Prescribing Q-curvature on higher dimensional spheres

We study the problem of prescribing a fourth order conformal invariant on higher dimensional spheres. Particular attention is paid to the blow-up points, i.e. the critical points at infinity of the corresponding variational problem. Using topological tools and a careful analysis of the gradient flow lines in the neighborhood of such critical points at infinity, we prove some existence results. 2000 Mathematics Subject Classification: 35J60, 53C21, 58J05.


Introduction
Let (M, g) be a smooth Riemannian manifold of dimension n ≥ 4, with scalar curvature R g and Ricci curvature Ric g .In 1983, Paneitz [30] introduced in dimension four the following fourth order operator where div g denotes the divergence and d the de Rham differential operator.This operator enjoys the analogous covariance property as the Laplacian in dimension two: under conformal change of metric g = e 2u g we have P 4  g = e −4u P 4 g .
In [11], Branson generalized the Paneitz operator to n-dimensional Riemannian manifolds, n ≥ 5.Such an operator is related to the Paneitz operator in dimension four in the same way the conformal Laplacian is related to the Laplacian in dimension two and is defined as: where Under the conformal change of metric g = u 4/(n−4) g, the conformal Paneitz operator enjoys the covariance property: and the closely related fourth order curvature invariant Q n g , called Q-curvature, satisfies 4) on M. (1.1) For more details about the properties of the Paneitz operator, see for example [12], [13], [15], [16], [18], [17], [19], [21], [26], [33].
A problem naturally arises when looking at equation (1.1): the problem of prescribing the Q-curvature, that is, given a smooth function f : M → R, does there exist a metric g conformally equivalent to g such that Q n g = f ?From equation (1.1), the problem is equivalent to finding a smooth solution u of the equation 4) , u > 0 on M. (1. 2) The requirement about the positivity of u is necessary for the metric g to be Riemannian.Problem (1.2) is the analogue of the classical scalar curvature problem to which a wide range of activity has been devoted in the last decades (see for example the monograph [1] and references therein).On the other hand, to the author's knowledge, problem (1.2) has been studied in [8], [9], [15], [22], [23] [24], [25], [33], [32] only.
In this paper, we are interested in the case where a noncompact group of conformal transformations acts on the equation so that Kazdan-Warner type conditions give rise to obstructions, as in the scalar curvature problem, see [21] and [32].The situation is the following: let (S n , g) be the standard sphere, n ≥ 5, endowed with its standard metric.In this case our problem is equivalent to finding a solution u of the equation where c n = 1 2 (n 2 −2n−4), d n = n−4 16 n(n 2 −4) and where K is a given function defined on S n .
Our aim is to give sufficient conditions on K such that problem (1.3) admits a solution.Our approach uses dynamical and topological methods involving the study of critical points at infinity of the associated variational problem, see Bahri [3].Precisely, we extend the topological tools introduced by Bahri [4] to the framework of such higher order equations.Our method relies on the use of the invariant introduced by Bahri [4], which we extend to prove some existence results for problem (1.3).The main idea is to use the difference of topology between the level sets of the function K to create a critical point of the Euler functional J associated to (1.3) and the main issue is under our conditions on K, a topological accident between the level sets of K induces a topological accident between the level sets of J.Such an accident is sufficient to prove the existence of a critical point of J.This then implies the existence of solution (1.3) in our statements.To state our main results, we need to introduce the assumptions that we will use and some notations.(A 1 ) We assume that K is a positive C 3 -function on S n and which has only nondegenerate critical points y 0 , y 1 , ..., y s with K(y 0 ) = max K, −∆K(y 0 ) > 0; −∆K(y 1 ) > 0; −∆K(y i ) < 0 for i ≥ 2 and index(K, y 1 ) = n.
Let Z be a pseudo gradient of K of Morse-Smale type, that is, the intersections of the unstable and stable manifolds of the critical points of K are transverse.We denote by (n − k) the Morse index of y 1 and we set where W s (y 1 ) is the stable manifold of y 1 for Z.Let us define where δ x denotes the Dirac mass at x.For a ∈ S n and λ > 0, let , which is a solution of the problem (see [27]) We notice that problem (1.3) has a variational structure.The corresponding functional is defined on the unit sphere Σ of H 2 2 (S n ) equipped with the norm: We set Σ + = {u ∈ Σ | u > 0} and for λ large enough, we introduce a map f λ : B 2 (X) → Σ + , defined by Then, B 2 (X) and f λ (B 2 (X)) are manifolds in dimension 2k + 1, that is, their singularities arise in dimension 2k−1 and lower, see [4].Recall that k satisfies k = n − index(K, y 1 ) and therefore the dimension of X is equal to k.Let ν + be a tubular neighborhood of X in S n .We denote by ν + (y), for y ∈ X, the fibre at y of this tubular neighborhood.For ε 1 > 0, z 1 , z 2 ∈ X such that z 1 = z 2 and −∆K(z i ) > 0 for i = 1, 2, we introduce the following set where v is defined in Lemma 2.3 (see below) and where (V 0 ) is the following conditions: (V 0 ) : v, ϕ i P = 0 for i = 1, 2 and every (1.6) for some system of coordinates (a i ) 1 , ..., (a i ) n on S n near (1.7) We also assume that (A 2 ) z 1 and z 2 are distinct of y 0 , or if one is y 0 , the other one is y 1 .
For λ large enough, we define the intersection number (modulo 2) of where Therefore, the number τ (z 1 , z 2 ) is well defined (see [29]).Our main result is the following.
The aim of the next result is to give some conditions on the function K which allow us to have τ (z 1 , z 2 ) = 1 for some couple (z 1 , z 2 ) and thus, we obtain a solution for (1.3) by Theorem 1.1.Let z 1 , z 2 ∈ X be such that −∆K(z i ) > 0. We choose ν + (z i ) such that K(z i ) = max ν + (z i ) K and z i is the unique critical point of K on ν + (z i ).
Theorem 1.2: Let n ≥ 9.There exist positive constants C 0 , C 1 such that, if, for two points z 1 and z 2 of X, the following conditions hold: 2. For some positive constant ρ 0 , for each i = 1, 2, and for each (a 1 , a 2 )

inf
Remark 1.3: i) For more details regarding the assumption n ≥ 9, see Remark 2.6.
ii) To see how to construct an example of a function K satisfying our assumptions, we refer the interested reader to [2] and [20].
The rest of the present paper is organized as follows.In Section 2, we recall some preliminaries, introduce some definitions and the notations needed in the proof of our results.In Section 3, we characterize the critical points at infinity.Then, we prove our results in Section 4. Lastly, in the Appendix we perform an expansion of the Euler functional associated to (1.3) and its gradient near the potential critical points at infinity.

Preliminaries
Solutions of problem (1.3) correspond, up to some positive constant, to critical points of the following functional defined on the unit sphere of H 2 2 (S n ) by The exponent 2n/(n − 4) is critical for the Sobolev embedding H 2 2 (S n ) → L q (S n ).As this embedding is not compact, the functional J does not satisfy the Palais-Smale condition and therefore standard variational methods cannot be applied to find critical points of J.In order to describe the sequences failing the Palais-Smale condition, we need to introduce some notations.For p ∈ N * and ε > 0, we set Let w be a nondegenerate solution of (1.3).We also set and |α 0 J(u) n/8 − 1| < ε The failure of the Palais-Smale condition can be described, following the ideas introduced in [14], [28], [31], as follows: Proposition 2.1: Let (u j ) ∈ Σ + be a sequence such that ∇J(u j ) tends to zero and J(u j ) is bounded.Then, there exist an integer p ∈ N * , a sequence ε j > 0, ε j tends to zero, and an extracted sequence of u j 's, again denoted u j , such that u j ∈ V (p, ε j , w) where w is zero or a solution of (1.3).
The following lemma defines a parametrization of the set V (p, ε).It follows from the corresponding statements in [4] and [5].

Lemma 2.3:[8]
Assuming the ε ij 's are small enough and J(u) where ε is a fixed small positive constant depending only on p.Moreover, we have the following estimate ) .
Note that Lemma 2.2 extends to the more general situation where the sequence (u j ) of Σ + , described in Proposition 2.1, has a nonzero weak limit, a situation which might occur if K is the Q-curvature (up to a positive constant) of a metric conformal to the standard metric g.Notice that such a weak limit is a solution of (1.3).Denoting by w a nondegenerate solution of (1.3), we then have the following lemma which follows from the corresponding statement in [4].

., p and every
Now, following Bahri [4], we introduce the following definitions and notations.

Definition 2.5:
A critical point at infinity of J on Σ + is a limit of a flowline u(s) of equation ∂u ∂s = −∇J(u) with initial data u 0 ∈ Σ + such that u(s) remains in V (p, ε(s), w) for large s.Here w is zero or a solution of (1.3), p ∈ N * , and ε(s) is some function such that ε(s) tends to zero when the flow parameter s tends to +∞.By Lemma 2.4, we can write such u(s) as Denoting a i = lim s→+∞ a i (s), we call (a 1 , ..., a p , w) ∞ a critical point at infinity of J.If w = 0, (a 1 , ..., a p , w) ∞ is called a mixed type of critical points at infinity of J.
Remark 2.6: Notice that for n ≥ 9 any configuration containing a solution w of (1.3) and a collection of critical points y i of K having −∆K(y i ) > 0 gives rise to a critical point at infinity of J.This is not true for n ≤ 7.In dimension 8, we have a balance phenomenon; that is, the self-interaction of the functions failing the Palais-Smale condition and the interaction of one of those functions with the solution w are of the same size.
In the sequel, we denote by A the set of w such that w is a critical point or a critical point at infinity of J in Σ + not containing y 0 in its description.We also denote by A q the subset of A such that the Morse index of the critical point (at infinity) is equal to q. Definition 2.7: (A family of pseudogradients F) A decreasing pseudogradient V for J is said to belong to F if the following properties hold: -the set of critical points at infinity of J on Σ + does not change if we take V instead of −∇J in the definition 2.5, -V is transverse to f λ (B 2 (X)), -for any w ∈ A, (y 0 , w) ∞ is a critical point at infinity with the following property: Here and below i(ϕ 1 , ϕ 2 ) denotes the intersection number for V of ϕ 1 and ϕ 2 (see [29] and [4]) where ϕ i is any critical point or a critical point at infinity of J. Definition 2.8: Given a decreasing pseudogradient V for J.We denote by ϕ(s, .) the associated flow.A critical point at infinity z ∞ is said to be dominated by Near the critical points at infinity, a Morse Lemma can be completed (see Proposition 3.4 and (3.11) below) so that the usual Morse theory can be extended and the intersection can be assumed to be transverse.Thus the above condition is equivalent to (see Proposition 7.24 and Theorem 8.2 of [6]) Definition 2.9: z ∞ is said to be dominated by another critical point at infinity If we assume that the intersection is transverse, then Given w 2k+1 ∈ A 2k+1 and V ∈ F, we denote by the intersection number (modulo 2) of W u ((y 0 , w 2k+1 ) ∞ ) and C δ .
In order to compute this intersection number, one can perturb V (not necessarily in F) so as to bring W u ((y 0 , w 2k+1 ) ∞ ) ∩ C δ to be transverse.This number is the same for all such small perturbations (just as in degree theory).Notice that the dimension of W u ((y 0 , w 2k+1 ) ∞ ) is equal to 2k +2 and the codimension of C δ is 2k + 2. Then (y 0 , w 2k+1 ) ∞ .C δ is also well defined, because the closure of W u ((y 0 , w 2k+1 ) ∞ ) only adds to W u ((y 0 , w 2k+1 ) ∞ ) the unstable manifolds of critical points of index less than or equal to 2k + 1.These manifolds are then of dimension 2k +1 at most.Since the codimension of C δ is equal to 2k + 2, these manifolds can be assumed to avoid C δ .Now, for w 2k+1 ∈ A 2k+1 and V ∈ F, we denote by the intersection number of f λ (B 2 (X)) and W s (w 2k+1 ).We notice that the dimension of f λ (B 2 (X)) is equal to 2k + 1 and the codimension of W s (w 2k+1 ) is equal to 2k +1.Then, the intersection number, defined in (2.2) is well defined because V is transverse to f λ (B 2 (X)) outside f λ (B 1 (X)), which cannot dominate critical points of index 2k + 1.Furthermore, W s (w 2k+1 ) adds to W s (w 2k+1 ) stable manifolds of critical points of an index larger than or equal to 2k + 2. Since f λ (B 2 (X)) is of dimension 2k + 1, these manifolds can be assumed to avoid it.Lastly, we set for each V ∈ F Notice that 2.3 was introduced by Bahri in [4] where he proved that I(V ) is independent on V ∈ F. Namely, he showed in [4] that I(V ) = 0, for each V ∈ F for the scalar curvature problem on S n with n ≥ 7. We will prove that the same holds for the Q-curvature equation when n ≥ 9.

Characterization of the critical points at infinity
In this section, we provide the characterization of the critical points at infinity.First, we construct a special pseudogradient for the associated variational problem for which the Palais-Smale condition is satisfied along the decreasing flow lines, as long as these flow lines do not enter in the neighborhood of critical points y i of K such that −∆K(y i ) > 0. As a by product of the construction of such a pseudogradient, we are able to determine the critical points at infinity for our problem.
Proposition 3.1: For p ≥ 2, there exists a pseudogradient W so that the following holds.
There is a constant c > 0 for each i and the only case where the maximum of the λ i 's increases along W is when each point a i is close to a critical point y j i of K with −∆K(y j i ) > 0 and j i = j r for i = r.
Proof.We order the λ i 's, for the sake of simplicity we can assume that: where C 1 and M are two positive large constants.Set Using Proposition 5.2, we derive that Observe that, if j ∈ I 2 then Using also the fact that i ∈ I 1 , thus, (3.1) becomes Now, we will distinguish two cases.case 1 I 1 ∩ I 2 = ∅.In this case, we define where M 1 is a large constant and m 1 is a small constant.Using Proposition 5.2, we derive Now, we define 3) and (3.4), we derive that

K. El Mehdi
Observe that, since I 1 ∩ I 2 = ∅, we can make 1/λ 2 k appear, for k ∈ I 2 , in the lower bound of (3.5) and therefore all the λ −2 i 's can appear in the lower bound of (3.5).Notice that for i / ∈ I 1 , we have In this case, for each i ∈ I 2 , the point a i is close to a critical point y k i of K. We claim that k i = k j for i = j that is each neighborhood B(y, ρ), for ρ small enough, contains at most one point a i with i ∈ I 2 .Indeed, arguing by contradiction, let us suppose that there exist i, j ∈ I 2 such that a i , a j ∈ B(y, ρ).Since y is nondegenerate we derive that |∇K(a k )| ≥ c|y − a k | for k = i, j and therefore (we assume that , a contradiction with the fact that λ i and λ j are of the same order.Thus our claim follows.
Let us introduce 1st subcase I 3 = ∅.In this case we define Using Proposition 5.2 we derive Observe that, if i, j ∈ I 2 , we have For Z 5 = Z 4 + Z 1 , using (3.3), (3.7), (3.8) and choosing M 1 ≤ M , we obtain 2nd subcase I 3 = ∅.In this case we define Using Proposition 5.2, as in the above subcase, we derive that The vector field W will be a convex combination of all Z 3 , Z 5 and Z 6 .Thus the proof of claim (a) is completed.From the definition, W is bounded and we have |dλ i (W )| ≤ cλ i for each i.Observe that, the only case where the maximum of the λ i 's increases is when I 2 = {1, ..., p} and I 1 = I 3 = ∅, it means each a i is close to a critical point y j i of K with j i = j r for i = r and −∆K(y j i ) > 0 for each i.Hence claim (c) follows.Finally, arguing as in Appendix B of [7], claim (b) follows from claim (a) and Lemma 2. (y 0 ) ∞ , (y 1 ) ∞ and (y 0 , y 1 ) ∞ .

Proof. Using Proposition 2.1, we derive that
where c is a positive constant which depends only on ε.It only remains to see what happens in ∪ p≥1 V (p, ε).From Proposition 3.1, we know that the only region where the maximum of the λ i 's increases along the pseudogradient W , defined in Proposition 3.1, is the region where each a i is close to a critical point y j i of K with −∆K(y j i ) > 0 and j i = j r for i = r.In this region, arguing as in [4], we can find a change of variables: (a 1 , ...a p , λ 1 , ..., λ p ) −→ (ã 1 , ..., ãp , λ1 , ..., λp ) := (ã, λ) such that ∆K(y j i ) where η is a small positive constant and c = c 2 (n − 4)/n K(y , where c 2 is defined in Proposition 5.1.This yields a split of variables ã and λ.Thus it is easy to see that if the α i 's are in their maximum and a i = y j i for each i, only the λ i 's can move.To decrease the functional J, we have to increase the λ i 's, thus we obtain a critical point at infinity only in this region.It remains to compute the Morse index of such critical points at infinity.For this purpose, we observe that −∆K(y j i ) > 0 for each i and the function Ψ admits in the variables α i 's an absolute degenerate maximum with one dimensional nullity space and an absolute minimum in the variable v. Then the Morse index of such critical point at infinity is equal to (p − 1 + p i=1 (n − index(K, y j i ))).Thus our result follows. 2 In Proposition 3.2, we have assumed that J has no critical point in Σ + .When such an assumption is removed, new critical points at infinity of J appear.Indeed, we have the following result: Proposition 3.3: Let n ≥ 9. Let w be a nondegenerate solution of (1).Then, (y 0 , w) ∞ , (y 1 , w) ∞ and (y 0 , y 1 , w) ∞ are critical points at infinity.The Morse index of these critical points are respectively equal to index(w) + 1, index(w) + index((y 1 ) ∞ ) + 1 and index(w) + index((y 1 ) ∞ ) + 2.
The proof of this proposition immediately follows from Proposition 3.2 and the following result:

Proposition 3.4:
There is an optimal (v, h) and a change of variables v − v → V and h − h → H such that J reads as Furthermore, we have the following estimates: ) .
Before giving the proof of Proposition 3.4, we need to prove the following lemma: Lemma 3.5: The following Claims hold true: , and v satisfies (W 0 )}.
Proof.Claim (b) follows immediately, since h ∈ T w (W u (w)).Next we are going to prove claim (a).We split T w (W s (w)) into E γ ⊕ F γ where E γ and F γ are orthogonal for , P and as well as for the quadratic form associated to w and such that We choose γ small enough such that 0 < γ < ᾱ/4, where ᾱ is the first eigenvalue This implies that It remains to study the term Observe that v is orthogonal to span{ δi , λ i ∂ δi (3.13) Now, we write with v2 ∈ span{ δi , ∂ δi ∂λ i , ∂ δi ∂(a i ) j , i ≤ p, j ≤ n} ⊥ .Thus, we have (see [8]) Notice that Using (3.12)-(3.14),we obtain Thus, using (3.15), we derive that But Thus Since γ < ᾱ/4, claim (a) follows.The proof of our lemma is thereby completed.2 Proof of Proposition 3.4 By Proposition 5.1 the expansion of J with respect to h (respectively to v) is very close, up to a multiplicative constant, to . By Lemma 3.5 there is a unique maximum h in the space of h (respectively a unique minimum v in the space of v).Furthermore, it is easy to derive that ||h|| ≤ c||f 2 || = O( i λ ) and ||v|| ≤ c||f 1 ||.The estimate of v follows from Lemma 2.3.Then our result follows. 2 Let us start by proving the following results.
Proof.An abstract topological argument displayed in [4], pages 358-369, which extends to our framework, shows that the value of I(V ) is constant for any V ∈ F. Now, let ε > 0 and K ε = 1 + εK.Let J ε be the associated variational problem.As ε tends to zero, J ε tends to J 0 in the C 1 sense, where J 0 is the functional defined replacing K by 1 in (1.5).On the other hand, using Proposition 5.1, we see that where c is independent of ε and 2S 4/n is the level to which a critical point at infinity of 2 masses of K ε converges when ε → 0. Thus, we can assume ε is so small that all critical points at infinity of J ε (of two masses or more) are above f λ (B 2 (X)).Clearly, for ε small, C δ (z 1 , z 2 ) is above (2S 4/n + δ/2).We derive that Notice that, decreasing λ, we complete an homotopy of f λ (B 2 (X)) that increases the interaction of any masses, and therefore remains below C δ (z 1 , z 2 ).This implies that for each µ ∈ [1, λ] we have Recall that Thus, we need to compute f λ (B 2 (X)).w2k+1 for any w 2k+1 ∈ A 2k+1 .Let We can assume that F is a compact manifold in dimension 2k + 2. The singularity of F is ∪ λ µ=1 f µ (B 1 (X)) which is of a dimension less than (k + 1), this singularity cannot dominate w 2k+1 .We deduce that F ∩ Ws (w 2k+1 ) is a compact manifold of dimension one.Thus the cardinal of ∂(F ∩ Ws (w 2k+1 )) is equal to zero, where ∂ is the boundary homomorphism of S 2k+2 (Σ + ).Observe that It follows that Along this homotopy, the trace of f µ (B 2 (X)) might intersect, for some values, ∂ −1 (W s (w 2k+1 )), where ∂ −1 (W s (w 2k+1 )) is made of stable manifolds of critical points of index 2k + 2. Therefore the abstract argument of [4] (see pages 358-369) applies, and the invariant remains unchanged.For µ = 1 at the end of the homotopy B 2 (X) is mapped onto a single function and (f 1 (B 2 (X)).w2k+1 ) is therefore zero.Thus, I(V ) at the end of the homotopy is equal to zero, and the results follow. 2 Now, we are going to prove Theorem 1.1.Proof of Theorem 1.1 Arguing by contradiction, we assume that J has no critical point in Σ + .It follows from Proposition 3.2 that A 2k+1 = ∅.Therefore combining (4.1), Proposition 4.1 and the fact that τ = 1, we derive a contradiction.The proof of our result is thereby completed. 2 The sequel of this section is devoted to the proof of Theorem 1.2.Proof of Theorem 1.2 In the sequel, we denote by Π a the stereographic projection through a point a ∈ S n .This projection induces an isometry i : H 2 (S n ) → H(R n ) according to the following formula where Let a 1 , a 2 in S n and ρ 1 , ρ 2 be two positive constants (we choose ρ 1 and where v satisfies (V 0 ) which is defined in (1.6).
We now write down the expansion of J(u) = N/D with where R K,i satisfies Now, assuming λ i and λ i ρ i are large, we write Thus (4.5) , where β = S n 2 i=1 1/K(a i ) (n−4)/4 and where Notice that On the other hand, we know from Proposition 3.4 of [8] that the quadratic form is bounded below by α 0 ||v|| 2 , α 0 is a fixed constant, on all v's satisfying (V 0 ).now that Thus, if we assume that is small, then the quadratic form which comes out of the expansion is positive definite, bounded below by (α 0 /4)||v|| 2 for v satisfying (V 0 ).Therefore the functional has a unique minimum ṽ and we have ||ṽ|| = O(||f ||).
The function J(u) has in fact one more term depending on v which is . (4.12) J is twice differentiable.Therefore, this remainder term is also twice differentiable and its second differential is easily checked to be sup Thus, if we assume that (sup K)O(||f || 8/(n−4) ) ≤ c (for n ≥ 12) and (for n < 12) sup K(K(a 1 ) (n−12)/8 + K(a 2 ) (n−12)/8 )O(||f ||) ≤ c where c is a small constant, the functional will have a unique minimum v near the origin and it satisfies also ||v|| = O(||f ||).Let us introduce the following neighborhood V of functions v ∈ H 2 (S n ) such that v satisfies (V 0 ) and Requiring v to belong to V , we let by u = (1/K(a i ) (n−4)/8 ) δi + v. Then where Q is a positive definite form, bounded below by (α 0 /4)||v − v|| 2 on V .An expansion of J(u) is easily derived by setting v = v in the expansion of J(u) (see (4.5)) and using the estimate of v. Thus, As in Proposition 5.2 and in Appendix B of [7], we obtain Thus for β 1 , β 2 ≥ 0, β 1 + β 2 = 1 and using the estimate of ||f || (see (4.6)), we derive (4.17) This derivative will remain negative as long as, for a suitable universal constant c 1 , we have for i Taking c 1 to be smaller, if necessary, we derive that, under (4.18) and if v ∈ V , J(u) is bounded below as follows: To (4.18), other conditions which we used earlier are to be added, namely Finally, all the quantities involved in (4.18), up to the factor 1/β, should be small for the expansions to hold, which amounts to We will take ) small enough so that K(z 1 ) ≤ K(a 1 ) ≤ 2K(y 1 ).(4.24)We will ask that a 2 ∈ ν + (z 2 ), ν + (z 2 ) be small enough so that K(z 2 ) ≤ K(a 2 ) ≤ 2K(y 1 ).
(4.25) and that The third condition of (4.27) follows from the first one, since |D 2 K(a i )| dominates |∆K(a i )| (up to a modification of c 1 ).Thus (4.28)At this point, following the proof of [4], we explain how we will proceed with the proof of Theorem 1.2.We wish to compute Let us define g λ and f λ are homotopic (see [4]).Using also the fact that −∆K(z 1 ) and −∆K(z 2 ) are positive, we can choose δ so small that We can accordingly modify C δ (z 1 , z 2 ) as follows: where Clearly, C δ (z 1 , z 2 ) and Cδ (z 1 , z 2 ) can be deformed, one into another, using an isotopy above the level c ∞ (z 1 , z 2 ).Thus Computing τ (z 1 , z 2 ) now becomes a matter of defining a pseudogradient such that the Palais-Smale condition ((P-S) for short) is satisfied along decreasing flow lines away from the critical points at infinity and computing τ (z 1 , z 2 ) for

Appendix: expansion of the functional and its gradient
This appendix is devoted to a useful expansion of J and its gradient near a critical point at infinity.In order to simplify the notations, in the remainder we write δi instead of δ(a i ,λ i ) .First, we prove the following result: Using the fact that h belongs to the tangent space at w, we derive that ( Since v ∈ T w (W s (w)) and h ∈ T w (W u (w)), the linear form on v can be written as

3 . 2 Proposition 3 . 2 :
Let n ≥ 9. Assume that J has no critical point in Σ + .Under the assumptions (A 1 ) and (A 2 ), the only critical points at infinity under the level c ∞ (y 1 , y 1 ) are: