A Morse inequality for a fourth order elliptic equation on a bounded domain

2019, No.35, 1–20; https://doi.org/10.14232/ejqtde.2019.1.35 www.math.u-szeged.hu/ejqtde/

A Morse inequality for a fourth order elliptic equation on a bounded domain

Khadijah Sharaf

and Hichem Hajaiej

1Department of mathematics, King Abdulaziz University, P.O. 80230, Jeddah, Kingdom of Saudi Arabia

2California State University Los Angeles, 5151 University Drive, Los Angeles, USA

Received 10 February 2019, appeared 9 May 2019 Communicated by Patrizia Pucci

Abstract. This article deals with the existence of multiple solutions of a fourth order elliptic equation with a critical nonlinearity on a bounded domain of Rⁿ,n ≥ 5. We develop an approach to overcome the lack of compactness of the problem and we es- tablish under a generic hypothesis a Morse inequality providing a lower bound of the number of solutions.

Keywords: fourth order PDE, critical nonlinearity, variational problem, critical points at infinity.

2010 Mathematics Subject Classification: 35J60, 35J65.

1 Introduction and main results

Let Ωbe a smooth bounded domain ofRⁿ,n≥ 5 and letK: Ω→ _Rbe a given function. We are interested in constructing a smooth positive functionuon Ωsatisfying











∆²u=K(x)uⁿⁿ⁺⁻⁴⁴, u>0 inΩ,

∆u= u=0 on∂Ω.

(1.1)

Equation (1.1) is heavily connected to the celebrated problem of prescribing Q-curvature on closed Riemannian manifolds. See [3,9–11,14–17] and the references therein for details.

Problem (1.1) has a variational structure. The solutions correspond to positive critical points of the functional:

J(u) =

R

Ω(_∆u)² R

ΩK(x)uⁿ²ⁿ⁻⁴dxⁿ⁻_n⁴ defined on the function space:

Σ=u ∈H₂²(_Ω)∩H₁⁰(_Ω), s.t.kuk=1 ,

BCorresponding author. Email: kh-sharaf@yahoo.com

where

kuk=

Z

Ω(_∆u(x))²dx¹₂ .

One can see that,uis a critical point of J inΣ⁺=u∈_∑, u>₀ , if and only if J(u)ⁿ⁻⁸⁴.uis a solution of (1.1). Problem (1.1) is delicate from the variational viewpoint since the functional J does not satisfy the Palais–Smale condition on Σ⁺ (P.S. in short): There exist sequences along which J is bounded, its gradient goes to zero and the sequences do not converge. This is a consequence of the lack of compactness of the embedding H₂²(_Ω)∩H₀¹(_Ω) ,→ Lⁿ²ⁿ⁻⁴(_Ω). Consequently, challenging situations where critical points at infinity are limits of non-compact flow-lines of the gradient vector field(−∂J), occur.

In [18] and [25], the authors showed the existence of solutions of (1.1), provided K ≡ 1.

Their results hinge on the shape ofΩ. WhenK 6= 1, some existence results can be found for example in [1], [9], and [13].

Recently in [1] Abdelhedi, Chtioui and Hajaiej established compactness and existence re- sults for (1.1) under the following three conditions:

(A) ^∂K_∂ν(x)6=0, ∀x∈∂Ω.

Hereνis the unit outward normal vector on∂Ω.

(f)_β Kis aC¹-positive function onΩsuch that at any critical pointy ofK, there exists a real numberβ= β(y)satisfying

K(x) =K(y) +

∑

b_k|(x−y)_k|^β+o(|x−y|^β), ∀x∈ B(y,ρ0), whereρ₀ is a positive fixed constant,b_k = b_k(y)∈_R\ {0},∀k=1 . . . ,n, and

−ⁿ−4 n

c₁ K(y)

∑

b_k(y) +c2

n−4

2 H(y,y)6=0, ∀y∈ K_n−4, where K_n−4 := {y ∈ _Ω,∇K(y) = 0 andβ(y) = n−4}. Here c₁ = R

Rⁿ |z1|ⁿ⁻⁴ (1+|z|²)ⁿdz, c₂ = R

Rⁿ dz

(1+|z|²)ⁿ⁺²⁴ and H(·,·) is the regular part of the Green function G(·,·) of the bilaplacian under the Navier boundary condition and

(A’) β(y) =β∈(_1,n−₄]_{at any}ysuch that∇K(y) =_0.

Many interesting studies were dedicated to the problem (1.1) and its related Q-curvature problem on closed manifolds under the above(f)_β-condition. See for example [19], [14] and [12] on the standard n-dimensional spheren≥5, treating respectively the case ofβ∈]n−4,n[, β ∈]1,n−4]and β = n. Concerning the problem on bounded domains case, we refer to [1].

We point out that(f)_β-condition covers the famous non degeneracy condition corresponding to the case of β = 2 and used in several works on (1.1) and its related curvature, see for example [2], [8], [13], [17] and [16].

According to the above results, we observe that the flatness order β does not exceed the value of n; the dimension of the associated domain. In this paper, we provide new existence results to the problem and we establish a lower bound of the number of solutions thanks to a Morse inequality. Our results are new and important as it address the case of β-flatness

condition for any β ≥ n−4. To state our existence results, we need to introduce some notations and assumptions: Let

K⁺_n−4= (

y∈ K_n−4,−ⁿ−₄ n

c₁ K(y)

∑

b_k(y) +c₂n−₄

2 H(y,y)>₀ )

, and

K_>n−4={y ∈_Ω,∇K(y) =0, β(y)>n−4}.

For any p-tuple of distinct points τp = (y`<sub>1</sub>, . . . ,y`_p)∈ (K⁺_n−4∪ K_>n−4)^p, 1 ≤ p, we define a symmetric matrix M(τ_p) = (m_ij)_1≤i,j≤pdefined by:

m_ii= m(y`<sub>i</sub>,y`_i)

=















− ¹

K(y`_i)ⁿ⁻⁴⁴

n−4 n

c₁ K(y`_i)

∑

b_k(y`_i)−c₂n−4

2 H(y`<sub>i</sub>,y`_i)

!

if β(y`_i) =n−4, n−4

2

c2

K(y`<sub>i</sub>)<sup>n−44 H</sup>(y`_i,y`<sub>i</sub>) if β(y`_i)>n−4,

∀i=_{1, . . . ,}p and

m_ij =m(y`<sub>i</sub>,y`_j) =−ⁿ−4

2 c₂ G(y`<sub>i</sub>,y`_j)

K(y`<sub>i</sub>)K(y`_j)

n−4 8

, for 1≤ i6=j≤ p.

(B) Assume that the least eigenvalue ρ(τ_p) of M(τ_p) is non zero for any τ_p ∈ (K_n⁺₋₄∪ K_>n−4)^p,p≥1.

For anyτ_p = (y`<sub>1</sub>, . . . ,y`_p)∈ (K_n−4∪ K_>n−4)^p,p ≥1, we define i(τp) = p−1+

∑

n−ei(y`_i), whereei(y) =]{b_k(y), 1≤k≤n, s.t.b_k(y)<0}.

We now state our multiplicity result.

Theorem 1.1. Let K : Ω → _Rbe a function satisfying(A),(B)and(f)_β,β ∈ [n−4,∞). If there exists an integer k₀∈_Nsuch that

(i) i(τ_p)6=k0+1, ∀τ_p ∈ K^∞, where

K^∞ :=ⁿ(y`<sub>1</sub>, . . . ,y`_p)∈(K⁺_n−4∪ K_>n−4)^p,p≥1, y`<sub>i</sub> 6=y`_j,∀i6= j andρ(y`<sub>1</sub>, . . . ,y`_p)>0o . (ii) All the critical points of J of indices≤k0+₁are non degenerate. Then

N_k0+1≥

1−

∑

τp∈K^∞,i(τp)≤k0

(−1)^i(τp) ,

where N_k0+1is the number of solutions of (1.1)having their Morse indices≤k₀+1.

We point out that Morse inequalities for Morse functions provide a lower bound for the number of the associated critical points. Therefore, Theorem 1.1 can be considered as a sort of Morse type inequality, since it provides a lower bound of the number of solutions and consequently a lower bound of the number of critical points of J. Notice also by the Sard–

Smale theorem, see [23], the critical points of J are non degenerate for genericK. In the sense that for anyC¹-functionK₀, there exits aC¹-functionKclose to K₀(in the C¹ sense) such that J has only non degenerate critical points.

An immediate corollary of Theorem 1.1is the following result which prove the existence of at least one solution without assuming that (1.1) has only non degenerate solutions.

Theorem 1.2. Assume that K satisfies (A),(B), (f)_β, β ∈ [n−4,∞)and the condition (i)of the above theorem. If

∑

τp∈K^∞,i(τp)≤k0

(−1)^i(τp) 6=1, then(1.1)has a solution of index≤k₀+1.

Observe that the integerk₀ =max{i(τ_p),τ_p ∈ K^∞}satisfies the condition(i)of the above Theorems. Therefore, the following two results are consequences of Theorem 1.1 and Theo- rem1.2.

Theorem 1.3. Assume(A),(B)and(f)_β,β∈[n−4,∞). For generic K it holds N≥1−

∑

τp∈K^∞

(−1)^i(τp), where N is the number of solutions of (1.1).

Theorem 1.4. Under the assumptions(A),(B)and(f)_β, β∈ [n−4,∞). If

∑

τp∈K^∞

(−1)^i(τp)6=1, then(1.1)has at least one solution.

Our method is inspired by Bahri’s principle of critical points theory at infinity [4]. The most important novelty of the present work is the extension of existence and multiplicity results of [1,14] and [19], to any order of flatness larger than n−4. The main analysis diffi- culty in our statement comes from the divergence of integrals for β large. This leads to get new estimates for the the associated Euler–Lagrange functional and its derivatives. Using these estimates, we construct a suitable pseudo-gradient, completely different from the one of [1] allowing us to describe the lack of compactness of our problem and identify the critical points at infinity of the associated variational structure. We then use topological arguments to prove our results. In the next section, we will state some preliminaries related to the vari- ational structure associated to problem (1.1). In Section 3, we will study the concentration phenomenon of the problem and identify the critical points at infinity of J and in Section 4, we will prove our existence results.

2 Variational structure

In this section, we state some preliminary tools of the variational structure associated to (1.1).

Fora ∈_Ωandλ>_{0, let}

δ_a,λ(x) =cn

λ 1+λ²|x−a|²

ⁿ⁻₂⁴

, (2.1)

wherecnis a positive constant chosen such that δ_a,λis the family of solutions of the following problem (see [22]):

∆²u=|u|ⁿ⁻⁸⁴u, u>0 inRⁿ. (2.2) LetPδ_a,λ the unique solution of

(∆²Pδ_a,λ= δ_a,λ

n+4

n−4 in Ω

Pδ_a,λ= _∆Pδa,λ =0 on ∂Ω.

For ε > 0 and p ∈ _N^∗, we define the following set of potential critical points at infinity associated to J:

V(p,ε) =











u∈_Σ⁺,s.t, ∃ a₁, . . . ,ap ∈_Ω,∃λ₁, . . . ,λp >ε⁻¹ and α₁, . . . ,αp>0 with

u−_∑i^p₌₁α_iPδ_ai,λi

<ε, ε_ij <ε ∀i6= j, λ_id_i >ε⁻¹ and

Jⁿ⁻ⁿ⁴(u)α

n−84

i K(a_i)−1

<ε ∀i=1, . . . ,p.

Here,d_i =d(a_i,∂Ω)andε_ij = ^λi

λj +^λj

λi +λ_iλ_j|a_i−a_j|²⁴⁻²ⁿ. Letwbe a critical point ofJ inΣ⁺. Define

V(p,ε,w) =

u∈ _Σ⁺, s.t. there existsα0>0 satisfyingu−α0w∈V(p,ε) and

α

8 n−4

0 J(u)ⁿ⁻ⁿ⁴ −1 <e

.

The following proposition describes the failure of the (P.S.)-condition ofJ.

Proposition 2.1([5,24]). Let(u_k)_kbe a sequence inΣ⁺such that J(u_k)is bounded and∂J(u_k)goes to zero. Then there exists a positive integer p, a sequence(ε_k)withε_k →0as k→+_∞and an extracted subsequence of (u_k)_k’s, again denoted (u_k)_k, such that u_k ∈ V(p,ε_k,w),∀k, where w is a solution of (1.1)or zero.

The following proposition gives a parametrization ofV(p,ε,w).

Proposition 2.2 ([5]). For all p ∈ _N^∗, there exists ε_p > 0 such that for any ε ≤ ε_p and any u in V(p,ε,w), the problem

min (

u−

∑

α_iPδ_ai,λi −α₀(w+h)

, α_i >0,λ_i >0,a_i ∈_Ω,h∈T_w(W_u(w)) )

. admits a unique solution(α,λ,a,h). Thus, we can uniquely write u as follows

u=

∑

α_iPδ_ai,λi+α₀(_w+_h) +_v, where v∈ H₂²(_Ω)∩H¹₀(_Ω)∩T_w(W_s(w))and satisfies

(V₀)

v,ψ

=0 forψ∈

w,h,Pδ_i,∂Pδ_i

∂λ_i ,∂Pδ_i

∂a_i ,i=1, . . . ,p

.

Here, Pδ_i = Pδ_ai,λi andh·,·idenotes the inner product on H₂²(_Ω)defined by hu,vi=

Z

Ω∆u∆v.

The following proposition deals with the v-part of u and shows that is negligible with respect to the concentration phenomenon.

Proposition 2.3 ([4,5]). There is a C¹-map which to each (α_i,a_i,λ_i,h) such that ∑_i^p=1α_iPδ_ai,λi + α0(w+h)belongs to V(p,ε,w)associates v=v(α_i,a_i,λ_i,h)such that v is the unique solution of the following minimization problem

min (

J

∑

α_iPδ_ai,λi +α0(w+h) +v

!

, v satisfies(V0) )

. In addition, there exists a change of variables v−v→V such that

J

∑

α_iPδ_ai,λi+α₀(w+h) +v

!

= J

∑

α_iPδ_ai,λi+α₀(w+h) +v

!

+kVk². The estimate ofkv¯kis given in the following lemma.

Lemma 2.4([14, p. 3020]). There exists c>₀independent of u such that the following holds

kvk ≤c

∑

1 λ

n 2

i

+ ¹ λ^β_i

+|∇K(a_i)|

λ_i + (logλ_i)ⁿ²ⁿ⁺⁴ λ

n+4 2

i

+c











k

∑

ε

n+4 2(n−4) k r

logε⁻_kr^{1 n}_2n⁺⁴

, if n≥12

k

∑

ε_{k r}

logε⁻_kr^{1 n−}_n⁴

, if n<_12.

We now state the definition of critical point at infinity.

Definition 2.5([4]). A critical point at infinity of J is a limit of a non-compact flow line u(s) of the gradient vector field(−∂J). By Propositions2.1and2.2,u(s)can be written as:

u(s) =

∑

α_i(s)Pδ_ai(s),λi(s)+v(s).

Denoting byy_i =lim_s→+_∞a_i(s) and α_i =lim_s→+_∞α_i(s), we then denote by

∑

α_iPδ_yi,∞ or (y₁, . . . ,y_p)_∞ such a critical point at infinity.

3 Concentration phenomenon and critical points at infinity

In this section, we study the concentration phenomenon of the problem and we provide the description of the critical points at infinity under(f)_β-condition,β∈[n−4,∞).

Theorem 3.1. Assume(A),(B)and(f)_β,β∈[n−4,∞). There exists a decreasing pseudo-gradient W in V(p,ε)satisfying the following

(i) h∂J(u),W(u)i ≤ −c

∑

1 λ^min_i ^(n,β)

+|∇K(a_i)|

λ_i

! +

∑

j6=i

ε_ij

! , (ii)

∂J(u+v¯),W(u) + ^∂v¯

∂(α_i,a_i,λ_i)(W(u))

≤ −c p

i

∑

1 λ^min_i ^(n,β)

+ |∇K(a_i)|

λ_i

+

∑

j6=i

ε_ij

. In addition, W is bounded and the only case whereλ_i(s),i=1, . . . ,p,tend to∞is when a_i(s)goes to y`<sub>i</sub>,∀i=1, . . . ,p such that(y`₁, . . . ,y`_p)∈ K^∞.

The proof of Theorem3.1 is based on the following sequence of lemmas which describe the concentration phenomenon in particular regions of V(p,ε)and hint the concentration of the required pseudo-gradientW. Letδ>0 small enough, setting:

V₁(p,ε) =

u=

∑

α_iPδ_(ai,λi)+v∈V(p,ε), a_i ∈ B(y`<sub>i</sub>,ρ<sub>0</sub>),λ<sup>n</sup><sub>i</sub><sup>−4</sup>|a<sub>i</sub>−y`_i|^β <δ,∀i=1, . . . ,p, with(y`<sub>1</sub>, . . . ,y`_p)∈ K^∞

, V₂(p,ε) =

u=

∑

α_iPδ_(ai,λi)+v∈V(p,ε), a_i ∈ B(y_{`i</sub>,ρ<sub>0</sub>),∇K(y<sub>`i}) =0,λⁿ_i⁻⁴|a_i−y<sub>`i</sub>|<sup>β</sup> <δ, m(y`_i,y`<sub>i</sub>)>0,∀i=1, . . . ,p, y`_i 6=y`<sub>j</sub> ∀j6=i, andρ(y`₁, . . . ,y`_p)<0

, V₃(p,ε) =

u=

∑

α_iPδ_(ai,λi)+v∈V(p,ε), a_i ∈ B(y_{`i</sub>,ρ<sub>0</sub>),∇K(y<sub>`i}) =0,λⁿ_i⁻⁴|a_i−y_`i|^β <δ,

∀i=1, . . . ,p, y`<sub>i</sub>6=y`_j∀j6=i, and there existsi₁∈ {1, . . . ,p}, s.t.m(y`_i

1,y`_i

1)<0

, V₄(p,ε) =

u=

∑

α_iPδ_(ai,λi)+v∈V(p,ε)_, a_i ∈ B(y_{`i</sub>,ρ<sub>0</sub>)<sub>,</sub>∇K(y<sub>`i}) =_0,∀i=_{1, . . . ,}p, y`<sub>i</sub> 6=y`_j ∀j6=i, and there existsi₁ ∈ {1, . . . ,p}, s.t.λⁿ_i⁻⁴

1 |a_i1 −y`_i

1|^β ≥ δ,

, V5(p,ε) =V(p,ε)\ ∪⁴_i=1V_i(p,ε).

Lemma 3.2. There exists a pseudo-gradient W₁ in V₁(p,ε)such that for any u= _∑i^p₌₁α_iPδ_(ai,λi) ∈ V₁(p,ε), we have

h∂J(u),W₁(u)i ≤ −c

∑

1 λⁿ_i⁻⁴

+|∇K(a_i)|

λ_i

! +

∑

j6=i

ε_ij

! .

W₁ is bounded and the concentration componentsλ_i(s)of the associated flow lines increase and go to +_∞,i=_{1, . . . ,}p.

Proof. Letu =_∑i^p₌₁α_iPδ_(ai,λi) ∈ V(p,ε). We increase all the λ_i,i= 1, . . . ,p with respect to the differential equation

λ˙_i =λ_i, ∀i=1, . . . ,p.

The corresponding vector field is

W₁(u) =

∑

α_i

∂Pδ_(ai,λi)

∂λ_i λ˙_i.

Recall that the variation of J with respect to λ_i,i= 1, . . . ,p was given in ([9, Proposition 3.3]) under the so-called non-degeneracy condition. In the same way, we state here this variation under(f)_β-condition,β∈[n−4,∞). We have the following two estimates.

∂J(u),α_iλ_i∂Pδ_ai,λi

∂λ_i

=2c₂J(u)

∑

j6=i

α_iα_j −λ_i∂ε_ij

∂λ_i

− ⁿ−₄ 2

H(a_i,a_j) (λ_iλ_j)ⁿ⁻²⁴

!

+2α²_iJ(u)









 n−4

n c₁∑ⁿ_k=1b_k(y`_i) K(a_i)λ^β

(y`_i) i

−c2n−4 2

H(y`<sub>i</sub>,y`_i) λⁿ_i⁻⁴

, ifβ(y`_i) =n−4

−c₂n−4 2

H(y`<sub>i</sub>,y`_i) λⁿ_i⁻⁴

, ifβ(y`_i)> n−4 +O

|a_i−y`_i|^β+o

∑

j6=i

ε_ij+ ^H(a_i,a_j) (λ_iλ_j)ⁿ⁻²⁴

!!

+o

∑

1

(λ_jd(a_j,∂Ω))ⁿ⁻⁴

!

(3.1) and

∂J(u),α_iλ_i∂Pδ_ai,λi

∂λ_i

=2c₂J(u)

∑

j6=i

α_iα_j −λ_i

∂ε_ij

∂λ_i − ⁿ−4 2

H(a_i,a_j) (λ_iλ_j)ⁿ⁻²⁴

!

+O

[min(n,β)]

∑

|a_i−y_`i|^β−j λ^j_i

!

+O 1 λ_i^β

!

+o

∑

j6=i

ε_ij+ ^H(a_i,a_j) (λ_iλ_j)ⁿ⁻²⁴

!!

. (3.2)

Herec₁andc₂are defined in the first section. The complete proof of (3.1) and (3.2) was given in [1]. Observe that for anyu∈V₁(p,ε)we have

|a_i−y`_i|^β =o 1 λⁿ_i⁻⁴

!

, asδ small.

−λ_i

∂ε_ij

∂λ_i = ⁿ−4 2

1

(|a_i−a_j|²λ_iλ_j)ⁿ⁻²⁴ +o 1 (λ_iλ_j)ⁿ⁻²⁴

!

, since|a_i−a_j| ≥ρ0. Therefore,

−λ_i∂ε_ij

∂λ_i −ⁿ−4 2

H(a_i,a_j)

(λ_iλ_j)ⁿ⁻²⁴ = ⁿ−4 2

1

|a_i−a_j|ⁿ⁻⁴ −H(a_i,a_j) 1

(λ_iλ_j)ⁿ⁻²⁴ +o 1 (λ_iλ_j)ⁿ⁻²⁴

!

= ⁿ−4 2

"

1

|y_{`i</sub> −y<sub>`j}|ⁿ⁻⁴ −H(y`<sub>i</sub>,y`_j)

# 1

(λ_iλ_j)ⁿ⁻²⁴ +o 1 (λ_iλ_j)ⁿ⁻²⁴

!

= ⁿ−4 2

G(y`<sub>i</sub>,y`_j)

(λ_iλ_j)ⁿ⁻²⁴ +o 1 (λ_iλ_j)ⁿ⁻²⁴

! . Therefore,

h∂J(u),W₁(u)i=2J(u)

∑

∑

j6=i

α_iα_jn−4

2 c₂G(y`<sub>i</sub>,y`_j) (λ_iλ_j)ⁿ⁻²⁴

+2J(u)

∑

α²_i









 n−4

n c₁∑ⁿk=1b_k(y_`i) K(a_i)λ^β

(y`_i) i

−c₂n−4 2

H(y_{`i</sub>,y<sub>`i}) λⁿ_i⁻⁴

, ifβ(y_`i) =n−4

−c₂n−₄ 2

H(y`<sub>i</sub>,y`_i) λⁿ_i⁻⁴

, ifβ(y_`i)> n−₄ +o

∑

1 λⁿ_i⁻⁴

! . Since J(u)ⁿⁿ⁻⁴α

8 n−4

i K(a_i) =1+o(1),∀i=1, . . . ,p, we get h∂J(u),W₁(u)i= −2J(u)⁴⁻⁴ⁿ

" _p

i

∑

∑

j6=i

m(y`<sub>i</sub>,y`_j) (λ_iλ_j)ⁿ⁻²⁴ +

∑

m(y`<sub>i</sub>,y`_i) λⁿ_i⁻⁴

# +o

∑

1 λⁿ_i⁻⁴

!

= −2J(u)⁴⁻⁴ⁿ



 1 λ

n−4 2

1

, . . . , 1 λ

n−4

p2



M(y`<sub>1</sub>, . . . ,y`_p)



 1 λ

n−4 2

1

, . . . , 1 λ

n−4

p2





t

+o

∑

1 λⁿ_i⁻⁴

! .

HereM(y`<sub>1</sub>, . . . ,y`_p)is defined in the first section. Using now the fact thatρ(y`<sub>1</sub>, . . . ,y`_p)is the least eigenvalue of M(y`<sub>1</sub>, . . . ,y`_p), we derive that

h∂J(u),W₁(u)i ≤ −ρ(y`<sub>1</sub>, . . . ,y`_p)

∑

1 λⁿ_i⁻⁴

≤ −c

∑

1 λⁿ_i⁻⁴

+|∇K(a_i)|

λ_i

! +

∑

j6=i

ε_ij

! , since ρ(y`<sub>1</sub>, . . . ,y`_p) > 0, ^|∇K_λ^(ai)|

i = o ¹

λⁿ_i⁻⁴

and ε_ij ∼ ¹

(λ_iλ_j)ⁿ⁻2⁴. This concludes the proof of Lemma3.2.

Lemma 3.3. There exists a pseudo-gradient W₂ in V₂(p,ε) such that for any u=_∑i^p₌₁α_iPδ_(ai,λi)

∈V₂(p,ε), we have

h∂J(u)_,W₂(u)i ≤ −c

∑

1 λⁿ_i⁻⁴

+|∇K(a_i)|

λ_i

! +

∑

j6=i

ε_ij

! . W₂is bounded andmax₁≤i≤pλ_i(s)remains bounded along the associated flow lines.

Proof. Letu= _∑i^p₌₁α_iPδ(a_i,λ_i) ∈V2(_p,ε). We set in this regionW₂¹ =−_∑^p_i=1α_iλ_i^∂Pδ_∂λ^(ai,λi)

i . Using the same techniques of Lemma3.2, we have:

h∂J(u),W₂¹(u)i=2J(u)⁴⁻⁴ⁿ



 1 λ

n−4 2

1

, . . . , 1 λ

n−4

p2



M(y`<sub>1</sub>, . . . ,y`_p)



 1 λ

n−4 2

1

, . . . , 1 λ

n−4

p2





t

+o

∑

1 λⁿ_i⁻⁴

! .

Lete= (e₁, . . . ,e_p)∈_R^pbe a unit eigenvector associated toρ(y`<sub>1</sub>, . . . ,y`_p). Forγ>0 small enough, we denote byB(e,γ)the ball inS^p−1of centereand radiusγsatisfying∀X∈ B(e,γ):

XM(y`<sub>1</sub>, . . . ,y`_p)X^t < ¹

2ρ(y`<sub>1</sub>, . . . ,y`_p). In the next, we denote by

Γ=



 1 λ

n−4 2

1

, . . . , 1 λ

n−4

p2



. Thus, if _k^Γ_Γk ∈ B(e,γ), we get

h∂J(u),W₂¹(u)i ≤ ¹

2ρ(y`<sub>1</sub>, . . . ,y`_p)

1 λ

n−4 2

1

, . . . , 1 λ

n−4

p2

!

2

≤ −c

∑

1 λⁿ_i⁻⁴

≤ −c

∑

1 λⁿ_i⁻⁴

+|∇K(a_i)|

λ_i

! +

∑

j6=i

ε_ij

! .

Therefore, we takeW₂=W₂¹ in this region as the required vector field.

If _k^Γ_Γk 6∈ B(e,γ), in this case we move _k^Γ_Γk along the pathc(t) = _k(⁽¹₁⁻₋^t_t⁾₎^Γ_Γ⁺₊^te_tek. Observe that allλ_i,i= 1, . . . ,p remain bounded along this path. Therefore, the Palais–Smale condition is satisfied along this piece of flow line. Let in this case

W₂(u) =

∑

α_iλ˙_i

∂Pδ(a_i,λi)

∂λ_i . where

λ˙_i =− ⁴

n−4k_Γkλ

n 2

i

k_Γke_i−_Γi

kc(0)k − ^ci(0)<k_Γke−_Γ,c(0)>

kc(0)k³

.

Herec_i ande_i are thei^th component ofcande respectively. Notice that we can chooseesuch that e_i > 0,∀i = 1, . . . ,p. This is due to the fact that m(y`<sub>i</sub>,y`_i) > 0,∀i = 1, . . . ,p. Using the estimates (3.1), we have

h∂J(u),W₂(u)i ≤ ¹

2k_Γk^{2 ∂}

∂t

c(t)M(y_{`1</sub>, . . . ,y<sub>`p})c(t)^t kc(t)k²

/t=0

≤ −ck_Γk²_, since _∂t^{∂ c(t)M(y`1,...,y`p)c(t)}

t

kc(t)k²

/t=0≤ −c, see [7, p. 650]. Therefore, h∂J(u),W₂(u)i ≤ −c

p i

∑

1 λⁿ_i⁻⁴

+ |∇K(a_i)|

λ_i

+

∑

j6=i

ε_ij

.

Lemma 3.4. There exists a pseudo-gradient W₃ in V₃(p,ε) such that for any u=_∑i^p₌₁α_iPδ_(ai,λi)

∈V3(p,ε), we have

h∂J(u),W₃(u)i ≤ −c p

i

∑

1 λⁿ_i⁻⁴

+ |∇K(a_i)|

λ_i

+

∑

j6=i

ε_ij

. W₃is bounded andmax₁≤i≤pλ_i(s)remains bounded along the associated flow lines.

Proof. Letu= _∑i^p₌₁α_iPδ_(ai,λi) ∈V₃(p,ε)and leti₁, . . . ,i`be the indices such thatm(y_ij,y_ij)<0.

We point out that the only cases where m(y,y)is negative is when β(y) = n−4. Otherwise m(y,y)∼ H(y,y)is therefore positive. Define

I =

i, 1≤i≤ p, λ_i ≥ ¹ 2 min

1≤j≤`λi_j

and J ={1, . . . ,p} \I.

Let MJ = (m_ij)_1≤i,j≤]J be the matrix defined by:

m_ii =mm(y`<sub>i</sub>,y`_i), ∀i∈ J and m_ij =mm(y`<sub>i</sub>,y`_j), ∀1≤i6= j≤]J.

Observe thatm_ii is positive∀i∈ J. Thus, we can apply the arguments of Lemmas3.2 and3.3.

Letρ(M_J)be the least eigenvalue ofM_J. Define form>0 and small W₃¹=m

(₁+_signρ(M_J))W₁

∑

i∈J

α_iPδ_(ai,λi)

+ (₁−_signρ(M_J))W₂

∑

i∈J

α_iPδ_(ai,λi)

, where signρ(M_J) = 1 if ρ(M_J) > 0 and signρ(M_J) = −1 if ρ(M_J) < 0. Using Lemmas 3.2 and3.3we have

h∂J(u),W₃¹(u)i ≤ −c

∑

i∈J

1 λⁿ_i⁻⁴

+ |∇K(a_i)|

λ_i

+

∑

j6=i,i,j∈J

ε_ij

+O

∑

i∈J,j∈I

ε_ij

.

Observe that our upper bound is limited to those indicesi ∈ J. We must add the indices i∈ I. For this let

W₃²(u) =−

∑

α_ijλ_ij

∂Pδ_(a

ij,λ_ij)

∂λ_ij . Using (3.1) and the fact thatm(y_ij,y_ij)<0,∀1≤ j≤ `, we get

h∂J(u),W₃²(u)i ≤ −c _`

j

∑

1 λⁿ_i⁻⁴

j

+ |∇K(a_ij)|

λ_ij

+

∑

∑

k6=ij

ε_ijk

≤ −c

∑

i∈I

1 λⁿ_i⁻⁴

+ |∇K(a_i)|

λ_i

+

∑

i∈I,j∈J

ε_ij

.

Therefore, formsmall, we derive that h∂J(u),W₃(u)i ≤ −c

p i

∑

1 λⁿ_i⁻⁴

+ |∇K(a_i)|

λ_i

+

∑

j6=i

ε_ij

,

whereW₃(u) =W₃¹(u) +W₃²(u)_.