 0 -W δε(t),ξ ) + o(ε), (3.2)
as ε → 0, uniformly with respect to t in compact subsets of R * + and points ξ ∈ M (we will show in Lemma 5.2 that, when 8 ≤ n ≤ 13, this estimate holds C 1 uniformly with respect to t and ξ). Indeed, we have

I ε (t, ξ) -J ε (u 0 -W δε(t),ξ ) = u 0 -W δε(t),ξ -i * (f ε (u 0 -W δε(t),ξ )), φ δε(t),ξ Pg + O( φ δε(t),ξ 2
u 0 -W δε(t),ξ -i * (f ε (u 0 -W δε(t),ξ )), φ δε(t),ξ Pg + O( φ δε(t),ξ 2
Pg ) = O(ε 2 | ln ε| 2 ) = o(ε).
Now, the proposition is reduced to estimate J ε (u 0 -W δε(t),ξ ). We will focus on C 0 -estimates. The C 1 -estimates can be obtained using the same argument as in Lemma 4.1 of [START_REF] Maria Micheletti | Blow-up solutions for asymptotically critical elliptic equations on Riemannian manifolds[END_REF]. Since u 0 is a solution of (1.5), we have

J ε (u 0 -W δε(t),ξ ) = 1 2 M u 2 * 0 dV + 1 2 M (∆ g W δε(t),ξ ) 2 dV + 1 2 M A g (∇ g W δε(t),ξ , ∇ g W δε(t),ξ )dV + 1 2 M hW 2 δε(t),ξ dV - M f ε (u 0 )W δε(t),ξ dV - M F ε (u 0 -W δε(t),ξ )dV.
Using a Taylor expansion with respect to ε, we get

1 2 M u 2 * 0 dV - 1 2 * -ε M u 2 * -ε 0 dV = 1 2 M u 2 * 0 dV - 1 2 * (1 + ε 2 * ) M u 2 * 0 (1 -ε ln u 0 )dV + O(ε 2 ) = ( 1 2 - 1 2 * ) M u 2 * 0 dV + ε 2 * M u 2 * 0 (ln u 0 - 1 2 * )dV + O(ε 2 )
Thus from the two previous equalities, we obtain

J ε (u 0 -W δε(t),ξ ) = ( 1 2 - 1 2 * ) M u 2 * 0 dV + ε 2 * M u 2 * 0 (ln u 0 - 1 2 * )dV + I 1,ε,t,ξ + I 2,ε,t,ξ + I 3,ε,t,ξ + O(ε 2 ), (3.4)
where

I 1,ε,t,ξ = 1 2 M (∆ g W δε(t),ξ ) 2 dV + 1 2 M A g (∇ g W δε(t),ξ , ∇ g W δε(t),ξ )dV + 1 2 M hW 2 δε(t),ξ dV - M F ε (W δε(t),ξ )dV, I 2,ε,t,ξ = M f ε (W δε(t),ξ )u 0 dV, and
I 3,ε,t,ξ = - M F ε (u 0 -W δε(t),ξ ) -F ε (u 0 ) -F ε (W δε(t),ξ ) + f ε (u 0 )W δε(t),ξ + f ε (W δε(t),ξ )u 0 dV. (3.5)
We begin by estimating I 3 . Using Taylor expansion (cf (2.20)) and rough estimations, we have

|I 3,ε,t,ξ | ≤ (F ε (u 0 -W δε(t),ξ ) -F ε (W δε(t),ξ ) + f ε (W δε(t),ξ )u 0 )1 B( √ δε(t)) L 1 + (F ε (u 0 -W δε(t),ξ ) -F ε (u 0 ) + f ε (u 0 )W δε(t),ξ )1 M \B( √ δε(t)) L 1 + F ε (u 0 )1 B( √ δε(t)) L 1 + f ε (u 0 )W δε(t),ξ 1 B( √ δε(t)) L 1 + F ε (W δε(t),ξ )1 M \B( √ δε(t)) L 1 + u 0 f ε (W δε(t),ξ )1 M \B( √ δε(t)) L 1 ≤ u 2 0 W 2 * -2-ε δε(t),ξ 1 B( √ δε(t)) L 1 + u 2 * -2-ε 0 W 2 δε(t),ξ 1 M \B( √ δε(t)) L 1 + F ε (W δε(t),ξ )1 M \B( √ δε(t)) L 1 + u 0 f ε (W δε(t),ξ )1 M \B( √ δε(t)) L 1 + F ε (u 0 )1 B( √ δε(t)) L 1 + f ε (u 0 )W δε(t),ξ 1 B( √ δε(t)) L 1 ≤ C u 2 0 W 2 * -2-ε δε(t),ξ 1 B( √ δε(t)) L 1 + C u 2 * -2-ε 0 W 2 δε(t),ξ 1 M \B( √ δε(t)) L 1 +O(δ ε (t) n 2 )
Therefore estimating the last two terms and using the definition of δ, we obtain

|I 3,ε,t,ξ | ≤    O(δ ε (t) n 2 ) = O(ε n 4 ) = o(ε 2 ) if n > 8 O(δ ε (t) 4 | ln δ|) = O(ε 2 | ln ε|) if n = 8 O(δ ε (t) n-4 ) = O(ε 2 ) if n < 8. (3.6)
Now, let us estimate I 2,ε,t,ξ . We recall that the Cartan expansion of the metric gives

|g|(x) = 1 - 1 6 Ric ij x i x j - 1 12 ∇ k Ric ij x i x j x k + O(|x| 4 ), (3.7)
where |g| stands for the determinant of the metric g in geodesic normal coordinates. Then, using a change of variables, Taylor expansion and by symmetry, we have

I 2,ε,t,ξ = u 0 (ξ)ω n-1 α n+4 n-4 -ε n δ ε (t) n-4 2 (1+ε) × r 0 2δε (t) 0 r n-1 (1 + r 2 ) n+4 2 -ε n-4 2 (1 + O(δ 2 r 2 ))dr + O(δ ε (t) n 2 + ε 2 | ln δ ε (t)|) = 2u 0 (ξ)ω n-1 α n+4 n-4 n δ ε (t) n-4 2 n(n + 2) + O(δ ε (t) n 2 + ε 2 | ln δ ε (t)|) = 2 n+1 u 0 (ξ)K -n 4 n ω n-1 δ ε (t) n-4 2 n(n + 2)α n ω n + O(δ ε (t) n 2 + ε 2 | ln δ ε (t)|), (3.8)
where α n is defined in (2.1). Finally, we use the computations of section 4 of [START_REF] Esposito | Mountain pass critical points for Paneitz-Branson operators[END_REF] and the estimate (4.2) of [START_REF] Deng | Blow-up solutions for Paneitz-Branson type equations with critical growth[END_REF] to estimate I 1,ε,t,δ . We notice, using (3.7) and by symmetry, that the remaining in equation

(4.2) of [8] (namely o(δ ε (t) 2 ) ) is actually in O(δ ε (t) 4
). We thus have

I 1,ε,t,δ = 2 n K -n 4 n 1 -C n ε - (n -4) 2 8 ε ln δ + (n -1) (n -6)(n 2 -4) (T r g (A g -A paneitz )δ ε (t) 2 1 n≥8 ) + o(ε) + O(δ ε (t) 4 ) , (3.9)
where

C n = 2 n-4 (n -4) 2 ω n-1 ω n ∞ 0 r n-2 2 ln(1 + r) (1 + r) n dr + (n -4) 2 8(n -2) (1 - 1 2 ln n(n -4)(n 2 -4)). (3.10) Thus, combining (3.4), (3.6), (3.8
) and (3.9), we obtain

J ε (u 0 -W δε(t),ξ ) = ( 1 2 - 1 2 * ) M u 2 * 0 + ε 2 * M u 2 * 0 (ln u 0 - 1 2 * )dV + 2 n K -n 4 n 1 -C n ε - (n -4) 2 8 ε ln δ ε (t) + (n -1) (n -6)(n 2 -4) (T r g (A g -A paneitz )δ ε (t) 2 + 2 n+1 u 0 (ξ)K -n 4 n ω n-1 δ ε (t) n-4 2 n(n + 2)α n ω n + o(ε). (3.11)
The lemma follows from (3.2) and (3.11).

The next proposition shows that, in order to construct a solution to (1.5), we only need to find a critical point for the reduced energy I ε . Proposition 3.2. Given two positive real numbers a < b, for ε small, if

(t ε , ξ ε ) ∈ (a, b) × M is a critical point of I ε , then the function u 0 -W δε(tε),ξε + φ δε(tε),ξε is a solution of (1.5).
Proof. Let (ξ α ) α be a sequence of points of M and suppose that (t α ) α and (ε α ) α are two sequences of real numbers such that ε α -→ α→∞ 0, a ≤ t α ≤ b and (t α , ξ α ) is a critical point of I εα for all α ∈ N. To simplify notations, we set, for i = 1, . . . , n, Z 0,α = Z δε α (tα),ξα and Z i,α = Z δε α (tα),ξα,e i .

Since φ δε α (tα),ξα is a solution of (2.6) by Proposition 2.1, there exist real numbers λ i,α , i = 0, . . . , n such that

DJ εα (u 0 -W δε α (tα),ξα + φ δε α (tα),ξα ) = n i=0 λ i,α Z i,α , . Pg . (3.12)
Using the previous equality, we see that

∂I εα ∂t (t α , ξ α ) = n i=0 λ i,α Z i,α , ∂ ∂t (-W δε α (tα),ξα + φ δε α (tα),ξα ) Pg . (3.13)
A simple computation gives

∂ ∂t (W δε α (tα),ξα )| t=tα = Cn t α Z 0,α , (3.14)
where Cn = α n if n < 8 and Cn = α n (n -4) 4 if n ≥ 8 (see (2.1) for the definition of α n ). Taking the derivative of Z δε α (tα),ξα , φ δε α (tα),ξα Pg = 0 with respect to t, we obtain

∂ ∂t Z δε α (tα)|t=t α ,ξα , φ δε α (tα),ξα Pg = -Z δε α (tα),ξα , ∂ ∂t φ δε α (tα),ξα | t=tα Pg . ( 3
∂I εα ∂t (t α , ξ α ) = - Cn t α λ 0,α ∆ eucl V 0 2 L 2 (R n ) + o( n i=0 λ i,α ), (3.16)
where o(1) -→ α→+∞ 0. Arguing the same way and noting that

∂ ∂y i (W δε α (tα),exp ξα (y) )| y=0 = α n (n -4) δ εα (t α ) Z i,α + R i,α
where R i,α -→ α→+∞ 0 in H 2 (M ), and

∂ ∂y i Z j,δε α (tα),exp ξα (y) | y=0 Pg = O( 1 δ εα (t α ) ),
we obtain

δ εα (t α ) ∂I εα ∂y i (t α , exp ξα (y))| y=0 = -λ i,α ∆ eucl V i 2 L 2 (R n ) + o( n i=0 λ i,α ). (3.17)
Therefore, from (3.12), (3.16) and (3.17), it follows that if (t α , ξ α ) is a critical point of I εα then u 0 -W δε α (tα),ξα + φ δε α (tα),ξα is a solution of (1.5).

We are now in position to prove the theorems.

4 Proof of the theorems.

We begin by proving Theorem 1.1.

Proof of Theorem 1.1. We set G : R * + × M → R the function defined by

G(t, ξ) = -c 4 (n) ln t + c 1 (n)ϕ(ξ)t,
where c 4 (n), c 1 (n) and ϕ(ξ) are defined in (3.1). From Proposition 3.1, we have

lim ε→0 1 ε (I ε (t, ξ) -c 5 (n, u 0 ) -c 2 (n, u 0 )ε -c 3 (n)ε ln ε) = G(t, ξ), (4.1)
C 1 uniformly with respect to ξ ∈ M and t in compact subset of R * + . We will consider two cases depending on the dimension of the manifold. We argue as in [START_REF] Maria Micheletti | Blow-up solutions for asymptotically critical elliptic equations on Riemannian manifolds[END_REF]. Let ξ 0 be the C 1 stable critical point of ϕ such that ϕ(ξ 0 ) > 0 and set

t 0 = c 4 (n) c 1 (n)ϕ(ξ 0 ) > 0.
Identifying the tangent space at ξ with R n we define the map By the invariance of the Brower degree via homotopy, we have that (t 0 , ξ 0 ) is a C 1 stable critical point of G. From Proposition 3.1 and standard properties of the Brower degree (see e.g. [START_REF] Fonseca | Degree theory in analysis and applications[END_REF]), there exists a couple (t ε , ξ ε ) of critical points of I ε converging to (t 0 , ξ 0 ).

H from [0, 1] × R + × R n into
Second case : 5 ≤ n < 8 and n > 13.

Since c 4 (n) and c 1 (n) are positive, we have

lim t→0 + G(t, ξ) = lim t→∞ G(t, ξ) = +∞,
uniformly in ξ ∈ M . Therefore, from (4.1) we deduce that, for ε small enough,there exists a couple (t ε , ξ ε ) which is a minimum for the functional I ε in (a, b) × M where a, b are positive constants not depending on ε. This implies from Proposition 3.2 that u 0 -W δε(tε),ξε -φ δε(tε),ξε is a solution of (1.5). Thus Theorem 1.1 is established.

Finally, we prove Theorem 1.2.

Proof of Theorem 1.2. The proof of Theorem 1.2 will follow closely the proof of Theorem 1.1 therefore we will only sketch it. We restrict ourselves to the case where 9 ≤ n ≤ 11 (the case 5 ≤ n ≤ 8 is contained in Theorem 1.1). The main difference is that here we will take δ ε (t ε ) = (t ε ε)

2 n-4 , for 9 ≤ n ≤ 11. We will only point out the impact of this choice in the two key estimates, namely the estimate of φ δε(t),ξ in Proposition 2.1 (given in Lemma 5.1) and the estimate of the reduced energy (see Proposition 3.1). Let us first consider the error estimate i.e. Lemma 5.1. With our new choice of δ(t ε ), it is immediate to check that the leading term in the expansion of Lemma 5.1 will be given by the term f 0 (W δε(t),ξ ) -P (W δε(t),ξ ) L 2n n+4 . This implies that Lemma 5.1 will rewrite as

i * (f ε (u 0 -W δε(t),ξ )) -u 0 + W δε(t),ξ Pg = 0(δ ε (t) 2 ) = 0(ε 4 n-4 ). (4.2)
Therefore we deduce that φ δε(t),ξ Pg = 0(ε

4 n-4 ), (4.3)
where f ε (u) = |u| 2 * -2-ε u. The triangular inequality yields to

i * (f ε (u 0 -W δε(t),ξ )) -u 0 + W δε(t),ξ Pg ≤ C f ε (u 0 -W δε(t),ξ ) -f ε (u 0 ) + f ε (W δε(t),ξ ) L 2n n+4 + C f ε (u 0 ) -P g (u 0 ) L 2n n+4 + C f ε (W δε(t),ξ ) -P g (W δε(t),ξ ) L 2n n+4 ≤ C(I 1 + I 2 + I 3 ). (5.2)
We first estimate I 1 . By triangular inequality we get

I 1 ≤ (f ε (u 0 -W δε(t),ξ ) + f ε (W δε(t),ξ ))1 B ξ ( √ δε(t)) L 2n n+4 + (f ε (u 0 -W δε(t),ξ ) -f ε (u 0 ))1 M \B ξ ( √ δε(t)) L 2n n+4 + f ε (W δε(t),ξ )1 M \B ξ ( √ δε(t)) L 2n n+4 + f ε (u 0 )1 B ξ ( √ δε(t)) L 2n n+4 . (5.3)
From Taylor expansion (e.g. using (2.20)) and Young inequality, we obtain

(f ε (u 0 -W δε(t),ξ ) + f ε (W δε(t),ξ ))1 B ξ ( √ δε(t)) L 2n n+4 ≤ C u 0 W 2 * -2-ε δε(t),ξ 1 B ξ ( √ δε(t)) L 2n n+4 + C u 2 * -1-ε 0 1 B ξ ( √ δε(t)) L 2n n+4
, as well as

(f ε (u 0 -W δε(t),ξ ) -f ε (u 0 ))1 M \B ξ ( √ δε(t)) L 2n n+4 ≤ C u 2 * -2-ε 0 W δε(t),ξ 1 M \B ξ ( √ δε(t)) L 2n n+4 + C W 2 * -1-ε δε(t),ξ 1 M \B ξ ( √ δε(t)) L 2n n+4
.

Using polar coordinates and a change of variables we deduce that:

I 1 =      O(δ n+4 4 ε (t)) = O(ε n+4 8 ) if n > 12, O(δ 4 ε (t)| ln δ ε (t)| 2 3 ) = O(ε 2 | ln ε| 2 3 ) if n = 12, O(δ n-4 2 ε (t)) = O(ε) if n < 12.
Concerning I 2 we easily get from Taylor's expansion that

I 2 = f ε (u 0 ) -f 0 (u 0 ) L 2n n+4 = O(ε)
. We now estimate I 3 . First we recall that with the help of the exponential map we can identify B ξ (R 0 ) with a neighborhood of the origin in R n . Therefore with this chart we may define χ ξ,δε(t) (.) := χ(d(.δ ε (t), ξ)). Using triangular inequality and a change of variables, we then get

I 3 ≤ Cδ n-4 2 ε χ 2 * -1-ε ξ,δε(t) (U 2 * -1-ε -U 2 * -1 ) L 2n n+4 + C (δ n-4 2 ε χ 2 * -1-ε ξ,δε(t) -χ 2 * -1 ξ,δε(t) )U 2 * -1 δε(t),ξ L 2n n+4 + f 0 (W δε(t),ξ ) -P g (W δε(t),ξ ) L 2n n+4 .
Following the computation in the proof of lemma 2.3 of [START_REF] Deng | Blow-up solutions for Paneitz-Branson type equations with critical growth[END_REF] we obtain these three estimates:

χ 2 * -1-ε ξ,δε(t) (U 2 * -1-ε -U 2 * -1 ) L 2n n+4 = O(ε), (δ n-4 2 ε (t)χ 2 * -1-ε ξ,δε(t) -χ 2 * -1 ξ,δε(t) )U 2 * -1 L 2n n+4 = O(ε| ln δ ε (t)|), and
f 0 (W δε(t),ξ ) -P g (W δε(t),ξ ) L 2n n+4 ≤ C    δ 2 ε (t) = O(ε) if n > 8, δ 2 ε (t)| ln δ ε (t)| = O(ε| ln ε|) if n = 8, δ n-4 2 ε (t) = O(ε) if n < 8.
This concludes the proof.

Finally, let us prove that (3.2) holds C 1 uniformly with respect to t in compact subsets of R * + and ξ ∈ M when 8 ≤ n ≤ 13.

Lemma 5.2. If 8 ≤ n ≤ 13, we have

I ε (t, ξ) = J ε (u 0 -W δε(t),ξ ) + o(ε)
C 1 uniformly with respect to t in compact subsets of R * + and ξ ∈ M .

Proof. To simplify notations, we set, for i = 1, . . . , n, Z 0 = Z δε(t),ξ and Z i = Z δε(t),ξ,e i .

We recall that

∂ ∂t (W δε(t),ξ ) = Cn t Z 0 ,
where Cn = α n (n -4) 4 (see (2.1) for the definition of α n ). Taking the derivative with respect to t to I ε (t, ξ) -J ε (u 0 -W δε(t),ξ ), we obtain We claim that |λ i | = O(ε ln ε), for all i = 0, . . . , n. Using (2.10), to prove the claim, we just need to show that DJ(u 0 -W δε(t),ξ + φ δε(t),ξ )[Z i ] = O(ε ln ε), for all i = 0, . . . , n. Since φ δε(t),ξ ∈ K ⊥ δε(t),ξ , using Hölder inequality, (2.7), Lemma 5.1 and rough estimates, we have DJ ε (u 0 -W δε(t),ξ + φ δε(t),ξ )[Z i ] = M P g (u 0 -W δε(t),ξ )Z i dV -M f ε (u 0 -W δε(t),ξ + φ δε(t),ξ )Z i dV = M (P g (u 0 -W δε(t),ξ ) -f ε (u 0 -W δε(t),ξ ))Z i dV -M (f ε (u 0 -W δε(t),ξ + φ δε(t),ξ ) -f ε (u 0 -W δε(t),ξ ))Z i dV ≤ P g (u 0 -W δε(t),ξ ) -f ε (u 0 -W δε(t),ξ )

∂I ε ∂t (t, ξ) - ∂J ε ∂t (u 0 -W δ(t),ξ ) = M P g (φ δε(t),ξ ) ∂ ∂t W δε(t),ξ dV - M (f ε (u 0 -W δε(t),ξ + φ δε(t),ξ ) -f ε (u 0 -W δε(t
L 2n n+4 Z i L 2 * + f ε (u 0 -W δε(t),ξ + φ δε(t),ξ ) -f ε (u 0 -W δε(t),ξ ) L 2n n+4 Z i L 2 *
≤ O( P g (u 0 -W δε(t),ξ ) -f ε (u 0 -W δε(t),ξ )

Lemma 2 . 1 .

 21 There exists a positive constant C a,b such that for ε small, for any t ∈ [a, b], any ξ ∈ M and any φ

  ε → 0. Using Lemma 5.1 and Proposition 2.1, we get

. 15 )

 15 Since a straight forward computation gives ∂ ∂t Z δε α (tα),ξα | t=tα Pg = O(1), from (2.7), (3.13), (3.14) and (3.15), we deduce that

First case : 8

 8 ≤ n ≤ 13.

3 2 4 =

 34 ε ln ε).Combining the previous estimates, we getDJ ε (u 0 -W δε(t),ξ + φ δε(t),ξ ) ∂φ δε(t),ξ ∂t = O(ε 2 (ln ε) 2 ),(5.9)andDJ ε (u 0 -W δε(t),ξ + φ δε(t),ξ ) ∂φ δε(t),exp ξ (y) ∂y i | y=0 = O(ε (ln ε) 2 ).(5.10) Now let us estimate I 2 and I 5 . Noticing that, if 8 ≤ n ≤ 13,(u 0 -W δε(t),ξ ) 2 * -3-ε Z i L n O(ε -14 ),

  ),ξ )) ∂W δε(t),ξ ∂t dV+ DJ ε (u 0 -W δε(t),ξ + φ δε(t),ξ )[ I 3 = DJ ε (u 0 -W δε(t),ξ + φ δε(t),ξ )[ δε(t),exp ξ (y) )| y=0 = α n (n -4) δ ε (t) Z i + R δε(t),ξ ,where R δε(t),ξ Pg = O(δ ε (t) 2 ) (see(6.13) of[START_REF] Maria | The role of the scalar curvature in a nonlinear elliptic problem on Riemannian manifolds[END_REF]) and using (2.7), we find(Z i ) -f ε (u 0 -W δε(t),ξ )Z i )φ δε(t),ξ dV -M f ε (u 0 -W δε(t),ξ + φ δε(t),ξ ) -f ε (u 0 -W δε(t),ξ ) -f ε (u 0 -W δε(t),ξ )φ δε(t),ξ Z i dV + DJ ε (u 0 -W δε(t),ξ + φ δε(t),ξ )[ ∂φ δε(t),exp ξ (y)Arguing the same way as in Proposition 3.2, we have DJ(u 0 -W δε(t),ξ + φ δε(t),ξ )[

	= In the same way, recalling that Cn t ∂ ∂y i ∂y i (t, exp ξ (y))| y=0 -∂J ε ∂y i = α n (n -4) δ ε (t) (W ∂I ε	∂φ δε(t),ξ ∂t ∂φ δε(t),ξ ] ∂t (u 0 -W δ(t),exp ξ (y) )| y=0 ]. ∂y i ]| y=0 ∂φ δε(t),ξ ∂t ] = O φ δε(t),ξ L	2n n+4	n i=0	|λ i | ,	(5.7)
	and
	DJ ε (u 0 -W δε(t),ξ +φ δε(t),ξ )[	∂φ δε(t),exp ξ (y) ∂y i	]| y=0 = O	φ δε(t),ξ L δ ε (t) 2n n+4	n i=0 |λ i |	.

M

(P g (Z 0 ) -f ε (u 0 -W δε(t),ξ )Z 0 )φ δε(t),ξ dV

-M f ε (u 0 -W δε(t),ξ + φ δε(t),ξ ) -f ε (u 0 -W δε(t),ξ ) -f ε (u 0 -W δε(t),ξ )φ δε(t),ξ Z 0 dV + DJ ε (u 0 -W δε(t),ξ + φ δε(t),ξ )[ ∂φ δε(t),ξ ∂t ] = I 1 + I 2 + I 3 ,

(5.4)

where

I 1 = Cn t M (P g (Z 0 ) -f ε (u 0 -W δε(t),ξ )Z 0 )φ δε(t),ξ dV,

(5.5

)

I 2 = -Cn t M (f ε (u 0 -W δε(t),ξ + φ δε(t),ξ ) -f ε (u 0 -W δε(t),ξ ) -f ε (u 0 -W δε(t),ξ )φ δε(t),ξ )Z 0 dV,

(5.6)

M (P g + O( R δε(t),ξ Pg φ δε(t),ξ Pg ) = I 4 + I 5 + I 6 + o(ε),

(5.8)

where

I 4 = α n (n -4) δ ε (t) M (P g (Z i ) -f ε (u 0 -W δε(t),ξ )Z i )φ δε(t),ξ dV, I 5 = -α n (n -4) δ ε (t) M (f ε (u 0 -W δε(t),ξ + φ δε(t),ξ ) -f ε (u 0 -W δε(t),ξ ) -f ε (u 0 -W δε(t),ξ )φ δε(t),ξ )Z i dV, I 6 = DJ ε (u 0 -W δε(t),ξ + φ δε(t),ξ )[ ∂φ δε(t),exp ξ (y) ∂y i ]| y=0 .

We begin by estimating the terms I 3 and I 6 . We recall that DJ ε (u 0 -W δε(t),ξ + φ δε(t),ξ )[.] = n i=0 λ i Z i , . Pg .

Finite dimensional reduction.Let (ξ α ) α be a sequence of points of M . In all the following, we will suppose up to extracting a subsequence that, for α large enough, all the points ξ α belong to a small open set Ω of M in which there exists a smooth orthogonal frame. Thus, we will identify the tangent spaces T ξ M with R n for all ξ ∈ Ω. We recall that we suppose that P g is coercive.

U (δ -1 exp -1 ξ (x)),

Acknowledgements : The authors would like to thank F. Robert for his comments and suggestions on a preliminary version of this paper. Additionally, the authors would like to thank the anonymous referee for pointing out relevant references.

The second author was supported by the CNPq (Brazil) project 501559/2012-4.

where φ δε(t),ξ is the function defined in Proposition 2.1. Now, let us consider the changes that occur in Proposition 3.1. Using (3.3), (4.2) and (4.3), we obtain that, for 9 ≤ n ≤ 11,

Then, it only remains to compute J ε (u 0 -W δε(t),ξ ). Being a bit careful with the different remainings apppearing in the proof of Proposition 3.1 and using that A g = A paneitz , we see that

Using this last estimate, we can argue exactly as in the case 5 ≤ n < 8 of the proof of Theorem 1.1. This concludes the proof of Theorem 1.2.

5 Appendix.

In this section, we will give an estimate of the error R ε,δε(t),ξ (see Proposition 2.1) and complete the proof of Proposition 3.1 by showing that (3.2) holds C 1 uniformly with respect to t in compact subsets of R * + and ξ ∈ M when 8 ≤ n ≤ 13. Let us begin with the estimate of the error. Lemma 5.1. Given two positive real numbers a < b, there exists a positive constant C a,b such that for ε small, for any real number t ∈ [a, b] and any point ξ ∈ M , there holds

All the estimates will be uniform in t, ξ and ε. Since i * is continuous, we have

(5.1) we obtain, using (2.19), for i = 0, . . . , n,

(5.11)

Finally, let us estimate I 1 and [START_REF] Deng | Blow-up solutions for Paneitz-Branson type equations with critical growth[END_REF], inequality (4.17)) and since, using rough estimates,

we obtain (5.12)

The lemma now follows from (5.4), (5.8), (5.9), (5.10), (5.11) and (5.12).


Laurent Bakri
email: laurent.bakri@gmail.com.

Jean-Baptiste Casteras
email: baptiste.casteras@univ-brest.fr

Some non-stability results for geometric Paneitz-Branson type equations

Keywords: Paneitz-Branson type equations, blow up solutions, Liapunov-Schmidt reduction procedure Mathematics Subject Classification (2010) : 35J30, 35J60, 35B33, 35B35

Let (M, g) be a compact riemannian manifold of dimension n ≥ 5. We consider two Paneitz-Branson type equations with general coefficients

and

where A g and B g are smooth symmetric

and ε is a small positive parameter. Under suitable assumptions, we construct solutions u ε to (??) and (??) which blow up at one point of the manifold when ε tends to 0. In particular, we extend the result of Deng and Pistoia 2011 (to the case where A g is the one defined in the Paneitz operator) and the result of Pistoia and Vaira 2013 (to the case n = 8 and (M, g) locally conformally flat).

Introduction and statements of the results

In this paper, we will study the stability of Paneitz type equations in the geometric case for two kinds of perturbations. The Paneitz operator, which is a conformally covariant fourth order operator defined on any pseudo-Riemannian manifold, has been introduced by Paneitz in [?]. Branson [?] discovered that this operator describes the conformal transformation of a curvature quantity, the Q-curvature. It turns out that this curvature appears in a lot of geometric and physics problems. We refer to the articles of Branson and Gover [?], Chang [?], [?], Chang and Yang [?], and Gursky [?] (and the references therein) for more details on the geometric and physics aspects associated to the notion of Q-curvature. Let (M, g) be a compact riemannian manifold of dimension n ≥ 5. We will be interested in solutions u ∈ C 4,θ (M ), θ ∈ (0, 1), of the following equation

P g u := ∆ 2 g u -div g (A g du) + hu = |u| 2 * -2 u, (1.1)
where h ∈ C ∞ (M ), 2 * = 2n n -4 and A g a smooth symmetric (2, 0)-tensor

given by

A g := (n -2) 2 + 4 2(n -1)(n -2) R g g - 4 n -2 Ric g , (1.2)
where R g (resp. Ric g ) stands for the scalar curvature (resp. Ricci curvature)

with respect to the metric g. When h is given by h = n -4 2 Q g where Q g is the Q-curvature with respect to the metric g then P g is the so-called Paneitz-Branson operator and equation (??) is refered to as the Paneitz-Branson equation. We recall that the Q-curvature is defined by

Q g = 1 2(n -1) ∆ g R g + n 3 -4n 2 + 16n -16 8(n -1) 2 (n -2) 2 R 2 g - 2 (n -2) 2 |Ric g | 2 g .
It is well known that the Paneitz operator is conformally invariant, i.e. if g = ϕ 4 n-4 g then, for all u ∈ C ∞ (M ), we have

P n g (uϕ) = ϕ n+4 n-4 P n g (u)
. We point out that if (M, g) is Einstein (Ric g = λg, λ ∈ R), then the Paneitz-Branson operator takes the form

P g u = ∆ 2 g u + b∆ g u + cu, (1.3)
where b = n 2 -2n -4 2(n -1) λ and c = n(n -4)(n 2 -4) 16(n -1) 2 λ. More generally, following the terminology introduced in [?], when P g is of the form given by (??) (respectively by (??)) for arbitrary smooth (2, 0) tensor A g and h ∈ C ∞ (M ) (respectively for arbitrary real numbers b and c), the operator P g is referred to as a Paneitz-Branson type operator with general coefficients (respectively Paneitz-Branson type operator with constant coefficients). A lot of attention has been devoted to the study of existence and compactness of solution to (??) (see for example [?, ?, ?, ?, ?, ?] and the references therein). Here, we will be interested in the stability of (??). In this paper, we will consider two kinds of stability for (??) : the stability with respect to the tensor A g and the stability with respect to the power of the right-side term of (??). More precisely, we say that (??) is exponentstable if, for any sequences of real positive numbers (ε α ) α such that ε α -→ α→∞ 0 and for any sequences of solutions (u α ) α ∈ C 4,θ (M ), θ ∈ (0, 1), of (1.4) bounded in H 2 (M ), then up to a subsequence, u α converges in C 4 (M ) to some smooth function u solution of (??). Respectively we say that (??) is A g -stable if the functions u α are in fact solutions of

∆ 2 g u α -div g (A g du α ) + hu α = |u α | 2 * -2-εα u α ,
∆ 2 g u α -div g ((A g + ε α B)du α ) + hu α = |u α | 2 * -2 u α , (1.5)
where B is a smooth symmetric (2, 0) tensor. We point out that a related notion of A g -stability has been first introduced by Hebey and Robert in [?]. Before, stating more precisely their results, we introduce some notations. We let λ i (A g ) x , x ∈ M , i = 1, . . . , n, be the eigenvalues of A g (x) (with respect to the metric g) repeated with their multiplicity. We define

λ 1 = inf x,i λ i (A g ) x , λ 2 = max x,i λ i (A g ) x and S w = [λ 1 , λ 2 ].
In particular, it is proved in [?] that if (M, g) is locally conformally flat (l.c.f.) and P g is a Paneitz-Branson type operator with strictly positive constant coefficients satisfying c-b 2 4 < 0, then

(??) is A g -stable whenever a. b / ∈ S w and n = 6, b. b = T rA g n if n = 9 or n = 7, c. b < T rA g n if n = 8.
We point out that the results obtained in [?] are stronger than the ones quoted above. In fact, they show stability of the equation with respect to both A g and h. They also obtained the stability when n = 5 under the hypothesis that the mass of the Green function associated to P g is strictly positive. To the authors' knowledge, it is the most refined positive stability result known presently. Concerning non-stability, the first result has been obtained by Deng and Pistoia in [?]. There, they show that, when A g is replaced by some arbitrary smooth (2, 0) tensor B g , equation (?

?) is not exponent-stable if a. n ≥ 7, T r g (B g -A g ) is not constant and min M T r g (B g -A g ) > 0, b. or n ≥ 8 and ξ 0 ∈ M a C 1 stable critical point of T r g (B g -A g ) such that T r g (B g -A g )(ξ 0 ) > 0.
A related result has been obtained by the authors in [?] where sign changing blowing-up solutions have been constructed in arbitrary dimensions. We refer to [?] for more details. Recently, Pistoia and Vaira [?] studied the A gstability of (??) when P g is the Paneitz-Branson operator. They proved that equation (??) is not A g -stable, under the following conditions : (M, g) is not conformally flat, n ≥ 9 and there exists ξ 0 ∈ M a C 1 stable critical point

(see below for the definition) of the function ξ → T r g B(ξ) |W eyl g (ξ)| g , such that T r g B(ξ 0 ) > 0. For a function φ ∈ C 1 (M ), we recall that a critical point ξ 0 of φ is said C 1 stable if
there exists an open neighborhood Ω of ξ 0 such that, for any point ξ ∈ Ω, there holds ∇ g φ(ξ) = 0 if and only if ξ = ξ 0 and such that the Brower degree deg(∇ g φ, Ω, 0) = 0. Our first theorem extends [?] to the case where B g = A g . More precisely, we have Theorem 1.1. Let (M, g) be a compact riemannian manifold of dimension n, the function h be such that P g is coercive and let Φ be defined by

Φ := - n 2 -4n -4 96(n -1)(n -3) |W eyl g | 2 g + 1 n -4 h - n -4 2 Q g . (1.6)
Assume either that : a. n ≥ 8 and that Φ is such that min x∈M Φ(x) > 0.

b. or n ≥ 11 and there exists ξ 0 ∈ M a C 1 stable critical point of Φ.

Then (??) is not exponent-stable.

As usual for this kind of result, we obtain the previous theorem by constructing a family of solutions (u ε ) ε of (??) which blows-up at some point ξ ∈ M when ε goes to 0. More precisely, the family of solutions we construct is of the form

u ε = ϕBBl ε + o(1),
where o(1) -→ ε→0 0, and ϕ is a conformal factor, the purpose of which will be precised later (see (??)), and

BBl ε (x) = [n(n -4)(n 2 -4)] n-4 8 µ ε µ ε + d g (x, x ε ) 2 n-4 2 , where x, x ε ∈ M and µ ε ∈ R + is such that µ ε -→ ε→0 0.
This form has been introduced by Esposito and Robert in [?]. In our second result we extend the result of [?] to dimension n = 8 using the same family of blowing-up solutions as the one described above. More precisely, we have

Theorem 1.2. Let (M, g) be a compact riemannian manifold of dimension n = 8. Assume that min M {|W eyl g (ξ)| g : T r g (B)(ξ) > 0} > 0. Then (??) is not A g -stable.
We would like to make some comments on this theorem. As it was pointed out in [?] (see Remark 3.1), with our approach we are also able to recover the case n > 8. The approach used in [?] consisted in taking ϕ ≡ 1 but adding an higher-order term to the standard bubble BBl ε . The method we use here is more simple but on the other hand, it seems more rigid. Finally, in our last result, we investigated the A g -stability when (M, g) is l.c.f. Before stating more precisely our theorem, we introduce some notations. We let i g be the injectivity radius of (M, g) and r 0 ∈ R * + such that r 0 < i g . Since we are assuming that (M, g) is l.c.f., there exists a family (g ξ ) ξ∈M of smooth conformal metrics to g such that g ξ is flat in the geodesic ball B ξ (r 0 ). We let G g be the Green's function of the Paneitz operator P g . We will assume that G g is of the form

G g ξ (exp ξ y, ξ) = 1 β n |y| n-4 + A ξ + 0 (4) (|y|), (1.7)
where

β n = (n -2)(n -4)ω n-1 , ω n-1 = |S n-1 |, A ξ > 0 depending only (M, g)
and on ξ (being smooth with respect to ξ) and f = O (k) (r m ) denotes any quantity satisfying

|∇ j f (exp ξ y)| ≤ C j |y| m-j ,
for 1 ≤ j ≤ k. We have :

Theorem 1.3. Let (M, g) be a locally conformally flat manifold of dimension n ≥ 6. Assume that h = Q g , (??) holds and that max M T r g (B) > 0. Then (??) is not A g -stable. More precisely, for ε > 0, there exists a family of solutions u ε of (??) which blows-up, when ε → 0, at some point ξ 0 so that

E(ξ 0 ) = max M E(ξ) where E(ξ) = h(ξ) A 2 n-4 ξ
. Moreover if n ≥ 7, for any isolated critical point ξ 0 of E with non-trivial degree and T r g (B)(ξ 0 ) > 0, for ε > 0, there exists a family of solutions u ε of (??) which blows-up, when ε → 0, atξ 0 .

The method used in order to prove the previous theorem is inspired by the one of Esposito, Pistoia and Vétois [?] where a similar result has been proved for the Yamabe equation. The main idea consists in modifying slightly the shape of the family (u ε ) ε of blowing-up solutions we are looking for by multiplying the standard bubble BBl ε by a function depending on the Green function. Finally, we point out that the assumption (??) is very natural. Gursky and Malchiodi in [?] (see Theorem 2.9) proved that if Q g is semi positive, R g ≥ 0, (M, g) locally conformally flat but not conformally equivalent to the round sphere, then (??) holds (see also some recent preprints of Hang and Yang for improved results [?]).

The proof of the theorems relies on a well known Lyapunov-Schmidt reduction procedure which permits to reduce the problem to a finite dimensional one for which we defined a reduced energy. The solutions to (??) will then be obtained as critical points of this reduced energy. We refer to [?] and the references therein for more information on the Lyapunov-Schmidt reduction procedure.

The plan of this paper is the following : in section 2, we give some preliminaries. Section 3 is devoted to the proof of Theorems ?? and ?? where the proofs of these two theorems are done in parallel. We begin by giving an estimate of the error and then give an estimate of the reduced energy. Finally, in Section 4, we prove Theorem ??.

Preliminaries.

Let (ξ α ) α be a sequence of points of M . In all the following, we will suppose up to extracting a subsequence that, for α large enough, all the points ξ α belong to a small open set Ω of M in which there exists a smooth orthogonal frame. Thus, we will identify the tangent spaces T ξ M with R n for all ξ ∈ Ω. We recall that we suppose that P g is coercive.

In all the following, we will denote by ., . Pg , the scalar product, for

u, v ∈ H 2 (M ), u, v Pg = M ∆ g u∆ g vdV + M A g (∇ g u, ∇ g v)dV + M huvdV,
where here and in the following dV stands for the volume element with respect to the metric g. We will denote . Pg the associated norm which is equivalent to the standard norm of H 2 (M ). We denote by i * :

L 2n n+4 (M ) → H 2 (M ) the adjoint operator of the embedding i : H 2 (M ) → L 2n n-4 (M ), i.e. for all w ∈ L 2n n+4 (M ), the function u = i * (w) ∈ H 2 (M ) is the unique solution of ∆ 2 g u -div g (A g du) + hu = w.
Using this notation, we see that equations (??) and (??) can be rewritten as, for u ∈ H 2 (M ),

u = i * (f ε (u)),
where f ε (u) = |u| 2 * -2-ε u for (??) and f ε (u) = |u| 2 * -2 u -εdiv g (B(∇u) for (??). Before proceeding we recall some basic facts. It is well known (see [?]) that all solutions u ∈ H 2 (R n ) of the equation

∆ 2 eucl u = u 2 * -1 = u n+4 n-4 in R n are given by U δ,y (x) = δ 4-n 2 U x -y δ , δ > 0, y ∈ R n where U (x) = [n(n -4)(n 2 -4)] n-4 8 1 1 + |x| 2 n-4 2 = α n 1 1 + |x| 2 n-4 2 . (2.1)
It is also well known (see [?]) that all solutions v ∈ H 2 (R n ) of

∆ 2 eucl v = (2 * -1)U 2 * -2 v are linear combinations of V 0 (x) = α n n -4 2 |x| 2 -1 (1 + |x| 2 ) n-2 2 and V i (x) = α n (n -4) x i (1 + |x| 2 ) n-2 2 , i = 1, . . . , n.
Let us fix N > n and ξ ∈ M , it is well known that there exists g = ϕ

4 n-2 g, ϕ > 0 is a smooth function on M , such that Ric g(ξ) = 0, ∇R g(ξ) = 0, ∆ gR g(ξ) = 1 6 |W eyl g (ξ)| 2 g (2.2)
and

dV g = 1 + O(r N ) dV R n .
We notice that, in this system of coordinates, we have

Q g (ξ) = 1 2(n -1) ∆ g R g (ξ) = 1 12(n -1) |W eyl g (ξ)| 2 g .
We also give the expression of the Paneitz operator for radial function in the previous metric which will be useful in the sequel :

Lemma 2.1. [?, lemma 2.6 p.13] If u is a radial function, then we have

P g(u) = ∆ 2 u + 1 n -2 ∇ l ∇ k R g (0)x k x l A(u) + n -4 24(n -1) |W eyl g | 2 g (0)u + O r 3 |u | + O r 2 |u | + O(r)u + O r N -1 u + O r N -2 u + O r N -3 u ,
where u = ∂u ∂r and

A(u) = u r 2(n -1) n -2 - (n -1)(n -2) 2 + 6 -n -u n -2 2 + 2 n -2 .
3 Proofs of Theorems ?? and ??.

This section will be devoted to the proof of Theorems ?? and ??. Since the form of the solution we construct will be the same in both cases, we will prove Theorems ?? and ?? in parallel . We define, for any real δ strictly positive, ξ ∈ M and x ∈ M ,

W δ,ξ (x) = χ (d g (x, ξ)) ϕ(x)δ 4-n 2 U δ -1 exp -1 ξ (x) ,
where d g (x, ξ) stands for the distance from x to ξ with respect to the metric g, exp ξ is the exponential map with respect to the metric g and χ :

R → R is a smooth cutoff function such that 0 ≤ χ ≤ 1, χ(x) = 1 if x ∈ -r 0 2 , r 0 2 and χ(x) = 0 if x ∈ R\(-r 0 , r 0 ). We also define, for any real δ strictly positive, ξ ∈ M and x ∈ M , Z δ,ξ (x) = χ(d g (x, ξ))δ n-4 2 d g (x, ξ) 2 -δ 2 (δ 2 + d g (x, ξ) 2 ) n-2 2 , and, for ω ∈ T ξ M , Z δ,ξ,ω (x) = χ(d g (x, ξ))δ n-2 2 exp -1 ξ x, ω g (δ 2 + d g (x, ξ) 2 ) n-2 2 .
We denote by Π δ,ξ respectively Π ⊥ δ,ξ the projection of H 2 (M ) onto

K δ,ξ = span {Z δ,ξ , (Z δ,ξ,e i ) i=1..n } respectively K ⊥ δ,ξ = φ ∈ H 2 (M )/ φ, Z δ,ξ Pg = 0 and φ, Z δ,ξ,ω Pg = 0, ∀ω ∈ T ξ M .
(3.1) We are looking for solution u to (??) of the form

u = W δε(tε),ξε + φ δε(tε),ξε ,
where φ δε(tε),ξε ∈ K ⊥ δε(tε),ξε and δ ε (t ε ) ∈ R + is defined below. It is easy to see that equations (??) (respectively (??)) are equivalent to the following system

Π δε(t),ξ W δε(t),ξ + φ δε(t),ξ -i * f ε (W δε(t),ξ + φ δε(t),ξ ) = 0, (3.2) and Π ⊥ δε(t),ξ W δε(t),ξ + φ δε(t),ξ -i * f ε (W δε(t),ξ + φ δε(t),ξ ) = 0, (3.3)
where

f ε is defined by f ε (u) = |u| 2 * -2-ε u (respectively by f ε (u) = |u| 2 * -2 u - εdiv g (B(∇u))). We define δ ε (t ε ), for t ε > 0 by δ ε (t ε ) = (t ε ε) 1 4 , if n ≥ 9, t ε l -1 (ε), if n = 8 (3.4)
where l : (0, e -1 2 ) → (0, e -1 2 ) is defined by

l(δ) = -δ 4 ln δ if f ε (u) = |u| 2 * -2-ε u and by l(δ) = -δ 2 ln δ if f ε (u) = |u| 2 * -2 u -εdiv g (B(∇u)).
Let us now give the plan of this section. We will begin by solving (??) in Section 3.1. In Section 3.2, we will solve (??) by proving an estimate of the reduced energy (see Propositions ?? and ??) and give the proof of the theorems. Finally, in Section 3.3, we finish the proof of Proposition ?? by showing that the estimate of the reduced energy holds C 1 -uniformly when n ≥ 11.

Finite dimensional reduction.

We begin by solving (??). The following proposition is well known and we refer to [?] and [?] for a proof of it.

Proposition 3.1. Given two real numbers a < b, there exists a positive constant C a,b such that for ε small, for any t ∈ [a, b] and any ξ ∈ M , there exists a unique function φ δε(t),ξ ∈ K ⊥ δε(t),ξ which solves equation (??) and satisfies

φ δε(t),ξ Pg ≤ C a,b i * (f ε (W δε(t),ξ )) -W δε(t),ξ Pg .
(3.5)

Moreover, φ δε(t),ξ is continuously differentiable with respect to t and ξ.

The next two lemma are devoted to estimate i * (f ε (W δε(t),ξ )) -W δε(t),ξ Pg in term of ε. We begin by the case f ε (u) = |u| 2 * -2-ε u (i.e. by (??)).

Lemma 3.2. Assume that n ≥ 8 and f ε (u) = |u| 2 * -2-ε u. Given two positive real numbers a < b, there exists a positive constant C a,b such that for ε small, for any real number t ∈ [a, b] and any point ξ ∈ M , there holds

i * (f ε (W δε(t),ξ )) -W δε(t),ξ Pg ≤ C a,b ε| ln δ ε (t)| + δ ε (t) n-4 2 , if n < 12, δ ε (t) 4 | ln δ ε (t)|, if n ≥ 12 .
Proof. All the estimates will be uniform in t, ξ and ε. Since i * is continuous, we have

i * (f ε (W δε(t),ξ )) -W δε(t),ξ Pg = O (f ε (W δε(t),ξ )) -P g (W δε(t),ξ ) L 2n n+4
.

(3.6)

The triangular inequality yields to

i * (f ε (W δε(t),ξ )) -W δε(t),ξ Pg ≤ C f ε (W δε(t),ξ ) -f (W δε(t),ξ ) L 2n n+4 + C f (W δε(t),ξ ) -P g (W δε(t),ξ ) L 2n n+4 ≤ C(I 1 + I 2 ). (3.7)
It is easy to see (see for instance inequality (3.28) of [?]) that

I 1 = f ε (W δε(t),ξ ) -f 0 (W δε(t),ξ ) L 2n n+4 = O(ε| ln δ ε (t)|). (3.8)
Using that the Paneitz operator P paneitz , (i.e. for which h = n -4 2 ) is a conformal operator and denoting Wδε(t),ξ = W δε(t),ξ ϕ , we have

f (W δε(t),ξ ) -P paneitz g (W δε(t),ξ ) = ϕ 2 * -1 (f ( Wδε(t),ξ ) -P paneitz g ( Wδε(t),ξ )),
where g is the metric defined in (??). Now, since Wδε(t),ξ is a radial function, using Lemma ??, we have

f (W δε(t),ξ ) -P g (W δε(t),ξ ) = f (W δε(t),ξ ) -P paneitz g (W δε(t),ξ ) + n -4 2 Q -h W δε(t),ξ = ϕ 2 * -1 (f ( Wδε(t),ξ ) -∆ 2 R n ( Wδε(t),ξ )) + O(r 2 ∂ rr Wδε(t),ξ + r∂ r Wδε(t),ξ + Wδε(t),ξ ) = O(r 2 ∂ rr Wδε(t),ξ + r∂ r Wδε(t),ξ + Wδε(t),ξ ). Direct computations give max Wδε(t),ξ L 2n n+4 , r∂ r Wδε(t),ξ L 2n n+4 , r 2 ∂ rr Wδε(t),ξ L 2n n+4 ≤ C      δ ε (t) 4 , if n ≥ 13, δ ε (t) 4 | ln δ ε (t)|, if n = 12, δ ε (t) n-4 2 , if 5 ≤ n ≤ 11.
Therefore, we deduce that

|I 2 | ≤ C      δ ε (t) 4 , if n ≥ 13, δ ε (t) 4 | ln δ ε (t)|, if n = 12, δ ε (t) n-4 2 , if 5 ≤ n ≤ 11.
(3.9)

Combining (??), (??) and (??), the proof of the lemma follows.

Next we prove the equivalent of the previous lemma for equation (??).

Lemma 3.3. Assume that n = 8 and f ε (u) = |u| 2 * -2 u -εdiv g (B(∇u)).

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

i * (f ε (W δε(t),ξ )) -W δε(t),ξ Pg ≤ C a,b (δ ε (t) 2 + εδ ε (t) 2 | ln δ ε (t)| 3 4 ).
Proof. Using the continuity of i * and the triangular inequality, we have

i * (f ε (W δε(t),ξ )) -W δε(t),ξ Pg ≤ ε div g (B(∇W δε(t),ξ )) L 2n n+4 + C f (W δε(t),ξ ) -P g (W δε(t),ξ ) L 2n n+4 ≤ C(I 1 + I 2 ).
(3.10)

From (??), we know that

I 2 = δ ε (t) 2 . Since 1 r W δε(t),ξ L 2n n+4 + W δε(t),ξ L 2n n+4 ≤ O(δ ε (t) 2 | ln δ ε (t)| 3 4 ),
we deduce that

I 1 = ε div g (B(∇W δε(t),ξ )) L 2n n+4 = O(εδ ε (t) 2 | ln δ ε (t)| 3 4 )).
The proof follows from the previous estimates.

The reduced problem.

In this section, we will solve equation (??) and give the proof of Theorems ?? and ??. We begin with the case f ε (u) = |u| 2 * -2-ε u (i.e. equation (??)).

For ε > 0 small enough, we define the energy associated to (??) by, for u ∈ H 2 (M ),

J ε (u) = 1 2 M (∆ g u) 2 + 1 2 M A g (∇ g u, ∇ g u)dV + 1 2 M hu 2 dV - M F ε (u)dV, (3.11)
where

F ε (u) = u 0 f ε (s)ds. We set I ε (t, ξ) = J ε (W δε(t),ξ + φ δε(t),ξ ), t ∈ R * +
and ξ ∈ M where φ δε(t),ξ ∈ K ⊥ δε(t),ξ is the function defined in Proposition ??. In the next proposition, we give the expansion of I ε with respect to ε. Proposition 3.4. Assume that n ≥ 8 and f ε (u) = |u| 2 * -2-ε u. There exist constants c i (n), i = 1, . . . , 5 depending on n, such that

I ε (t, ξ) = c 5 (n)+c 2 (n)ε+c 3 (n)ε ln ε-c 4 (n)ε ln(t)+c 1 (n)Φ(ξ)εt+o(ε) (3.12)
as ε → 0, C 0 uniformly with respect to t in compact subsets of R * + and with respect to ξ ∈ M and C 1 uniformly if n ≥ 11. Moreover, we have that

c 4 (n) > 0, c 1 (n) > 0 and Φ = - n 2 -4n -4 96(n -1)(n -3) |W eyl g | 2 g + 1 n -4 (h - n -4 2 Q g ).
Proof. We begin by proving that

I ε (t, ξ) = J ε (W δε(t),ξ ) + o(ε), (3.13)
as ε → 0, uniformly with respect to t in compact subsets of R * + and points ξ ∈ M (we will show in Lemma ?? that this estimate holds C 1 uniformly with respect to t and ξ when n ≥ 12). Indeed, we have

I ε (t, ξ) -J ε (W δε(t),ξ ) = W δε(t),ξ -i * (f ε (W δε(t),ξ )), φ δε(t),ξ Pg + O( φ δε(t),ξ 2 Pg ) (3.14)
when ε → 0. Using Lemma ??, Proposition ?? and the definition of δ ε (t) (??), we get

W δε(t),ξ -i * (f ε (W δε(t),ξ )), φ δε(t),ξ Pg + O( φ δε(t),ξ 2 Pg ) = O( φ δε(t),ξ 2 Pg ) = O ε 2 | ln δ ε (t)| 2 + δ ε (t) n-4 if n < 12 δ ε (t) 8 | ln δ ε (t)| 2 if n ≥ 12 = o(ε).
Now, the proposition is reduced to estimate J ε (W δε(t),ξ ). We will focus on C 0 -estimates. The C 1 -estimates can be obtained using the same argument as in Lemma 4.1 of [?]. We use the computations of section 6 of [?] and the estimate (4.2) of [?] to estimate I 1,ε,t,δ . Using that J ε is conformally invariant and using (??), we have

M P g (W δε(t),ξ )W δε(t),ξ dV = α 2 n n(n -4)(n 2 -4)ω n 2 n +      ω n (n -1)(n -3)(n -4)Φ(ξ) 2 n-4 (n -6)(n -8) δ ε (t) 4 + o(δ ε (t) 4 ), if n ≥ 9, ω n-1 (n -4)Φ(ξ)δ ε (t) 4 | ln δ ε (t)| + o(δ ε (t) 4 | ln δ ε (t)|), if n = 8, (3.15) where Φ is given by Φ = - n 2 -4n -4 96(n -1)(n -3) |W eyl g | 2 g + 1 n -4 (h - n -4 2 Q g ).
Next, we define I q p = ∞ 0 r q (1 + r) p dr for any p, q integers such that q < p -1.

In the sequel, we will use that

I q p = q p -q -1 I q-1 p = p p -q -1 I q p+1 ,
and

I n-2 n = 1 n -1 , I n-2 2 n = ω n 2 n-1 ω n-1 .
We also have that

I n-2 2 n-n-4 2 ε = I n-2 2 n + n -4 2 Ĩ n-2 2 n ε + O(ε 2 ),
where Ĩq p = ∞ 0 r q ln(1 + r)

(1 + r) p dr. Therefore, we obtain that

1 2 * -ε M W 2 * -ε δε(t),ξ dV = α 2 * -ε n 2 * -ε (δ ε (t)) n-4 2 ε ω n-1 r 0 δε(t) 0 1 1 + r 2 n-n-4 2 ε r n-1 (1 + O(r n ))dr = n -4 2n K -n/4 n 1 + n -4 2 ε ln(δ ε (t)) + n -4 2n n Ĩ n-2 2 n I n-2 2 n + n(1 -n 2 ln n(n -4)(n 2 -4)) n -2 ε + o(δ ε (t) 5 ).
(3.16)

Then the proposition follows from (??) and (??).

Next we consider equation (??). For ε > 0 small enough, we define the energy associated to (??) by, for u ∈ H 2 (M ),

Jε (u) = 1 2 M (∆ g u) 2 + 1 2 M (A g + εB)(∇ g u, ∇ g u)dV + 1 2 M hu 2 dV - M F 0 (u)dV, (3.17)
where F 0 (u) = u 0 f 0 (s)ds. We set Ĩε (t, ξ) = Jε (W δε(t),ξ + φ δε(t),ξ ), t ∈ R * + and ξ ∈ M where φ δε(t),ξ ∈ K ⊥ δε(t),ξ is the function defined in Proposition ??. In this case, we obtain : Proposition 3.5. Assume that n = 8 and f ε (u) = |u| 2 * -2 u -εdiv g (B(∇u)). There exist constants c i (n), i = 1, . . . , 3 depending on n, such that

Ĩε (t, ξ) =c 1 (n) + ε 2 | ln ε| -1 c 2 (n)t 2 T r g B(ξ) -c 3 (n)t 4 |W eyl g | 2 g + o(ε 2 | ln ε| -1 ), (3.18)
as ε → 0 C 0 uniformly with respect to t in compact subsets of R * + and with respect to ξ ∈ M . Moreover, we have that c 2 (n) > 0 and c 3 (n) > 0.

Proof. Using Lemma ??, Proposition ?? and (??), we get

W δε(t),ξ -i * (f ε (W δε(t),ξ )), φ δε(t),ξ Pg + O φ δε(t),ξ 2 Pg = O δ ε (t) 4 .
We have

1 2 * M W 2 * δε(t),ξ dV = ω n 2 n + o(δ ε (t) 5 ),
and

1 2 ε M B(∇W δε(t),ξ , ∇W δε(t),ξ )dV = (n -4) 2 2n εδ ε (t) 2 α 2 n ω n-1 T r g B(ξ) r 0 δε(t) 0 r n+1 (1 + r 2 ) n-2 dr + O(δ ε (t) 5 ) = εδ ε (t) 2 2(n -1) n(n -6)(n 2 -4) K -n 4 n T r g B(ξ) + O(δ ε (t) 5 ).
Therefore, from the two previous estimates and (??), we have

Jε (W δε(t),ξ ) = α 2 n n(n -4)(n 2 -4)ω n 2 n + εδ ε (t) 2 2(n -1) n(n -6)(n 2 -4) K -n 4 n T r g B(ξ) - n 2 -4n -4 96(n -1)(n -3) |W eyl g | 2 g δ ε (t) 4 | ln δ ε (t)| + o δ ε (t) 4 | ln δ ε (t)| .
Finally, we recall the following proposition (see [?] and [?] for a proof) which shows that to obtain a solution of (??) (respectively (??)), we only need to find a critical point for Ĩε (respectively I ε ).

Proposition 3.6. Given two positive real numbers a < b, for ε small, if (t ε , ξ ε ) ∈ (a, b) × M is a critical point of Ĩε (respectively I ε ), then the function W δε(tε),ξε + φ δε(tε),ξε is a solution of (??) (respectively (??)).

We are now in position to prove Theorems ?? and ??. We restrict ourselves to prove Theorem ?? (Theorem ?? can be obtained in the same way).

Proof of Theorem ??. We set G : R * + × M → R the function defined by

G(t, ξ) = -c 4 (n) ln t + c 1 (n)ϕ(ξ)t,
where c 4 (n), c 1 (n) and ϕ(ξ) are defined in (??). From Proposition ??, we have

lim ε→0 1 ε (I ε (t, ξ) -c 5 (n) -c 2 (n)ε -c 3 (n)ε ln ε) = G(t, ξ), (3.19)
C 0 (and C 1 if n ≥ 11) uniformly with respect to ξ ∈ M and t in compact subset of R * + . We will consider two cases depending on the dimension of the manifold.

First case : n ≥ 11.

We argue as in [?]. Let ξ 0 be the C 1 stable critical point of ϕ such that ϕ(ξ 0 ) > 0 and set

t 0 = c 4 (n) c 1 (n)ϕ(ξ 0 ) > 0.
Identifying the tangent space at ξ with R n we define the map

H from [0, 1] × R + × R n into R n+1 by H(s, t, ξ) = s ∂G(t, exp ξ (y)) ∂t , ∂G(t, exp ξ (y)) ∂y 1 | y=0 , . . . , ∂G(t, exp ξ (y)) ∂y n | y=0 +(1 -s) t -t 0 , ∂(ϕ • exp ξ (y)) ∂y 1 | y=0 , . . . , ∂(ϕ • exp ξ (y)) ∂y n | y=0 .
By the invariance of the Brower degree via homotopy, we have that (t 0 , ξ 0 ) is a C 1 stable critical point of G. From Proposition ?? and standard properties of the Brower degree (see e.g. [?]), there exists a couple (t ε , ξ ε ) of critical points of I ε converging to (t 0 , ξ 0 ). Second case : 8 ≤ n ≤ 10.

Since c 4 (n) and c 1 (n) are positive, we have

lim t→0 + G(t, ξ) = lim t→∞ G(t, ξ) = +∞,
uniformly in ξ ∈ M . Therefore, from (??) we deduce that, for ε small enough,there exists a couple (t ε , ξ ε ) which is a minimum for the functional I ε in (a, b) × M where a, b are positive constants not depending on ε. This implies from Proposition ?? that W δε(tε),ξε + φ δε(tε),ξε is a solution of (??). Thus Theorem ?? is established.

3.3 C 1 uniform estimate for the reduced energy.

Finally, we end this section by proving that the estimate (??) holds C 1 uniformly if n ≥ 11.

Lemma 3.7. If n ≥ 11, we have

I ε (t, ξ) = J ε (W δε(t),ξ ) + o(ε),
C 1 uniformly with respect to t in compact subsets of R * + and ξ ∈ M .

Proof. To simplify notations, we set, for i = 1, . . . , n,

Z 0 = Z δε(t),ξ and Z i = Z δε(t),ξ,e i .
We recall that ∂ ∂t

(W δε(t),ξ ) = Cn t Z 0 = Cn δ ε (t) δ ε (t) Z 0 ,
where Cn = α n (n -4) 4 (see (??) for the definition of α n ). Taking the derivative with respect to t to I ε (t, ξ) -J ε (W δε(t),ξ ), we obtain

∂I ε ∂t (t, ξ) - ∂J ε ∂t (W δ(t),ξ ) = M P g (φ δε(t),ξ ) ∂ ∂t W δε(t),ξ dV - M (f ε (W δε(t),ξ + φ δε(t),ξ ) -f ε (W δε(t),ξ ))
∂W δε(t),ξ ∂t dV (3.20) where (3.22)

+ DJ ε (W δε(t),ξ + φ δε(t),ξ ) ∂φ δε(t),ξ ∂t = Cn t M (P g (Z 0 ) -f ε (W δε(t),ξ )Z 0 )φ δε(t),ξ dV - M f ε (W δε(t),ξ + φ δε(t),ξ ) -f ε (W δε(t),ξ ) -f ε (W δε(t),ξ )φ δε(t),ξ Z 0 dV + DJ ε (W δε(t),ξ + φ δε(t),ξ ) ∂φ δε(t),ξ ∂t =I 1 + I 2 + I 3 ,
I 1 = Cn t M (P g (Z 0 ) -f ε (W δε(t),ξ )Z 0 )φ δε(t),ξ dV, (3.21) I 2 = - Cn t M (f ε (W δε(t),ξ + φ δε(t),ξ ) -f ε (W δε(t),ξ ) -f ε (W δε(t),ξ )φ δε(t),ξ )Z 0 dV,
I 3 = DJ ε (W δε(t),ξ + φ δε(t),ξ ) ∂φ δε(t),ξ ∂t . (3.23)
In the same way, recalling that

∂ ∂y i (W δε(t),exp ξ (y) ) y=0 = α n (n -4) δ ε (t) Z i + R δε(t),ξ ,
where R δε(t),ξ Pg = O(δ ε (t) 2 ) (see (6.13) of [?]) and using (??), we find

∂I ε ∂y i (t, exp ξ (y)) y=0 - ∂J ε ∂y i (W δ(t),exp ξ (y) ) y=0 = α n (n -4) δ ε (t) M (P g (Z i ) -f ε (W δε(t),ξ )Z i )φ δε(t),ξ dV - M f ε (W δε(t),ξ + φ δε(t),ξ ) -f ε (W δε(t),ξ ) -f ε (W δε(t),ξ )φ δε(t),ξ Z i dV + DJ ε (W δε(t),ξ + φ δε(t),ξ ) ∂φ δε(t),exp ξ (y) ∂y i y=0 + O R δε(t),ξ Pg φ δε(t),ξ Pg = I 4 + I 5 + I 6 + o(ε), (3.24)
where

I 4 = α n (n -4) δ ε (t) M (P g (Z i ) -f ε (W δε(t),ξ )Z i )φ δε(t),ξ dV, I 5 = - α n (n -4) δ ε (t) M (f ε (W δε(t),ξ + φ δε(t),ξ ) -f ε (W δε(t),ξ ) -f ε (W δε(t),ξ )φ δε(t),ξ )Z i dV, I 6 = DJ ε (W δε(t),ξ + φ δε(t),ξ ) ∂φ δε(t),exp ξ (y) ∂y i y=0 .
We begin by estimating the terms I 3 and I 6 . By Proposition ??, there exist real numbers λ i , i = 0, . . . , n such that

DJ ε (W δε(t),ξ + φ δε(t),ξ )[.] = n i=0 λ i Z i , . Pg .
Arguing in the same way as in Proposition 2.2 of [?] (see in particular (4.23) and (4.24)), we have

DJ(W δε(t),ξ + φ δε(t),ξ ) ∂φ δε(t),ξ ∂t = O φ δε(t),ξ L 2n n+4 n i=0 |λ i | ,
and

DJ ε (W δε(t),ξ + φ δε(t),ξ ) ∂φ δε(t),exp ξ (y) ∂y i y=0 = O φ δε(t),ξ L 2n n+4 n i=0 |λ i | δ ε (t) . We claim that |λ i | = O P g (W δε(t),ξ ) -f ε (W δε(t),ξ ) L 2n n+4
, for all i = 0, . . . , n.

Using that Z i , Z j Pg → ∆ eucl V i 2 L 2 (R n ) δ ij , to prove the claim, we just need to show that DJ(W δε(t),ξ + φ δε(t),ξ )[Z i ] = O( P g (W δε(t),ξ ) -f ε (W δε(t),ξ ) L 2n n+4
), for all i = 0, . . . , n. Since φ δε(t),ξ ∈ K ⊥ δε(t),ξ , using Hölder inequality, (??), Lemma ?? and rough estimates, we have

DJ ε (W δε(t),ξ + φ δε(t),ξ )[Z i ] = M P g (W δε(t),ξ )Z i dV - M f ε (W δε(t),ξ + φ δε(t),ξ )Z i dV = M (P g (W δε(t),ξ ) -f ε (W δε(t),ξ ))Z i dV - M (f ε (W δε(t),ξ + φ δε(t),ξ ) -f ε (W δε(t),ξ ))Z i dV ≤ P g (W δε(t),ξ ) -f ε (W δε(t),ξ ) L 2n n+4 Z i L 2 * + f ε (W δε(t),ξ + φ δε(t),ξ ) -f ε (W δε(t),ξ ) L 2n n+4 Z i L 2 * ≤O P g (W δε(t),ξ ) -f ε (W δε(t),ξ ) L 2n n+4 + O φ δε(t),ξ L 2n n-4 ( W δε(t),ξ 2 * -2-ε L 2n n-4 + φ δε(t),ξ 2 * -2-ε L 2n n-4 ) ≤O P g (W δε(t),ξ ) -f ε (W δε(t),ξ ) L 2n n+4
.

Combining the previous estimates, we get

DJ ε (W δε(t),ξ + φ δε(t),ξ ) ∂φ δε(t),ξ ∂t = O P g (W δε(t),ξ ) -f ε (W δε(t),ξ ) 2 L 2n n+4 , (3.25) and DJ ε (W δε(t),ξ + φ δε(t),ξ )
∂φ δε(t),exp ξ (y)

∂y i | y=0 = O P g (W δε(t),ξ ) -f ε (W δε(t),ξ ) 2 L 2n n+4
δ ε (t) .

(3.26) Now let us estimate I 2 and I 5 . Noticing that, if n ≥ 11,

(W δε(t),ξ ) 2 * -3-ε Z i L n 4 = O(1), we obtain, for i = 0, . . . , n, M (f ε (W δε(t),ξ + φ δε(t),ξ ) -f ε (W δε(t),ξ ) -f ε (W δε(t),ξ )φ δε(t),ξ )Z i dV ≤ C M (W δε(t),ξ ) 2 * -3-ε φ 2 δε(t),ξ Z i dV, if n ≥ 12, M ((W δε(t),ξ ) 2 * -3-ε φ 2 δε(t),ξ + φ 2 * -1-ε δε(t),ξ Z i dV ), if n = 11, ≤ C      (W δε(t),ξ ) 2 * -3-ε Z i L n 4 φ δε(t),ξ 2 L 2n n-4 , if n ≥ 12, φ δε(t),ξ 2 L 2n n-4 (W δε(t),ξ ) 2 * -3-ε Z i L n 4 + Z i L 2n n-4 φ δε(t),ξ 2 * -1-ε L 2n n-4 , if n = 11, ≤ O( φ δε(t),ξ 2 L 2n n-4 ) when n ≥ 11. (3.27)
Finally, let us estimate I 1 and I 4 . Since

P g (Z i ) -f ε (W δε(t),ξ )Z i L 2n n+4 = O(δ ε (t)
2 ) (one can argue as in [?], inequality (4.17)), we obtain

M (P g (Z i ) -f ε (W δε(t),ξ )Z i )φ δε(t),ξ dV ≤ C P g (Z i ) -f ε (W δε(t),ξ )Z i L 2n n+4 φ δε(t),ξ L 2 * ≤ O δ ε (t) 2 φ δε(t),ξ L 2 * .
(3.28)

The lemma now follows from (??), (??), (??), (??), (??) and (??).

4 Case M is l.c.f. : Proof of Theorem ??.

In all this section, we assume that M is l.c.f. We will also assume that G g ξ , the Green function of P g ξ , is of the form

G g ξ (exp ξ y, ξ) = 1 β n |y| n-4 + A + 0 (4) (|y|),
where β n = (n -2)(n -4)ω n-1 (see the introduction for the definition of 0 (4) (|y|)). In the following, with an abuse of notation, we will identify the metric g and g ξ . In this section, we will modify the notation of the function W δε(t),ξ defined in the previous one. Here, we will be looking for a solution of the form

W δε(t),ξ (x) = G g (x, ξ) Ŵδε(t),ξ (x),
where Ŵδε(t),ξ is defined by

Ŵδε(t),ξ =            Ŵ in δε(t),ξ (x) := β n δ ε (t) 4-n 2 d(x, ξ) n-4 U d(x, ξ) δ ε (t) , if d(x, ξ) ≤ r 0 , Ŵ out δε(t),ξ (x) := β n δ ε (t) 4-n 2 r n-4 0 U r 0 δ ε (t) + γ ε(d(x, ξ) -r 0 )( Ŵ in δε(t),ξ ) (r 0 ), if d(x, ξ) > r 0 .
In the previous definition, γ ε : R → R, ε ∈ R + is a smooth function satisfying the following properties :

supp(γ ε) ⊂ [0, ε], γ(0) = 0, γ (0) = 1, |∇ i γ(r)| ≤ ε-i , ∀r ≥ 0 and i = 1, . . . , 4.
It is easy to check that Ŵδε(t),ξ ∈ H 2 (M ). We also define, for any real δ strictly positive, ξ ∈ M and x ∈ M ,

Z δ,ξ (x) = G g (x, ξ) Ẑδ,ξ (x), (4.1)
and, for ω

∈ T ξ M , Z δ,ξ,ω (x) = G g (x, ξ) Ẑδ,ξ,ω (x), (4.2)
where

Ẑδ,ξ (x) = d(x, ξ) n-4 χ(d g (x, ξ))δ n-4 2 d(x, ξ) 2 -δ 2 (δ 2 + d(x, ξ) 2 ) n-2 2 , and, for ω ∈ T ξ M , Ẑδ,ξ,ω (x) = d(x, ξ) n-4 χ(d g (x, ξ))δ n-2 2 exp -1 ξ x, ω g (δ 2 + d(x, ξ) 2 ) n-2 2 , χ is a smooth cut-off function such that 0 ≤ χ ≤ 1, χ ≡ 0 in [r 0 , ∞) and χ ≡ 1 in [0, r 0 2 ].
In this section, we choose

δ ε (t) = tε 1 n-6 if n ≥ 7 e -t ε if n = 6 . (4.3)
In view of the results of the previous section, it is easy to see that we only need to obtain an estimate of the error and of the reduced energy in order to prove Theorem ??. We begin with the error estimate.

Lemma 4.1. Assume that M is l.c.f. and f ε (u) = |u| 2 * -2 u -εdiv g (B(∇u)).

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

i * (f ε (W δε(t),ξ )) -W δε(t),ξ Pg ≤ C a,b      δ ε (t) n-4 + εδ ε (t) n-4 2 , if n < 8, δ ε (t) 4 | ln δ ε (t)| 3 4 + εδ ε (t) 2 | ln δ ε (t)| 3 4 , if n = 8, δ ε (t) n 2 + εδ ε (t) 2
if n > 8.

(4.4)

Proof. Integrating by parts, we find

i * (f ε (W δε(t),ξ )) -W δε(t),ξ , φ = M (f ε (W δε(t),ξ ) -P g W δε(t),ξ )φdV - ∂B ξ (r 0 ) (∆ g W out δε(t),ξ ∂ in φ + ∆ g W in δε(t),ξ ∂ out φ)dσ + ∂B ξ (r 0 ) ∂ in (∆ g W out δε(t),ξ ) + ∂ out (∆ g W in δε(t),ξ ) φdσ
where ∂B ξ (r 0 ) is the boundary of the geodesic ball with respect to g ξ of radius r 0 centered in ξ and ∂ ν in (resp. ∂ νout ) stands for the derivatives with respect to the inward (resp. outward), unit normal vectors of ∂B ξ (r 0 ) and dσ g is the volume element of ∂B ξ (r 0 ). Now, straight forward computations give, on ∂B ξ (r 0 ),

(∂ in ∆ g W out δε(t),ξ + ∂ out ∆ g W in δε(t),ξ ) = G g (∆ g Ŵ in δε(t),ξ -∆ g Ŵ out δε(t),ξ ) + G g (∆ g Ŵ in δε(t),ξ ) -(∆ g Ŵ out δε(t),ξ ) + 2G g (∇ Ŵ in δε(t),ξ ) -(∇ Ŵ out δε(t),ξ ) = O(δ ε (t) n 2 ), (4.5)
and

(∆W in δε(t),ξ -∆W out δε(t),ξ ) = G g ∆ g ( Ŵ in δε(t),ξ -Ŵ out δε(t),ξ ) = O(δ ε (t) n 2 ). (4.6)
We observe, since Ŵδε(t),ξ = O(δ ε (t)

n-4

2 ) on M \B ξ (r 0 ) and

P g G g (., ξ) = 0, that f ε (W δε(t),ξ -G g (., ξ)Γ) -P g (W δε(t),ξ -G g (., ξ)Γ) = O(δ ε (t) n+4 2 + εδ ε (t) n-4
2 ), on M \B ξ (r 0 ), (4.7)

where Γ(x) = γ ε(d(x, ξ) -r 0 )( Ŵ in δε(t),ξ ) (r 0 ). Next, using the properties of γ ε and the fact that n ≥ 5, it is easy to see that

M \B ξ (r 0 ) P g (G g (., ξ)Γ)φdV = B ξ (r 0 +ε)\B ξ (r 0 ) P g (G g (., ξ)Γ)φdV = O(ε), (4.8) and M \B ξ (r 0 ) f ε (G g (., ξ)Γ)φdV = O(ε).
Therefore, by Sobolev's and trace's embeddings, and choosing ε ≤ δ ε (t) n 2 , we deduce that

i * (f ε (W δε(t),ξ )) -W δε(t),ξ H 2 (M ) ≤ f ε (W δε(t),ξ ) -P g W δε(t),ξ L 2n n+4 (B ξ (r 0 )) + O(δ ε (t) n 2 ).
Since M is l.c.f., using the conformal invariance of P g and doing some computations, we find

P g (W δε(t),ξ ) = β n G g r n-4 δ ε (t) -n+4 2 U (δ -1 y) 2 * -1 (4.9) + 4A ξ δ ε (t) 2 β n α n δ ε (t) n-4 2 (n -4) 2 |y| n-8 (δ ε (t) 2 + |y| 2 ) n+2 2 × δ ε (t) 4 (18 -9n + n 2 ) -8δ ε (t) 2 (n -3)|y| 2 -(n -6)|y| 4 + h.o.t.
Here and in the following, h.o.t. stands for a term which is asymptotically smaller than one of the previous terms in the expansion as ε goes to 0. Therefore, we deduce that

f ε (W δε(t),ξ ) -P g W δε(t),ξ L 2n n+4 (B ξ (r 0 )) = O δ ε (t) n 2 |y| n-8 (δ ε (t) 2 + |y| 2 ) n+2 2 δ ε (t) 4 + δ ε (t) 2 |y| 2 + |y| 4 L 2n n+4 (B ξ (r 0 )) + ε div g B(∇W δε(t),ξ ) L 2n n+4 (B ξ (r 0 ))
. Now, rough estimates give

δ ε (t) n 2 |y| n-8 (δ ε (t) 2 + |y| 2 ) n+2 2 δ ε (t) 4 + δ ε (t) 2 |y| 2 + |y| 4 L 2n n+4 (B ξ (r 0 )) ≤ C      δ ε (t) n-4 , if n < 8, δ ε (t) 4 | ln δ ε (t)| 3 4 , if n = 8, δ ε (t) n 2 , if n > 8.
and

ε div g (B(∇W δε(t),ξ )) L 2n n+4 (B ξ (r 0 )) ≤ C    εδ ε (t) n-4 2 , if n < 8, εδ ε (t) 2 | ln δ ε (t)| n+4 2n , if n = 8, εδ ε (t) 2 , if n > 8.
Combining all the previous estimates, we conclude that

f ε (W δε(t),ξ ) -P g W δε(t),ξ L 2n n+4 (B ξ (r 0 )) =      δ ε (t) n-4 + εδ ε (t) n-4 2 , if n < 8, δ ε (t) 4 | ln δ ε (t)| 3 4 + εδ ε (t) 2 | ln δ ε (t)| 3 4 , if n = 8, δ ε (t) n 2 + εδ ε (t) 2 , if n > 8.
Finally, we give an estimate of the reduced energy Ĩε (t, ξ).

Proposition 4.2. Assume that M is l.c.f. and f ε (u) = |u| 2 * -2 u -εdiv g (B(∇u)). There exist constants c i (n), i = 1, 2, 3 depending on n, such that

Ĩε (t, ξ) = c 1 (n) + e -2t ε (c 2 (n)tT r g B(ξ) -c 3 A ξ ) + o(e -2t ε ), if n = 6, ε n-4 n-6 (c 2 (n)t 2 T r g B(ξ) -c 3 (n)t n-4 A ξ ) + o(ε n-4 n-6 ), if n ≥ 7
(4.10) as ε → 0 C 0 uniformly with respect to t in compact subsets of R * + and with respect to ξ ∈ M and C 1 uniformly if n ≥ 7. Moreover, we have that c 2 (n) > 0 and c 3 (n) > 0.

Proof. As previously, using Proposition ??, Lemma ?? and the definition of δ ε (t) (given in (??)), it is easy to see that

Ĩε (t, ξ) = Jε (W δε(t),ξ ) + o(δ ε (t) n-4 ). and ε B ξ (r 0 ) B(∇W δε(t),ξ , ∇W δε(t),ξ )dV = (n -4) 2 2n εδ ε (t) 2 α 2 n ω n-1 T r g (B) r 0 δε(t) 0 r n+1 (1 + r 2 ) n-2 dr + o(δ ε (t) n-4 ) = 1 2 (n -4) 2 2n εδ ε (t) 2 α 2 n ω n-1 T r g (B)        4n(n -1) (n -4)(n -6) , if n > 6, 1 2 ln(1 + 1 δ ε (t) 2 ), if n = 6 + o(δ ε (t) n-4 ).
Therefore, combining the three previous estimates, we finally obtain

Jε (W δε(t),ξ ) = 4 n α 2 * ω n + A ξ β n α 2 * n ω n-1 4(n -1)(n -2) δ ε (t) n-4 + 2(n -4) 2 A ξ β n α 2 n δ ε (t) n-4 ω n-1 2 × 2(n 2 -9n -18) (n -2)(n -3)(n -4) - (n + 2) (n -2) + (n -4) 2 8n εδ ε (t) 2 α 2 n ω n-1 T r g (B)    4n(n -1) (n -4)(n -6) , if n > 6, ln |δ ε (t)|, if = 6 + o(δ ε (t) n-4 ) =C 1 -C 2 A ξ δ ε (t) n-4 + C 3 T r g Bεδ ε (t) 2 + o(δ ε (t) n-4 ),
where C 3 > 0 and C 2 > 0.

The rest of the proof is devoted to prove that (??) holds C 1 -uniformly if n ≥ 7. This will be an immediate consequence of the following three lemmata. To simplify notation, in the following, we denote Z 0 = Z δ,ξ and Z i = Z δ,ξ,e i , where (e i ) i is a base of T ξ M (see (??) and (??) for the definitions of Z δ,ξ and Z δ,ξ,e i ).

Lemma 4.3. For any i = 1, . . . , n, we have

d dη i Ĩε (t, exp ξ η)| η=0 = C n λ i δ ε (t) ∆V i 2 L 2 + o n j=0 λ j , (4.11) and d dt Ĩε (t, ξ) = C n λ 0 δ ε (t) 4δ ε (t) ∆V 0 2 L 2 + o n j=0 λ j . (4.12)
where C n = α n (n -4) (see (??) for the definition of α n and see below for the definition of the λ i 's).

Proof. We only prove (??) (the proof of (??) following along the same line).

For any i = 1, . . . , n, using Proposition ??, we have that there exist λ i , i = 0, . .

d dη i W δε(t),ξ = O δ ε (t) n-2 .
Next, using (??) and (??), we obtain B ξ (r 0 ) P g W δε(t),ξ -f 0 (W δε(t),ξ )

d dη i W -C n 1 δ ε (t) Z i dV = O δ ε (t) n-2
r 0 0 r 2n-8 (δ ε (t) 2 + r 2 ) n-1 δ ε (t) 4 + δ ε (t) 2 r 2 + r 4 dr = O(δ ε (t) n-3 ),

  δε(t),exp ξ η + φ δε(t),exp ξ η )| η=0 .To simplify notation, we denoted dη i (W δε(t),exp ξ η )| η=0 = d dη iW (and we adopt the same convention for all functions). Integrating by parts, we have, for all i = 1, . . . , n and j = 0, . . . n,On the other hand, using that φ δε(t),ξ ∈ K ⊥ δε(t),ξ , Hölder inequality and Lemma ??, we deduce that O(δ ε (t) n-3 + εδ ε (t) 3 ln δ ε (t)), if n > 6. W δε(t),ξ -f ε (W δε(t),ξ ))(We also have, on ∂B ξ (r 0 ),∆ g W out δε(t),ξ ∂ in d dt W δε(t),ξ + ∆ g W in δε(t),ξ ∂ out d dt W δε(t),ξ = O(δ ε (t) n-3 δ ε (t) ),and∂ in (∆ g W out δε(t),ξ ) + ∂ out (∆ g W in δε(t),ξ ) d dt W δε(t),ξ = O(δ ε (t) n-3 δ ε (t) ).This proves (??). Now, let us show (??). Integrating by parts, we have∂ in (∆ g W out δε(t),ξ ) + ∂ out (∆ g W in δε(t),ξ )Proceeding exactly as in Lemma 6.1 of [?], we have for any y ∈ B 0 (r 0 ),d dη i W δε(t),exp ξ η (exp ξ y)| η=0 = C n δ ε (t) Z i (exp ξ y) + O |y|δ ε (t)For y ∈ M \B ξ (r 0 ) we haved dη i W δε(t),exp ξ η (exp ξ y)| η=0 = O δ ε (t)

	and
	. , n, such that Ĩε (t, exp ξ η)| η=0 = (W Z j , n j=0 λ j Z j , d dη i d dη i W = M P g Z j d dη i W -C n δ ε (t) Using Hölder's inequality and since d dη i d dη i W -M P g Z j d dη i W -C n δ ε (t) Z i dV ≤ P g Z j L 2n Z i dV + C n δ ε (t) Z i L 2 * C n δ ε (t) = o(1), we get Z i , Z j . n+4 d dη i W -C n δ ε (t) Z i L 2 * = o(1). Therefore, we deduce from the two previous lines that Z j , d dη i W = C n δ ε (t) ∆V i 2 L 2 δ ij + o(1). (4.13) d dη i Jε (W δε(t),exp ξ η )| η=0 = C n δ ε (t) D Jε (W δε(t),ξ )Z i + o(δ ε (t) 2 ), if n = 6, (4.16) Proof. Integrating by parts, we have d dt Jε (W δε(t),ξ ) -C n δ ε (t) 4δ ε (t) D Jε (W δε(t),ξ )Z 0 = d dt W δε(t),ξ -C n δ ε (t) 4δ ε (t) Z 0 dV -∂B ξ (r 0 ) (∆ g W out δε(t),ξ ∂ in d dt W δε(t),ξ + ∆ g W in δε(t),ξ ∂ out d dt W δε(t),ξ )dσ + ∂B ξ (r 0 ) Since d dt W δε(t),ξ -C n δ ε (t) 4δ ε (t) Z 0 = O(δ ε (t) n-6 2 δ ε (t) 1 M \B ξ ( r 0 2 ) ), using estimates obtained in Lemma ??, we find that d dt W δε(t),ξ -C n δ ε (t) 2δ ε (t) Z 0 )dV dη i Jε (W ) -C n 1 δ ε (t) D Jε (W δε(t),ξ )Z i = M (P g W δε(t),ξ -f ε (W δε(t),ξ ))( d dη i W -C n 1 δ ε (t) Z i )dV -∂B ξ (r 0 ) (∆ g W out δε(t),ξ ∂ in d dη i W + ∆ g W in δε(t),ξ ∂ out d dη i W )dσ + ∂B ξ (r 0 ) d dη i W dσ. n-4 2 (δ ε (t) 2 + |y| 2 ) n-4 2 . (4.17) n-4 2 , (4.18) here we assumed to simplify computations that the function γ ε ≡ 0, it is easy to check that this assumption is harmless since n ≥ 6. Using (??), one can show that = O δ d ∆ g W out δε(t),ξ ∂ in d dη i W + ∆ g W in δε(t),ξ ∂ out d dη i W = O δ
	Z j ,	d dη i	φ ≤	d dη i	Z j	Pg	φ δε(t),ξ Pg = O	φ δε(t),ξ Pg δ ε (t)	= o(1). (4.14)
	Combining (??) and (??), we get (??).
	Lemma 4.4. We have
	d dt	Jε (W δε(t),ξ ) = C n	δ ε (t) 4δ ε (t)	D Jε (W δε(t),ξ )Z 0
								+	o(δ ε (t) 2 ln δ ε (t)), O(δ ε (t) n-2 + εδ ε (t) n-4 ), if n > 6, if n = 6,
										(4.15)

M P g W δε(t),ξ -f ε (W δε(t),ξ ) ∂ in (∆ g W out δε(t),ξ ) + ∂ out (∆ g W in δε(t),ξ ) d dt W δε(t),ξ dσ. M (P g ε (t) (δ ε (t) n-3 + εδ ε (t) n-5 ) , if n ≥ 6. ε (t) n-2 ,

and

∂ in (∆ g W out δε(t),ξ ) + ∂ out (∆ g W in δε(t),ξ )

The second author was supported by the CNPq (Brazil) project 501559/2012-4.

Using (??) and (??), we have M (∆ g W δε(t),ξ ) 2 dV + M A g (∇W δε(t),ξ , ∇W δε(t),ξ )dV + P g (W δε(t),ξ )W δε(t),ξ dV + O(δ ε (t) n-2 ).

Here we also used (??), (??) and W δε(t),ξ = O(δ ε (t)

n-4

2 ) on ∂B ξ (r 0 ). Now using (??), we find

Integrating the previous formula, we find

We also have

Finally, using (??), we get

Combining the previous estimates, we obtain (??).

Lemma 4.5. For any i = 0, . . . , n, we have

In particular, we have, for all i = 0, . . . , n and n ≥ 7,

(4.20)

Proof. For any i = 0, . . . , n, we have

Independently, we obtain that

(4.22) Using Hölder inequality, we get

and, as in (??),

when 6 ≤ n.

(4.23) To conclude, we only have to estimate Z i -i * (f ε (W δε(t),ξ )Z i ) Pg . First, by Sobolev's embedding, we have

Straight forward computations using that M is l.c.f. and P g is conformally invariant give

and, for i = 1, . . . , n,

Using the two previous estimates and that

+ εδ ε (t) 2 , if i = 0, n ≥ 7.

(4.24)

Combining (??), (??), (??), (??) and using Lemma ??, we obtain (??). Finally (??) follows from the fact that the estimates we obtain for Jε (W δε(t),ξ ) in Lemma ??, are C 1 -uniform with respect to t in compact sets of R * + and ξ ∈ M .