A note on a second order PDE with critical nonlinearity

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A note on a second order PDE with critical nonlinearity

Khadijah Sharaf

^B

Department of Mathematics, King Abdulaziz University, P.O. 80230, Jeddah, Kingdom of Saudi Arabia Received 26 March 2018, appeared 14 February 2019

Communicated by Dimitri Mugnai

Abstract. In this work, we are interested in a nonlinear PDE of the form: −_∆u = K(x)uⁿⁿ⁺⁻²²,u > 0 on Ω and u = 0 on ∂Ω, where n ≥ 3 and Ω is a regular bounded domain of Rⁿ. Following the results of [K. Sharaf, Appl. Anal. 96(2017), No. 9, 1466–

1482] and [K. Sharaf, On an elliptic boundary value problem with critical exponent, Turk. J. Math., to appear], we provide a full description of the loss of compactness of the problem and we establish a general index account formula of existence result, when the flatness order of the functionKat any of its critical points lies in(1,∞).

Keywords: nonlinear PDE, variational problem, critical points at infinity.

2010 Mathematics Subject Classification: 35J60, 35J65.

1 Introduction and main results

In this work, we consider the existence of smooth solutions of











−_∆u=K(x)uⁿⁿ⁺⁻²², u>0 inΩ, u=0 on∂Ω,

(1.1)

wheren ≥3,Ωis a regular bounded domain ofRⁿ andKis a given function onΩ.

The original interest of such problem grew out of prescribing scalar curvature equations, see for example [1,3,7–9,11,12,15,16,22] and the references therein.

Equation (1.1) can be expressed as a variational problem in H₀¹(_Ω). However, the variational structure presents a loss of compactness since the exponent ⁿ_n⁺₋²₂is critical andH₀¹(_Ω),→ Lⁿ²ⁿ⁻²(_Ω)is not compact.

The first contributions to (1.1) concern the caseK = 1, where Bahri-Coron and Pohozaev proved that the resolution of (1.1) depends on the topology of the domainΩ, see [4] and [17].

For K 6= 1, many conditions onKwere provided to ensure existence of solutions of (1.1), see for example [6,13,14,18–21].

Recently in [19] and [21], we studied problem (1.1) and provided existence and compactness results under the following four conditions:

BEmail: kh-sharaf@yahoo.com

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(A) ^∂K

∂ν(x)<0, ∀x ∈_∂Ω.

Hereνis the unit outward normal vector on∂Ω.

(f)_β Kis aC¹-positive function such that at any critical pointyofK, there exists a real number β= β(y)satisfying the following expansion:

K(x) =K(y) +

∑

n k=1

b_k|(x−y)_k|^β+o(|x−y|^β),

for all x ∈ B(y,ρ₀) where ρ₀ is a positive fixed constant, b_k = b_k(y) ∈ _R\ {0}, ∀k = 1 . . . ,n, and











−ⁿ−2 n

c₁ K(y)

∑

n k=1

b_k(y) +c₂n−2

2 H(y,y)6=0, ∀y∈ K_n₋₂,

∑

n k=1

b_k(y)6=0, ∀y∈ K<n−2. Here

K_n₋₂ _:= {y ∈_Ω,∇K(y) =_{0 and}β(y) =n−₂}_, K_<_n₋₂ := {y ∈_Ω,∇K(y) =0 andβ(y)<n−2}, c₁=

Z

Rⁿ

|z₁|ⁿ⁻²

(1+|z|²)ⁿ^dz, ^c² =

Z

Rⁿ

dz (1+|z|²)ⁿ⁺²²^,

and H(·_,·) is the regular part of the Green function G(·_,·) _of (−_∆) under the zero- Dirichlet boundary condition. Let us denote also

K⁺_n₋₂:= (

y∈ K_n₋₂, −ⁿ−2 n

c₁ K(y)

∑

n k=1

b_k(y) +c₂n−2

2 H(y,y)>0 )

, K_>_n₋₂_:={y∈ _Ω,∇K(y) =_{0 and}β(y)>n−₂}_.

For anyτ_p:= (y_`₁, . . . ,y_`_p)∈(K_n⁺₋₂∪ K_>_n₋₂)^p,p≥1, such thaty_`_i 6=y_`_j,∀1≤i6= j≤ p, we set the matrixM(τp) = (m_ij)₁_≤_i,j_≤_pdefined by

m_ii=m(y`_i,y`_i)

=











− ¹

K(y`_i)ⁿ⁻²²

n−2 n

c₁ K(y`_i)

∑

n k=1

b_k(y`_i)−c₂n−2

2 H(y`_i,y`_i)

!

ifβ(y`_i) =n−2, n−2

2

c₂

K(y`_i)ⁿ⁻²²^H(y`_i,y`_i) ifβ(y`_i)> n−2,

∀i=1, . . . ,pand

m_ij =m(y`_i,y`_j) =−ⁿ−2 2 c2

G(y_`_i,y_`_j)

K(y`_i)K(y`_j)

n−2 4

, for 1≤i6= j≤ p.

(B) Assume that the least eigenvalueρ(τ_p)_of M(τ_p)is not zero.

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The last assumption is

(C) ^_







β(y)∈ (1,n−2] ∀ys.t.∇K(y) =0, or

β(y)∈ [n−_2,_∞) ∀ys.t.∇K(y) =_0.

Thus, it becomes of interest to study the equation (1.1) in the mixed case situation; that is when there exists some critical points y of K having β(y) < n−2 and other having β(y) ≥ n−2 and therefore get global compactness and existence results under (f)_β-condition forβ varies in (_1,_∞)_{. Define}

C_<^∞_n₋₂:= (

(y`₁, . . . ,y`_p)∈ K_<^p_n₋₂,p≥1, y`_i 6=y`_j,∀i6=jand −

∑

n k=1

b_k(y`_i)>0, ∀i=1, . . . ,p )

, C_≥^∞_n₋₂:=ⁿ(y`₁, . . . ,y`_p)∈(K⁺_n₋₂∪ K>n−2)^p,p≥1, y`_i 6=y`_j,∀i6= jandρ(y`₁, . . . ,y`_p)>0o

. The first result of this paper describes the loss of compactness and the concentration phenomenon of the problem (1.1).

For a ∈ _Ω _and λ _{1, let} Pδ₍_a,λ₎ be the almost solution of the Yamabe-type problem defined in the next section.

Theorem 1.1. Assume that (1.1) has no solution. Under conditions (A),(B) and(f)_β,β > 1, the critical points at infinity of the associated variational problem (see definition(2.1)) are:

(y`₁, . . . ,y`_p)_∞:=

∑

p j=1

1

K(y`_j)ⁿ⁻²² ^Pδ⁽^y^`^j^,∞⁾^,

where (y`₁, . . . ,y`_p) ∈ C_<^∞_n₋₂ ∪ C_≥^∞_n₋₂ ∪ C_<^∞_n₋₂ × C_≥^∞_n₋₂. The index of (y`₁, . . . ,y`_p)_∞ is i(y_`₁, . . . ,y_`_p)_∞= p−₁+_∑_i^p₌₁n−ei(y_`_i), whereei(y) =]{b_k(y)_{, 1}≤ k≤n, s. t. b_k(y)<₀}.

The above characterization allow us to derive a global index formula of existence.

Theorem 1.2. Let Ω be a regular bounded domain of Rⁿ,n ≥ 3 and let K : Ω → _R be a given function satisfying(A),(B)and(f)_β,β∈ (1,∞).If

(y`₁,...,y`_p)∈C_<^∞_n₋₂∪C

∑

_≥^∞_n₋₂∪(C_<^∞_n₋₂×C_≥^∞_n₋₂)

(−1)ⁱ⁽^y^`¹^,...,y^`^p⁾^∞ 6=1, then(1.1)has a solution.

Remark 1.3. For an explicit example of function Ksatisfying the hypotheses of Theorem1.2, let Ωbe the unit ball Bⁿ of Rⁿ,n ≥ 4 and let βbe a real larger than n−2. For anyX ∈ _Rⁿ, we define

f₁(X) =1−

∑

n k=1

|x_k|^β and f₂(x) =

∑

n k=1

|x_k|³². For any integer k₀ ≥2, we denotey_k₀ = _k¹

0, 0, . . . , 0

. Letθ be the cut-off function defined by:

θ(t) =1 ift < ¹

4k₀, θ(t) =0 ift> ¹

2k₀ and θ⁰(t)<0 if 1

4k₀ <t < ¹ 2k₀.

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Now letK:Bⁿ→R, such that∀X∈_Bⁿ:

K(X) =θ(kX−y_k₀k)f₁(X−y_k₀) +θ(kX+y_k₀k)f₁(X+y_k₀)

+θ(kXk)f₂(X)−1−θ(kX−y_k₀k)−θ(kX+y_k₀k)−θ(kXk)kXk².

Observe thatK admits three critical pointsy_k₀,−y_k₀ and 0_Rⁿ. By constructionK satisfies(f)_β condition near its critical points with

β(y_k₀) = β(−y_k₀) =β>n−₂ _and β(₀_Rn) = ³

2 <n−_2.

According to the result of Theorem 1.1, 0_Rⁿ does not give a critical point at infinity since

−_∑ⁿ_k₌₁b_k(₀_Rn) = −n< _{0. However}y_k₀ and−y_k₀ correspond to two critical points at infinity (y_k₀)_∞and(−y_k₀)_∞respectively. In addition, the pair(y_k₀,−y_k₀)corresponds to a critical point at infinity if and only ifρ(y_k₀,−y_k₀)>0 whereρis the least eigenvalue of the matrix

M = ⁿ−2 2

H(y_k₀,y_k₀) −G(y_k₀,−y_k₀)

−G(y_k₀,−y_k₀) H(−y_k₀,−y_k₀)

It is easy to see thatρ(y_k₀,−y_k₀)>0 if and only if

H(y_k₀,y_k₀)H(−y_k₀,−y_k₀)−G²(y_k₀,−y_k₀)>0,

since Tr(M) > 0. We know from [5, Remark 3, p. 72] , that G(X,Y) → −_∞ if |X−Y| → 0.

Thus fork₀large enough,ρ(y_k₀,−y_k₀)<0. Therefore, the only critical points at infinity in our statement are,

(y_k₀)_∞ and(−y_k₀)_∞withei(y_k₀) =ei(−y_k₀) =n.

It follows that the function K satisfies the index formula of Theorem 1.2 and the assumption(B). Concerning the assumption (A), observe that outside B y_k₀,_2k¹

0

∪B −y_k₀,_2k¹

0

∪ B 0_Rⁿ,_2k¹

0

, the functionKis equals to−kXk². Therefore,DK(X) =−2Xand on the boundary ofBⁿ,ν_X= Xand hence

∂K

∂ν(X) =hDk(X),ν_Xi=−2.

Our argument follows the critical points at infinity theory of A. Bahri [2]. In the next section, we will state the general framework of the variational structure of (1.1). After that we will characterize the critical points at infinity and prove Theorems1.1 and1.2.

2 General framework

Equation (1.1) is equivalent to finding the critical points of the following functional J(u) =

R

Ω|∇u|² R

ΩK(x)uⁿ²ⁿ⁻²dxⁿ⁻_n²

, u∈_Σ⁺.

Here

Σ= (

u∈ H₁⁰(_Ω), s.t.kuk_H0 1(_Ω) =

_Z

Ω|∇u(x)|²dx ¹₂

= 1 )

,

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and

Σ⁺=u∈_Σ_, u>₀ _.

It is known that J fails the Palais–Smale condition. The sequences which violate the (P. S) condition has been analyzed as follows. For a∈_Ωandλ>0, define

δ_a,λ(x) =c₀

λ

1+λ²|x−a|² ⁿ⁻₂²

, (2.1)

where c₀is a fixed positive constant. The familyδ_a,λ, a∈ _Ωandλ>0 are the only solutions of

(−_∆u=uⁿⁿ⁺⁻²²,

u>0 inRⁿ. (2.2)

DefinePδ_a,λ onΩbe the unique solution of











−_∆u= δ

n+2 n−2

a,λ

u>0 inΩ, u=0 on∂Ω.

(2.3)

By the maximum principal and regularity arguments, Pδ_a,λ is smooth and positive onΩ.

For ε > 0 and p ∈ _N^∗, let V(p,ε) be the set of all functions u ∈ _Σ⁺ such that there exists (a₁, . . . ,a_p)∈_Ω^p,λ₁, . . . ,λ_p> ε⁻¹andα₁, . . . ,α_p>0 satisfying

u−

∑

p i=1

α_iPδa_i,λ_i

<ε,

with

J(u)ⁿ⁻ⁿ²α

4 n−2

i K(a_i)−1

<εandε_ij = ^λ_λⁱ

j + ^λ^j

λ_i +λ_iλ_j|a_i−a_j|²

−(n−2)

2 < ε ∀i6=j.

For any sequence (u_k)_k _in _Σ⁺ failing the (P.S) condition, there exists an extracted subse- quence (u_k_`)_` such thatu_k_` ∈V(p,ε_k_`),∀`∈_{N. Here}p ∈_N^∗ andε_k_` →0 when `→+_{∞. See} [4] and [23].

The following parametrization ofV(p,ε)was given in [4]. For any u ∈ V(p,ε), u can be written as

∑

p i=1

¯

α_iPδ_a_¯_i_{, ¯}_λ_i+v, wherev ∈ H₀¹(_Ω)and satisfies

(V₀)

hv,ψi=0 forψ∈

Pδ_a_i_,λ_i,∂Pδ_a_i_,λ_i

∂λ_i ,∂Pδ_a_i_,λ_i

∂a_i , i=1, . . . ,p

,

h·,·idenotes the inner product onH₀¹(_Ω)associated to the normk · k, and ¯α_i, ¯a_i, ¯λ_i,i=1, . . . ,p are the unique solution of

∑^pi=₁α_iPδmin_ai_,λ_i∈V(p,ε)

u−

∑

p i=1

α_iPδ_a_i_,λ_i .

In the following, we show that thev-part ofuis negligible with respect to the concentration phenomenon. See [2,4].

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There is aC¹-map which to each (α_i,a_i,λ_i)such that∑_i^p=1α_iPδ_a_i_,λ_i belongs toV(p,ε)asso- ciatesv=v(α_i,a_i,λ_i)such thatvis the unique solution of the following minimization problem

min (

J

∑

p i=1

α_iPδ_a_i_,λ_i +v

!

, v∈ H₀¹(_Ω) and satisfies(V₀) )

. Moreover, there exists a change of variablesv−v→Vsuch that

J

∑

p i=1

α_iPδ_a_i_,λ_i+v

!

= J

∑

p i=1

α_iPδ_a_i_,λ_i+v

!

+kVk². The following definition is extracted from [2].

Definition 2.1([2]). 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 the above argument,u(s)can be written as:

u(s) =

∑

p i=1

α_i(s)Pδa_i(s),λ_i(s) +^v(s).

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

∑

p i=1

α_iPδy_i,∞ or(y₁, . . . ,yp)_∞ such a critical point at infinity.

3 Critical points at infinity

In this section we prove Theorems1.1and1.2. We start by the following result which describes the concentration phenomenon of the variational structure associated to the problem (1.1).

Theorem 3.1. Under the assumptions(A),(B)and(f)_β, β > 1. There exists a decreasing bounded pseudo-gradient W in V(p,ε),p≥1,satisfying the following:

There exists c>0such that for any u=_∑_i^p₌₁α_iPδa_i,λ_i ∈V(p,ε)we have (i) h∂J(u),W(u)i ≤ −c

∑

p i=1

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

+ |∇K(a_i)|

λ_i

! +

∑

j6=i

ε_ij

! . Moreover, the only case whereλ_i(t)_, i=_{1, . . . ,}p,tends to∞is when a_i(t)goes to y_`_i, ∀i=_{1, . . . ,}p such that(y`₁, . . . ,y`_p)∈ C_<^∞_n₋₂∪ C_≥^∞_n₋₂∪(C_<^∞_n₋₂× C_≥^∞_n₋₂).

Here(C_<^∞_n₋₂ andC_≥^∞_n₋₂)are defined in the first section.

Before presenting the proof of Theorem3.1, we recall the following result which describes the concentration phenomena of the problem whenβ∈ (_1,n−₂), see [19, Section 3].

Theorem 3.2([19]). Under the assumptions of Theorem3.1withβ∈]_1,n−₂[, there exists a decreasing bounded pseudo-gradient W₁ satisfying(i)of Theorem3.1, for any u = _∑_i^p₌₁α_iPδ_a_i_,λ_i ∈ V(p,ε) and the only case whereλ_i(t)goes to+_∞, i=1, . . . ,p is when a_i(t)goes to y`_i with(y`₁, . . . ,y`_p)∈ C_<^∞_n₋₂.

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Notice that the case of β= n−2 was handled also in [19].

Recently we proved the following result which describes the concentration phenomena in the case whereβ∈[n−2,+_∞).

Theorem 3.3 ([21]). Under the assumptions of Theorem 3.1 with β ∈ [n−2,∞), there exists a decreasing bounded pseudo-gradient W₂ satisfying(i)of Theorem3.1, for any u = _∑^p_i₌₁α_iPδa_i,λ_i ∈ V(_p,ε) and the only case, where λ_i(_t)_, _i = _{1, . . . ,}_{p goes to} +_∞ _{is when a}_i(_t) _{goes to y}_`_i _with (y`₁, . . . ,y`_p)∈ C_≥^∞_n₋₂.

The complete construction of the required pseudo-gradientW₂inV(p,ε)was given in [21].

We provide in the next the construction ofW₂in a specific region R^δ_>_n₋₂(p,ε)where R^δ_>_n₋₂(p,ε):=

u=

∑

p i=1

α_iPδ₍_a

i,λ_i) ∈V(p,ε),a_i ∈ B(y`_i,ρ0)

y_`_i ∈ K_>_n₋₂,λⁿ_i⁻²|a_i−y_`_i|^β⁽^y^`ⁱ⁾<δ,∀i=1, . . . ,pandy_`_i 6=y_`_j,∀i6= j

. Hereδis a small positive constant. Let u=_∑^p_i₌₁α_iPδ₍

ai,λ_i)∈ R^δ_>_n₋₂(p,ε).

Case 1: If ρ(y`₁, . . . ,y`_p) > 0. We use the expansion (3.1) below. Since J(u)ⁿⁿ⁻²α

4 n−2

j K(a_j) = 1+o(1),∀j=1, . . . ,p, (3.1) becomes

∂J(u),α_iλ_i

∂Pδ(a_i,λi)

∂λ_i

= −2c₂J(u)²⁻²ⁿ

"

∑

i6=j

1

(K(a_i)K(a_j))ⁿ⁻⁴² ^λⁱ

∂ε_ij

∂λ_j + ⁿ−2 2

H(a_i,a_j) (λ_iλ_j)ⁿ⁻²²

!

+ⁿ−2 2

1 K(a_i)ⁿ⁻²²

H(y`_i,y`_i) λⁿ_i⁻²

#

+O

|a_i−y`_i|^β⁽^y^`ⁱ⁾+o

∑

k6=j

ε_jk+ ^H(a_i,a_k) (λ_iλ_k)ⁿ⁻²²

!!

. Observe that asδ small we have,

|a_i−y`_i|^β⁽^y^`ⁱ⁾ =o 1 λⁿ_i⁻²

! . Moreover, since|a_i−a_j| ≥ρ₀,∀i6= j, we have

λ_i∂ε_ij

∂λ_i

=−ⁿ−2 2

1

(λ_iλ_j|a_i−a_j|²)ⁿ⁻²²(₁+o(₁))_. Therefore ,

λ_i∂ε_ij

∂λ_i

+ ⁿ−2 2

H(a_i,a_j)

(λ_iλ_j)ⁿ⁻²² =−ⁿ−2 2

G(y`_i,y`_j)

(λ_iλ_j)ⁿ⁻²² (₁+o(₁))_. Thus

∂J(u),α_iλ_i

∂Pδ₍_a_i_,λ_i₎

∂λ_i

=−2J(u)²⁻²ⁿ

"

∑

i6=j

m(y`_i,y`_j)

(λ_iλ_j)ⁿ⁻²² + ^m(y`_i,y`_i) λⁿ_i⁻²

# +o

∑

p k=1

1 λⁿ_k⁻²

! , where the coefficientsm(y_`_i,y_`_j)_{, 1}≤i,j≤ pare defined in the first section.

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For anyi=1, . . . ,pwe set ˙λ_i =λ_i. The corresponding pseudo-gradient is W₂(u) =

∑

p i=1

α_iλ_i

∂Pδ₍_a_i_,λ_i₎

∂λ_i . From the latest expansion,W₂ satisfies

D

∂J(u),W₂(u)^E

= −2J(u)²⁻²ⁿ



 1 λ

n−2 2

1

, . . . , 1 λ

n−2

p2



M(y`₁, . . . ,y`_p)



 1 λ

n−2 2

1

, . . . , 1 λ

n−2

p2





t

+o

∑

p k=1

1 λⁿ_k⁻²

!

≤ −ρ(y`₁, . . . ,y`_p)

∑

p i=1

1 λⁿ_i⁻²

,

sinceρ(y`₁, . . . ,y`_p)is the least eigenvalue of M(y`₁, . . . ,y`_p). Using the fact that

|∇K(a_j)|

λ_j ∼ |a_j−y`_j|^β⁻¹

λ_j =O 1

λⁿ_j⁻²

!

and for anyi6= j, we have

ε_ij ∼ ¹

(λ_iλ_j)ⁿ⁻²² =O 1 λⁿ_i⁻²

!

+O 1

λⁿ_j⁻²

! , we get

D

∂J(u),W₂(u)^E≤ −c

∑

p i=₁

|∇K(a_i)|

λ_i + ¹ λⁿ_i⁻²

! +

∑

i6=j

ε_ij

! . Case 2: Ifρ(y_`₁, . . . ,y_`_p)<0. This is the opposite situation od the case 1. Thus

W₂(u) =−

∑

p i=1

α_iλ_i

∂Pδ₍_a_i_,λ_i₎

∂λ_i , satisfies the requirement of Theorem3.3.

Proof of Theorem3.1. Letu =_∑_i^p₌₁α_iPδa_i,λ_i ∈ V(p,ε)_,p ≥1. Following the above two results, the only case that we will consider here is whenucan be written as

u=

∑

q i=₁

α_iPδa_i,λ_i+

∑

p i=_q+₁

α_iPδa_i,λ_i =:u₁+u₂, where 1≤q< p and

u₁ ∈R₁ :=

u=

∑

s i=1

α_iPδ₍_a_i_,λ_i₎∈V(s,ε),s≥1, s.t. a_i ∈ B(y_`_i,ρ₀),

with β(y`_i)<n−2,∀i=1, . . . ,s

,

(9)

u₂∈ R₂:=

u=

∑

s i=₁

α_iPδ₍_a_i_,λ_i₎ ∈V(s,ε),s≥1, s.t.a_i ∈ B(y`_i,ρ₀),

withβ(y`_i)≥n−2,∀i=1, . . . ,s

. Let us denote byW₁ the pseudo-gradient given by Theorem 3.2 andW₂ the pseudo-gradient given by Theorem3.3. In order to construct the required pseudo-gradientW of Theorem3.1, we distinguish three cases. Letδ be a fixed positive constant small enough.

•_{Case 1.}

u₁ ∈ (

u=

∑

s i=1

α_iPδ(a_i,λi)∈ R₁, s.t., λ_i|a_i−y`_i|<δ,∀i=1, . . . ,s, and (y`₁, . . . ,y`_s)∈ C_<^∞_n₋₂ )

and u₂ ∈

( u=

∑

s i=1

α_iPδ₍_a_i_,λ_i₎∈ R₂, s.t., λ_i|a_i−y_`_i|<_δ,∀i=_{1, . . . ,}s, and (y_`₁, . . . ,y_`_s)∈ C_≥^∞_n₋₂ )

. According to the construction of [19] and [21], the vector fieldsW₁ andW₂ in these regions are defined as follows:

W₁(u₁):=

∑

q i=1

α_iλ_i

∂Pδ₍_a_i_,λ_i₎

∂λ_i and W₂(u₂) =

∑

p i=q+1

α_iλ_i

∂Pδ₍_a_i_,λ_i₎

∂λ_i .

Observe that all the components λ_i of the corresponding flow lines satisfies the differential equation

λ˙ =λ_i, ∀i=1, . . . ,p.

In this case, we set

W(u) =We₁(u) +We₂(u),

where We₁(u) := W₁(u₁) andWe₂(u) := W₂(u₂). Following the computation of [19] and [20], we have

∂J(u)_,α_iλ_i∂Pδ_a_i_,λ_i

∂λ_i

= −2c₂ J(u) K(a_i)

∑

j6=i

α_iα_j λ_i∂ε_ij

∂λ_i

+ ^H(a_i,a_j) (λ_iλ_j)ⁿ⁻²²

!

+2α²_i J(u) K(a_i)









 n−₂

2 c₁∑ⁿk=₁b_k(y`_i) λ^β

(y`_i) i

, if β(y`_i)<n−2 n−2

2 c₁∑ⁿ_k=1b_k(y`_i) λ

β(y`_i) i

−c2

H(y`_i,y`_i) λⁿ_i⁻²

, if β(y`_i) =n−2

−c₂H(y`_i,y`_i) λⁿ_i⁻²

, if β(y`_i)>n−2 +O

|a_i−y`_i|^β+o

∑

j6=i

ε_ij+ ^H(a_i,a_j) (λ_iλ_j)ⁿ⁻²²

! + ¹

λ_iⁿ⁻²

!

. (3.1)

Since we have|a_i−a_j| ≥ρ₀,∀i6= j, we obtain λ_i∂ε_ij

∂λ_i = −ⁿ−2 2

1

|a_i−a_j|²λ_iλ_jⁿ⁻₂²

+o 1

(λ_iλ_j)ⁿ⁻²²

!

≤ −cε_ij. (3.2)

(10)

Using the fact thatKsatisfies(f)_β assumption around eachy`_i, we derive

∇K(a_i) λ_i = o

1 λ^β_i

and|a_i−y`_i|^β = o

1 λ^β_i

. We therefore have

h∂J(u),We₁(u)i ≤ −c

∑

q i=1

1 λ^β_i

+ |∇K(a_i)|

λ_i

!

+

∑

1≤j6=i≤q

ε_ij

!

+O

∑

1≤i≤q,q+1≤j≤p

ε_ij

!

, (3.4) and

D

∂J(u),We2(u)^E

≤ −c

∑

p i=q+₁

+|∇K(a_i)|

λ_i

!

+

∑

q+1≤j6=i≤p

ε_ij

!

+O

∑

q+₁≤i≤p, 1≤j≤q

ε_ij

!

. (3.5)

For any 1≤i≤qand for anyq+1≤ j≤ pwe claim that ε_ij =o

1 λ^β_i

+o

1 λⁿ_j⁻²

. (3.6)

Indeed, since|a_i−a_j| ≥ρ₀, we haveε_ij ∼ ₍ ¹

λ_iλ_j)ⁿ⁻₂². Let M1. Ifλ_i < _Mλ_j_{, then} 1

(λ_iλ_j)ⁿ⁻₂² ≤ ^M

n−2 2

λⁿ_i⁻²

=o

1 λ^β_i

, sinceβ<n−2 for 1≤i≤q. Ifλ_i > Mλ_j, then

1

(λ_iλ_j)ⁿ⁻₂² ≤ ¹ Mⁿ⁻²²

1 λⁿ_j⁻²

=o

1 λⁿ_j⁻²

, asM large.

Thus (3.6) follows. The inequalities (3.4) and (3.5) with estimates (3.6) yield h∂J(u)_,W(u)i ≤ −c

p i

∑

=1

+ |∇K(a_i)|

λ_i

+

∑

j6=i

ε_ij

.

Observe that throughW,λ_i(t)tends to∞,∀i=1, . . . ,p; it is a concentration phenomenon.

•Case 2.

u₁6∈

( u=

∑

s i=1

α_iPδ(a_i,λi)∈ R₁,s.t.,λ_i|a_i−y`_i|<δ,∀i=1, . . . ,s, and(y`₁, . . . ,y`_s)∈ C_<^∞_n₋₂ )

. Three possibilities may occur. Either there existsi₀, 1 ≤ i₀ ≤ qsuch that λ_i₀|a_i₀−y`_i

0| ≥δ or

there exists i₁, 1 ≤ i₁ ≤ qsuch that −_∑ⁿ_k₌₁b_k(y_i₁)< 0 or there existi 6= jsuch that y_`_i = y_`_j.

(11)

In all possibilities, it was constructed in [19], section 3, a pseudo-gradientW₁ along which the max₁≤i≤q(λ_i(s))remains bounded and satisfies

h∂J(u₁),W₁(u₁)i ≤ −c

∑

q i=1

1 λ^β_i

+|∇K(a_i)|

λ_i

!

+

∑

1≤j6=i≤q

ε_ij

!

. (3.7)

Therefore, forWe₁(u) =W₁(u₁), we set D

∂J(u),We₁(u)^E≤ −c

∑

q i=1

1 λ_i^β

+ |∇K(a_i)|

λ_i

!

+

∑

1≤j6=i≤q

ε_ij

!

+O

∑

1≤i≤q,q+1≤j≤p

ε_ij

!

. (3.8) Denote byi₁ an index such that

λ^β_i

1 =inf{λ^β_i,i=1, . . . ,q} and let us denote

L=

j, 1≤j≤ p; λ^β_j ≥ ¹ 2λ^β_i

1

. It is easy to see that we can appear all − ¹

λ_i^β, i ∈ L in the upper bound of (3.8). In order to make appear all−^|∇^K⁽^aⁱ^)|

λi , i∈L, let us recall the following estimate obtained in [19, Section 3].

∂J(u),α_i 1 λ_i

∂Pδ_a_i_,λ_i

∂(a_i)_k

= −(n−2)α²_iJ(u) ^b^k

λ_iK(a_i)^β^sign(a_i−y`_i)_k|(a_i−y`_i)_k|^β⁻¹ +O

[β] j

∑

=₂

|a_i−y`_i|^β⁻^j λ_i^j

! +O

1 λ_i^β

+O

∑

j6=i

1 λ_i

∂ε_ij

∂a_i

! .

(3.9)

Let

Y_i(u) =

∑

n k=1

b_k sign(a_i−y`_i)_k ¹ λ_i

∂Pδ₍_a_i_,λ_i₎

∂(a_i)_k ^.

For each indexi∈L\ {1, . . . ,q}, we move the concentration point a_i with respect toY_i. Using (3.9), the corresponding variation of J is given by:

h∂J(u),Y_i(u)i ≤ − ^c³ K(a_i)^α

2iJ(u)

∑

n k=1

b_k²|a_i−y`_i|^β⁻¹ λ_i +O

[β]

∑

j=2

|a_i−y_`_i|^β⁻^j λ^j_i

! +O

1 λ^β_i

+O

∑

j6=i

ε_ij

! .

(3.10)

For any j=2, . . . ,[β], we claim that

|a_i−y_`_i|^β⁻^j λ^j_i

=O

1 λ^β_i

+o |a_i−y_`_i|^β⁻¹ λ_i

!

. (3.11)

Indeed, let M 1. If|λ_i(a_i−y`_i)| ≤ M, we have

|a_i−y`_i|^β⁻^j λ^j_i

=O

1 λ^β_i

,