Prescribing <span class="mathjax-formula">$Q$</span>-curvature on higher dimensional spheres

BLAISE PASCAL

Khalil El Mehdi

Prescribing Q-curvature on higher dimensional spheres

Volume 12, n^o2 (2005), p. 259-295.

<http://ambp.cedram.org/item?id=AMBP_2005__12_2_259_0>

©Annales mathématiques Blaise Pascal, 2005, tous droits réservés.

L’accès aux articles de la revue « Annales mathématiques Blaise Pascal » (http://ambp.cedram.org/), implique l’accord avec les conditions générales d’utilisation (http://ambp.cedram.org/legal/). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copy- right.

Publication éditée par le laboratoire de mathématiques de l’université Blaise-Pascal, UMR 6620 du CNRS

Clermont-Ferrand — France

cedram

Article mis en ligne dans le cadre du

Centre de diffusion des revues académiques de mathématiques

Prescribing Q-curvature on higher dimensional spheres

Khalil El Mehdi

Abstract

We study the problem of prescribing a fourth order conformal in- variant on higher dimensional spheres. Particular attention is paid to the blow-up points, i.e. the critical points at infinity of the corre- sponding 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.

Key words: Variational problems, lack of compactness,Qcurvature, crit- ical points at infinity .

1 Introduction

Let(M, g)be a smooth Riemannian manifold of dimensionn≥4, with scalar curvature R_g and Ricci curvature Ric_g. In 1983, Paneitz [30] introduced in dimension four the following fourth order operator

P_g⁴= ∆²_g− div_g(2

3R_g−2Ric_g)◦d,

wherediv_g denotes the divergence and dthe de Rham differential operator.

This operator enjoys the analogous covariance property as the Laplacian in dimension two: under conformal change of metricg˜=e^2ugwe have

P_˜g⁴=e^−4uP_g⁴.

In [11], Branson generalized the Paneitz operator to n-dimensional Rie- mannian manifolds,n≥5. Such an operator is related to the Paneitz opera- tor in dimension four in the same way the conformal Laplacian is related to the Laplacian in dimension two and is defined as:

P_gⁿ= ∆²_g−div_g(a_nS_gg+b_nRic_g)◦d+n−4 2 Qⁿ_g,

where

a_n= (n−2)²+ 4

2(n−1)(n−2), b_n= −4 n−2 Qⁿ_g =− 1

2(n−1)∆_gS_g+n³−4n²+ 16n−16

8(n−1)²(n−2)² S_g²− 2

(n−2)²|Ric_g|². Under the conformal change of metric ˜g= u^4/(n−4)g, the conformal Paneitz operator enjoys the covariance property:

P_gⁿ(uϕ) =u(n+4)/(n−4)P_g˜ⁿ(ϕ) for allϕ∈C^∞(M), and the closely related fourth order curvature invariantQⁿ_g, called Q-curvature, satisfies

P_gⁿ(u) = n−4

2 Qⁿ_˜gu(n+4)/(n−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 theQ-curvature, that is, given a smooth functionf :M →R, does there exist a metric g˜conformally equivalent to g such that Qⁿ_g˜ = f ? From equation (1.1), the problem is equivalent to finding a smooth solution uof the equation

P_gⁿ(u) = n−4

2 f u(n+4)/(n−4)

, u >0 on M. (1.2)

The requirement about the positivity ofuis necessary for the metricg˜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(Sⁿ, g) be the standard sphere, n ≥5, endowed with its standard metric. In this case our problem is equivalent to finding a solutionuof the equation

Pu:= ∆²u−c_n∆u+d_nu=Kuⁿ⁻⁴ⁿ⁺⁴, u >0 onSⁿ, (1.3)

wherec_n= ¹₂(n²−2n−4),d_n = ⁿ⁻⁴₁₆ n(n²−4)and whereK is a given function defined onSⁿ.

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 functionK to create a critical point of the Euler functional J associated to (1.3) and the main issue is under our conditions onK, 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.

(A₁) We assume thatK is a positiveC³-function onSⁿand which has only nondegenerate critical pointsy₀, y₁, ..., y_s with

(K(y₀) = maxK,−∆K(y₀)>0; −∆K(y₁)>0;

−∆K(y_i)<0fori≥2 and index(K, y₁)6=n.

Let Z be a pseudo gradient of K of Morse-Smale type, that is, the inter- sections of the unstable and stable manifolds of the critical points ofK are transverse. We denote by(n−k)the Morse index ofy₁and we set

X =W_s(y₁), (1.4)

whereW_s(y₁) is the stable manifold ofy₁ forZ. Let us define B₂(X) ={α₁δ_x1+α₂δ_x2/α_i≥0, α₁+α₂= 1, x_i∈X}, whereδ_x denotes the Dirac mass atx. For a∈Sⁿ and λ >0, let

eδ_(a,λ)(x) = β_n 2ⁿ⁻⁴²

λⁿ⁻⁴²

1 +^λ2₂⁻¹(1−cosd(x, a))ⁿ⁻⁴₂ ,

where d is the geodesic distance on(Sⁿ, g) and β_n = [(n−4)(n−2)n(n+ 2)]^(n−4)/8. After performing a stereographic projectionΠwith the point−a as pole, the functionδe_(a,λ) is transformed into

δ_(0,λ)=β_n λⁿ⁻⁴² (1 +λ²|y|²)ⁿ⁻⁴² , which is a solution of the problem (see [27])

∆²u=uⁿ⁺⁴ⁿ⁻⁴, u >0 on Rⁿ.

We notice that problem (1.3) has a variational structure. The corresponding functional is

J(u) = Z

Sⁿ

K|u|^2n/(n−4)

(4−n)/n

(1.5) defined on the unit sphereΣ ofH₂²(Sⁿ)equipped with the norm:

||u||²=hu, ui_P = Z

Sⁿ

Pu·u= Z

Sⁿ

|∆u|²+c_n Z

Sⁿ

| ∇u|²+d_n Z

Sⁿ

u². We set Σ⁺ = {u ∈Σ | u >0} and for λ large enough, we introduce a map f_λ:B₂(X)→Σ⁺, defined by

(α₁δ_x1+α₂δ_x2)−→ α₁δ˜_(x1,λ)+α₂˜δ_(x2,λ)

||α₁δ˜_(x1,λ)+α₂˜δ_(x2,λ)||.

Then,B₂(X)andf_λ(B₂(X))are manifolds in dimension2k+ 1, that is, their singularities arise in dimension2k−1and lower, see [4]. Recall thatksatisfies k=n−index(K, y₁)and therefore the dimension ofX is equal tok.

Let ν⁺ be a tubular neighborhood of X in Sⁿ. We denote by ν⁺(y), for y ∈X, the fibre at y of this tubular neighborhood. For ε₁ > 0, z₁, z₂ ∈X such thatz₁6=z₂ and −∆K(z_i)>0 fori= 1,2, we introduce the following set

Γ_ε1 = ²

X

i=1

δ˜_(zi+hi,λi)

K(z_i+h_i)ⁿ⁻⁴⁸ +v|v∈H₂²(Sⁿ) satisfies(V₀),

||v−v||< ε₁, λ_i> ε⁻¹₁ fori= 1,2, h_i∈ν⁺(z_i), |h₁|²+|h₂|²< ε₁

, wherev is defined in Lemma 2.3 (see below) and where(V₀)is the following

conditions:

(V₀) : hv, ϕ_ii_P = 0 for i= 1,2 and every (1.6) ϕ_i =δe_(ai,λi), ∂eδ_(ai,λi)/∂λ_i, ∂eδ_(ai,λi)/(∂a_i)_j, j = 1, ..., n,

for some system of coordinates (a_i)₁, ...,(a_i)_n onSⁿ near (1.7) a_i:=z_i+h_i.

We also assume that

(A₂) z₁and z₂are distinct of y₀, or if one isy₀, the other one isy₁.

Forδ >0small, the boundary ofΓ_ε1 (defined by ||v−v||=ε₁, orλ₁=ε⁻¹₁ , orλ₂= ε⁻¹₁ , or |h₁|²+ | h₂ |²= ε₁) does not intersect J⁻¹(c_∞(z₁, z₂) +δ), where

c∞(z₁, z₂) = S_n

2

X

i=1

1 K(z_i)^(n−4)/4

!4/n

. (1.8)

We then set

C_δ :=C_δ(z₁, z₂) = Γ_ε1∩J⁻¹(c_∞(z₁, z₂) +δ). (1.9) For ε₁ and δ small enough, C_δ(z₁, z₂) is a closed Fredholm (noncompact) manifold without boundary of codimension2k+ 2.

Forλ large enough, we define the intersection number (modulo 2) of W_u(f_λ(B₂(X))) withC_δ(z₁, z₂) denoted by

τ(z₁, z₂) =W_u(f_λ(B₂(X))).C_δ(z₁, z₂), (1.10) where W_u(f_λ(B₂(X))) is the unstable manifold of f_λ(B₂(X)) for a decreas- ing pseudogradient V for J which is transverse to f_λ(B₂(X)). Notice that the dimension ofW_u(f_λ(B₂(X))) is equal to2k+ 2 and the codimension of C_δ(z₁, z₂) is equal to2k+ 2. Therefore, the number τ(z₁, z₂) is well defined (see [29]). Our main result is the following.

Theorem 1.1: Let n ≥ 9. If τ(z₁, z₂) = 1 for a couple (z₁, z₂) ∈ X² satisfying(A₂) and −∆K(z_i)>0fori= 1,2, then (1.3) has a solution.

The aim of the next result is to give some conditions on the function K which allow us to have τ(z₁, z₂) = 1 for some couple (z₁, z₂) and thus, we obtain a solution for (1.3) by Theorem 1.1. Let z₁, z₂ ∈ X be such that

−∆K(z_i)>0. We chooseν⁺(z_i)such thatK(z_i) = max_ν⁺_(zi)K andz_iis the unique critical point ofK onν⁺(z_i).

Theorem 1.2: Letn ≥9. There exist positive constants C₀, C₁ such that, if, for two pointsz₁ andz₂ ofX, the following conditions hold:

1. w(z₁, z₂) := ^K(z_2K(y^1)+K(z2)

1) −1≤C₀. 2. For some positive constantρ₀,

wⁿ⁻⁶ⁿ⁻⁴(a₁, a₂) 1

d(a₁, a₂)²+ 1 ρ²₀

+| ∇K(a_i)|²

K(a_i)² +w^1/2(a₁, a₂)|D²K(a_i)|

K(a_i) +w^1/3(a₁, a₂) sup

B_(ai,ρ0)

|D³K(x)|

K(a_i) 2/3

≤ C₁

1 + (^supK

K(y1))ⁿ⁻⁴⁸

−∆K(a_i) K(a_i)

for each i= 1,2, and for each (a₁, a₂)∈ν⁺(z₁)×ν⁺(z₂) such that c∞(a₁, a₂)≤c∞(y₁, y₁).

3. inf

∂(ν⁺(z1)×ν⁺(z2))

c_∞(a₁, a₂)≥c_∞(y₁, y₁),

then (1.3)has a solution. (Herec∞(a₁, a₂)(respc∞(y₁, y₁)is defined by (1.8) replacing(z₁, z₂) by(a₁, a₂) (resp(y₁, y₁))).

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 as- sumptions, we refer the interested reader to [2] and [20].

The rest of the present paper is organized as follows. In Section 2, we re- call 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.

2 Preliminaries

Solutions of problem (1.3) correspond, up to some positive constant, to crit- ical points of the following functional defined on the unit sphere of H₂²(Sⁿ) by

J(u) = Z

Sⁿ

K|u|^{n−42n 4−n}_n

.

The exponent 2n/(n−4) is critical for the Sobolev embedding H₂²(Sⁿ) ,→ L^q(Sⁿ). As this embedding is not compact, the functionalJ does not satisfy the Palais-Smale condition and therefore standard variational methods can- not be applied to find critical points ofJ. In order to describe the sequences failing the Palais-Smale condition, we need to introduce some notations. For p∈N^∗ andε >0, we set

V(p, ε) =

u∈Σ| ∃a₁, ..., a_p∈Sⁿ,∃λ₁, ..., λ_p> ε⁻¹,∃α₁, ..., α_p>0with

u−

p

X

i=1

α_ieδ_(ai,λi)

< ε, ε_ij< ε∀i6=j,

J(u)ⁿ⁻⁴ⁿ α

8 n−4

i K(a_i)−1

< ε∀i

, where

ε_ij = λ_i

λ_j + λ_j

λ_i + λ_iλ_j

2 (1−cosd(a_i, a_j))

(4−n)/2

. Letw be a nondegenerate solution of (1.3). We also set

V(p, ε, w) =

u∈Σ| ∃α₀>0 with(u−α₀w)∈V(p, ε) and |α₀J(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 denotedu_j, such thatu_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 2.2: For any p ∈ N^∗, there is ε_p > 0 such that if ε ≤ ε_p and u∈V(p, ε), then the following minimization problem

min

u−

p

X

i=1

α_iδ˜_(ai,λi)

, α_i>0, λ_i>0, a_i∈Sⁿ

has a unique solution (α, λ, a) = (α₁, ..., α_p, λ₁, ..., λ_p, a₁, ..., a_p). In particu- lar, we can write uas follows:

u=

p

X

i=1

α_iδ˜_(ai,λi)+v, where vbelongs to H₂²(Sⁿ)and satisfies (V₀).

Next, we recall the following result which deals with thev-part of u.

Lemma 2.3:[8] Assuming theε_ij’s are small enough andJ(u)ⁿ⁻⁴ⁿ α

8 rn−4K(a_r) is close to1fori6=jand forr=i, j, then there exists a uniquev=v(a, α, λ) which minimizes

J Pp

i=1α_iδ˜_(ai,λi)+v

with respect to v ∈ E_ε := {v | vsatisfies(V₀)and

|| v ||< ε}, where ε is a fixed small positive constant depending only on p.

Moreover, we have the following estimate

||v||≤c ^p

X

i=1

| ∇K(a_i)| λ_i + 1

λ²_i

+X

i6=j

ε^min(^1,_2(n−4)ⁿ⁺⁴ )

ij (logε⁻¹_ij )^min(ⁿ⁻⁴n ,ⁿ⁺⁴_2n ) .

Note that Lemma 2.2 extends to the more general situation where the sequence(u_j) ofΣ⁺, 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 metricg. Notice that such a weak limit is a solution of (1.3). Denoting byw a nondegenerate solution of (1.3), we then have the following lemma which follows from the corresponding statement in [4].

Lemma 2.4: For any p ∈ N^∗, there is ε_p > 0 such that if ε ≤ ε_p and u∈V(p, ε, w), then the following minimization problem

min

u−

p

X

i=1

α_i˜δ_(ai,λi)−α₀(w+h) , α_i >0, λ_i>0, a_i∈Sⁿ, h∈T_w(W_u(w))

has a unique solution (α, λ, a, h) = (α₁, ..., α_p, λ₁, ..., λ_p, a₁, ..., a_p, h). In par- ticular, we can write uas follows:

u=

p

X

i=1

α_iδ˜_(ai,λi)+α₀(w+h) +v,

wherevbelongs toH₂²(Sⁿ)∩T_w(W_s(w))and satisfies(W₀). HereT_w(W_u(w)) and T_w(W_s(w)) denote the tangent spaces at w of the unstable and stable manifolds ofw, and(W₀) are the following conditions:

(W₀) :



















hv, ϕ_iiP = 0 for i= 1, ..., p and every

ϕ_i=eδ_(ai,λi), ∂eδ_(ai,λi)/∂λ_i, ∂eδ_(ai,λi)/∂(a_i)_j, j = 1, ..., n, for some system of coordinates (a_i)₁, ...,(a_i)_n on Sⁿ near a_i, hv, wi= 0,

hv, h₁i= 0 ∀h₁∈T_w(W_u(w)).

Now, following Bahri [4], we introduce the following definitions and no- tations.

Definition 2.5: A critical point at infinity ofJ on Σ⁺ is a limit of a flow- lineu(s) of equation ^∂u_∂s =−∇J(u) with initial datau₀∈Σ⁺ such thatu(s) remains in V(p, ε(s), w) for large s. Here w is zero or a solution of (1.3), p∈N^∗, andε(s)is some function such thatε(s)tends to zero when the flow parameters tends to+∞. By Lemma 2.4, we can write such u(s) as

u(s) =

p

X

i=1

α_i(s)˜δ_{(ai(s),λi(s))}+α₀(s)(w+h(s)) +v(s).

Denotinga_i= lim_s→+∞a_i(s), we call(a₁, ..., a_p, w)_∞ a critical point at infin- ity of J. Ifw6= 0, (a₁, ..., a_p, w)_∞ is called a mixed type of critical points at infinity ofJ.

Remark 2.6: Notice that forn≥9any 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 ofJ. This is not true forn ≤ 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 solutionw are of the same size.

In the sequel, we denote byAthe set ofwsuch thatwis a critical point or a critical point at infinity ofJ inΣ⁺ not containingy₀in 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 toq.

Definition 2.7: (A family of pseudogradients F) A decreasing pseudogra- dientV forJ is said to belong toF if the following properties hold:

- the set of critical points at infinity ofJ onΣ⁺ does not change if we take V instead of−∇J in the definition 2.5,

-V is transverse tof_λ(B₂(X)),

- for any w ∈ A, (y₀, w)_∞ is a critical point at infinity with the following property:

i((y₀, w)_∞, w) = 1 ∀w∈A

i((y₀, w)∞, w⁰) = 0 ∀w⁰∈A, w⁰6=w, index(w⁰) = index(w) i((y₀, w)_∞,(y₀, w⁰)_∞) =i(w, w⁰) ∀w⁰∈A, index(w⁰) = index(w)−1.

Here and belowi(ϕ₁, ϕ₂)denotes the intersection number forV ofϕ₁andϕ₂ (see [29] and [4]) whereϕ_i is any critical point or a critical point at infinity ofJ.

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 byf_λ(B₂(X)) if

∪s≥0ϕ(s, f_λ(B₂(X)))∩W_s(z∞)6=∅.

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])

∪s≥0ϕ(s, f_λ(B₂(X)))∩W_s(z∞)6=∅.

Definition 2.9: z∞ is said to be dominated by another critical point at infinityz⁰_∞ if

W_u(z_∞⁰ )∩W_s(z_∞)6=∅.

If we assume that the intersection is transverse, then index(z_∞⁰ )≥index(z_∞) + 1.

Given w_2k+1∈A_2k+1 and V ∈ F, we denote by

(y₀, w_2k+1)_∞.C_δ (2.1)

the intersection number (modulo 2) ofW_u((y₀, 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((y₀, 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 ofW_u((y₀, w_2k+1)_∞)is equal to2k+2and the codimension of C_δ is 2k+ 2. Then (y₀, w_2k+1)_∞.C_δ is also well defined, because the closure ofW_u((y₀, w_2k+1)_∞) only adds to W_u((y₀, w_2k+1)_∞) the unstable manifolds of critical points of index less than or equal to 2k+ 1.

These manifolds are then of dimension2k+ 1at most. Since the codimension ofC_δ is equal to2k+ 2, these manifolds can be assumed to avoid C_δ.

Now, for w_2k+1∈A_2k+1 andV ∈ F, we denote by

f_λ(B₂(X)).w_2k+1:=f_λ(B₂(X)).W_s(w_2k+1) (2.2) the intersection number of f_λ(B₂(X)) and W_s(w_2k+1). We notice that the dimension of

f_λ(B₂(X)) is equal to2k+ 1and 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(B₂(X)) outside f_λ(B₁(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 to2k+ 2.

Sincef_λ(B₂(X)) is of dimension 2k+ 1, these manifolds can be assumed to avoid it.

Lastly, we set for eachV ∈ F I(V) =τ− X

w2k+1∈A_2k+1

((y₀, w_2k+1)_∞.C_δ)(f_λ(B₂(X)).w_2k+1). (2.3)

Notice that 2.3 was introduced by Bahri in [4] where he proved thatI(V) is independent on V ∈ F. Namely, he showed in [4] that I(V) = 0, for each V ∈ F for the scalar curvature problem onSⁿ with n ≥ 7. We will prove that the same holds for theQ-curvature equation when n≥9.

3 Characterization of the critical points at infinity

In this section, we provide the characterization of the critical points at infin- ity. First, we construct a special pseudogradient for the associated variational problem for which the Palais-Smale condition is satisfied along the decreas- ing 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 independent of u=Pp

i=1α_iδe_i∈V(p, ε) so that

(a) h−∇J(u), WiP ≥c ^p

X

i=1

| ∇K(a_i)| λ_i + 1

λ²_i +X

i6=j

ε_ij

. (b)

h−∇J(u+v), W+ ∂v

∂(α_i, a_i, λ_i)(W)iP ≥c ^p

X

i=1

| ∇K(a_i)| λ_i + 1

λ²_i +X

i6=j

ε_ij

. (c) | W | is bounded. Furthermore, |dλ_i(W)| ≤ cλ_i for each i and the only case where the maximum of theλ_i’s increases alongW is when each pointa_i is close to a critical pointy_ji ofK with−∆K(y_ji)>0 andj_i6=j_r fori6=r.

Proof. We order the λ_i’s, for the sake of simplicity we can assume that:

λ₁≤...≤λ_p. Let

I₁={i|λ_i| ∇K(a_i)|≥C₁⁰}, I₂={1} ∪ {i |λ_j≤M λ_j−1, for each j≤i}, where C₁⁰ andM are two positive large constants. Set

Z₁=X

i∈I1

1 λ_i

∂δe_i

∂a_i

∇K(a_i)

| ∇K(a_i)|.

Using Proposition 5.2, we derive that h−∇J(u), Z₁iP ≥cX

i∈I1

| ∇K(a_i)|

λ_i +O X

j∈I2

1 λ_i

∂ε_ij

∂a_i

!

+O



 X

i∈I1

1 λ²_i +X

j /∈I2

ε_ij



+R. (3.1)

Observe that, ifj∈I₂then 1

λ_i

∂ε_ij

∂a_i

=λ_j|a_i−a_j|ε(n−2)/(n−4)

ij =o(ε_ij). (3.2)

Using also the fact thati∈I₁, thus, (3.1) becomes h−∇J(u), Z₁i_P ≥cX

i∈I1

| ∇K(a_i)| λ_i + 1

λ²_i +O



 X

j /∈I2

ε_ij



+R. (3.3) Now, we will distinguish two cases.

case 1I₁∩I₂6=∅. In this case, we define Z₂=−M₁X

i /∈I2

2ⁱλ_i∂eδ_i

∂λ_i −m₁X

i∈I2

λ_i∂eδ_i

∂λ_i, whereM₁is a large constant andm₁ is a small constant.

Using Proposition 5.2, we derive h−∇J(u), Z₂i_P ≥cM₁X

i /∈I2

Xε_ij+O 1

λ²_i

+R

+m₁cX

i∈I2



 X

j∈I2

ε_ij+O



 1 λ²_i +X

j /∈I2

ε_ij



+R



. (3.4) Now, we defineZ₃=Z₁+Z₂. Using (3.3) and (3.4), we derive that

h−∇J(u), Z₃i_P

≥cX

i∈I1

| ∇K(a_i)| λ_i + 1

λ²_i +cX

j6=i

ε_ij+O X

i /∈I2

M₁ λ²_i +X

i∈I2

m₁ λ²_i

!

+R. (3.5)

Observe that, since I₁∩I₂ 6= ∅, we can make 1/λ²_k appear, for k ∈ I₂, in the lower bound of (3.5) and therefore all theλ⁻²_i ’s can appear in the lower bound of (3.5). Notice that fori /∈I₁, we haveλ_i | ∇K(a_i)|≤C₁⁰. Thus, if we chooseM₁≤M and m₁<< M^p, (3.5) becomes

h−∇J(u), Z₃i_P ≥c

p

X

i=1

| ∇K(a_i)| λ_i + 1

λ²_i +cX

j6=i

ε_ij. (3.6) case 2 I₁∩I₂ = ∅. In this case, for each i ∈ I₂, the point a_i is close to a critical point y_ki of K. We claim that k_i 6= k_j for i 6= j that is each neighborhoodB(y, ρ), forρsmall enough, contains at most one pointa_iwith i ∈ I₂. Indeed, arguing by contradiction, let us suppose that there exist i, j ∈I₂ 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 λ_i ≤ λ_j) λ_i|a_i−a_j| ≤c. This implies that ε_ij ≥c(λ_i/λ_j)^(n−4)/2, a contradiction with the fact that λ_i andλ_j are of the same order. Thus our claim follows.

Let us introduce

I₃={i∈I₂|∆K(a_i)>0}.

1st subcaseI₃6=∅. In this case we define Z₄=−X

i∈I3

λ_i∂δe_i

∂λ_i −M₁X

i /∈I2

2ⁱλ_i∂eδ_i

∂λ_i. Using Proposition 5.2 we derive

h−∇J(u), Z₄i_P ≥cX

i∈I3

1

λ²_i +OX ε_ij +M₁cX

i /∈I2

X

j6=i

ε_ij+O 1

λ²_i !

+R. (3.7)

Observe that, ifi, j∈I₂, we have|a_i−a_j| ≥cthen (sincen≥9) ε_ij=O λ⁻⁵_i +λ⁻⁵_j

. (3.8)

ForZ₅=Z₄+Z₁, using (3.3), (3.7), (3.8) and choosing M₁≤M, we obtain h−∇J(u), Z₅i ≥c

p

X

i=1

| ∇K(a_i)| λ_i + 1

λ²_i +cX

j6=i

ε_ij. (3.9)

2nd subcaseI₃=∅. In this case we define Z₆=X

i∈I2

λ_i∂eδ_i

∂λ_i −M₁X

i /∈I2

2ⁱλ_i∂eδ_i

∂λ_i +Z₁. Using Proposition 5.2, as in the above subcase, we derive that

h−∇J(u), Z₆i ≥c

p

X

i=1

| ∇K(a_i)| λ_i + 1

λ²_i +cX

j6=i

ε_ij. (3.10) The vector field W will be a convex combination of all Z₃, Z₅ and Z₆. 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 theλ_i’s increases is when I₂ = {1, ..., p} and I₁ = I₃ = ∅, it means each a_i is close to a critical point y_ji ofK withj_i6=j_r fori6=rand−∆K(y_ji)>0for eachi. Hence claim(c) follows.

Finally, arguing as in Appendix B of [7], claim(b)follows from claim(a)and

Lemma 2.3. 2

Proposition 3.2: Let n ≥ 9. Assume that J has no critical point in Σ⁺. Under the assumptions (A₁) and (A₂), the only critical points at infinity under the levelc∞(y₁, y₁) are:

(y₀)∞, (y₁)∞ and (y₀, y₁)∞.

Moreover, the Morse indices of such critical points at infinity are

n−index(K, y₀) = 0,n−index(K, y₁)and1 +n−index(K, y₁) respectively.

Proof. Using Proposition 2.1, we derive that| ∇J|≥cinΣ⁺\ ∪p≥1V(p, ε), wherecis a positive constant which depends only onε. It only remains to see what happens in∪p≥1V(p, ε). From Proposition 3.1, we know that the only region where the maximum of theλ_i’s increases along the pseudogradientW, defined in Proposition 3.1, is the region where each a_i is close to a critical point y_ji of K with −∆K(y_ji) > 0 and j_i 6= j_r for i 6= r. In this region, arguing as in [4], we can find a change of variables:

(a₁, ...a_p, λ₁, ..., λ_p)−→(˜a₁, ...,˜a_p,˜λ₁, ...,˜λ_p) := (˜a,˜λ)

such that J

^p X

i=1

α_i˜δ_(ai,λi)+v

(3.11)

= Ψ(ea,eλ) := S_n^4/nP α²_i

Pα

2n n−4 i K(ea_i)

ⁿ⁻⁴_n

1−(c−η)

p

X

i=1

∆K(y_ji) eλ²_iK(y_ji)ⁿ⁴

+|V |²,

where ηis a small positive constant and c=c₂(n−4)/n

PK(y_j^(4−n)/4_i −1

, where c₂ is defined in Proposition 5.1. This yields a split of variables ˜a and ˜λ. Thus it is easy to see that if the α_i’s are in their maximum and ea_i = y_ji for each i, only the eλ_i’s can move. To decrease the functional J, we have to increase the eλ_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_ji)> 0for 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 variablev. Then the Morse index of such critical point at infinity is equal to(p−1 +Pp

i=1(n−index(K, y_ji))). Thus our result follows. 2 In Proposition 3.2, we have assumed that J has no critical point inΣ⁺. 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,

(y₀, w)∞, (y₁, w)∞ and (y₀, y₁, w)∞

are critical points at infinity. The Morse index of these critical points are respectively equal to

index(w) + 1, index(w) +index((y₁)_∞) + 1 and index(w) +index((y₁)_∞) + 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 thatJ reads as

J(u) = S_nPp

i=1α²_i +α²₀||w||² (S_nPp

i=1α

2n n−4

i K(a_i) +α

2n n−4

0 ||w||²)ⁿ⁻⁴ⁿ

"

1−(n−4)c₂ nβ₀

p

X

i=1

4α

2n n−4 i

×∆K(a_i) λ²_i − c₁

2γ₀ X

i6=j

α_iα_jε_ij+o X

i6=j

ε_ij+

p

X

i=1

1 λ²_i

!#

+||V||²− ||H||². Furthermore, we have the following estimates:

||h|| ≤cX

i

1 λ^(n−4)/2_i

||v|| ≤c ^p

X

i=1

| ∇K(a_i)| λ_i + 1

λ²_i

+X

i6=j

ε^min(1,_2(n−4)ⁿ⁺⁴ )

ij (logε⁻¹_ij )^min(ⁿ⁻⁴n ,ⁿ⁺⁴_2n ) .

Before giving the proof of Proposition 3.4, we need to prove the following lemma:

Lemma 3.5: The following Claims hold true:

(a) Q₁(v, v) is a positive definite quadratic form in

E_ε⁰ ={v∈H²(Sⁿ)|v∈T_w(W_s(w)), and v satisfies(W₀)}.

(b) Q₂(h, h) is a negative definite quadratic form in T_w(W_u(w)).

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)) intoE_γ⊕F_γ whereE_γ and F_γ are orthogonal forh,iP and as well as for the quadratic form associated tow and such that

(||v||²−ⁿ⁺⁴_n−4R

Kw^8/(n−4)v²≥(1−γ)||v||² on F_γ dim(E_γ)<+∞.

We choose γ small enough such that 0 < γ < α/4, where¯ α¯ is the first eigenvalue of−∆−ⁿ⁺⁴_n−4δ˜^8/(n−4)_(a,λ) . Notice thatα¯ is independent ofδ˜_(a,λ). Since dim(E_γ)<∞we have

Z

δ˜^8/(n−4)_i v²₁=o(||v₁||²) ∀v₁∈E_γ, and∀i.

Now let

v=v₁+v₂, with v₁∈E_γ, v₂∈F_γ. (3.12) Then

Q₁(v, v) =||v₁||²+||v₂||²−

p

X

i=1

n+ 4 n−4

Z

δ˜_i^8/(n−4) v₁²+v₂²+ 2v₁v₂

−n+ 4 n−4

Z

Kw^8/(n−4) v₁²+v₂²+ 2v₁v₂

=||v₁||²+||v₂||²−

p

X

i=1

n+ 4 n−4

Z

δ˜_i^8/(n−4) v₁²+v₂²

−n+ 4 n−4

Z

Kw^8/(n−4) v₁²+v₂²

+o(||v₁||||v₂||) This implies that

Q₁(v, v)≥ ||v₁||²+ (1−γ)||v₂||²−

p

X

i=1

n+ 4 n−4

Z

˜δ^8/(n−4)_i v₂²

−n+ 4 n−4

Z

Kw^8/(n−4)v₁²+o ||v₁||||v₂||+||v₁||²

≥(1−γ)||v₂||²−

p

X

i=1

n+ 4 n−4

Z

δ˜^8/(n−4)_i v²₂+o ||v₂||²

+α⁰||v₁||². It remains to study the term

||v₂||²−

p

X

i=1

n+ 4 n−4

Z

˜δ^8/(n−4)_i v₂². Observe thatvis orthogonal to span{δ˜_i, λ_i∂λ^∂δ˜i

i,_λ¹

i

∂δ˜i

∂ai, 1≤i≤p}but notv₂. Sincev₁ belongs to a finite dimensional space, we have

∀ϕ∈ ∪i≤p{δ˜_i, λ_i∂δ˜_i

∂λ_i, 1 λ_i

∂δ˜_i

∂(a_i)_j}, | hv₁, ϕiP |≤ ||v₁||∞

Z

|∆²ϕ|=o(||v₁||).

(3.13) Now, we write

v₂= ¯v₂+X

i

A_iδ˜_i+X

i

B_iλ_i∂˜δ_i

∂λ_i+X

i,j

C_ij1 λ_i

∂δ˜_i

∂(a_i)_j, (3.14)

with¯v₂∈span{˜δ_i, ^∂δ˜i

∂λi, ^∂δ˜i

∂(ai)j, i≤p, j ≤n}^⊥. Thus, we have (see [8])

||¯v₂||²−

p

X

i=1

n+ 4 n−4

Z

˜δ^8/(n−4)_i ¯v²₂≥ α¯ 2||¯v₂||². Notice that

||v₂||²−

p

X

i=1

n+ 4 n−4

Z

δ˜_i^8/(n−4)v₂²=||¯v₂||²+O X

i

A²_i +B²_i +X

j

C_ij²

!

−

p

X

i=1

n+ 4 n−4

Z

δ˜_i^8/(n−4)v¯₂²+O ||¯v₂||(|A_i|+|B_i|+X

j

|C_ij|)

!

(3.15) Using (3.12)-(3.14), we obtain

A_i=o(||v₁||), B_i=o(||v₁||) and C_ij=o(||v₁||) for each i, j.

Thus, using (3.15), we derive that Q₁(v, v)≥ −γ||v₂||²+α¯

2||¯v₂||²+o ||v₁||²+||v₂||²

+α⁰||v₁||². But

||v₂||²=||¯v₂||²+O X

i

A²_i +B_i²+X

j

C_ij²

!

=||¯v₂||²+o(||v₁||²).

Thus

Q₁(v, v)≥α¯ 2 −γ

||v₂||²+α⁰||v₁||²+o ||v₁||²+||v₂||² .

Since γ < α/4, claim¯ (a) follows. The proof of our lemma is thereby com-

pleted. 2

Proof of Proposition 3.4 By Proposition 5.1 the expansion of J with respect toh(respectively tov) is very close, up to a multiplicative constant, to Q₂(h, h) +f₂(h) (respectively Q₁(v, v)−f₁(v)). 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||f₂|| = O(P

iλ^(4−n)/2_i )and||¯v|| ≤c||f₁||. The estimate ofvfollows from Lemma 2.3.

Then our result follows. 2