sup*inf inequalities for scalar curvature equation in dimension 4 and 5

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sup*inf inequalities for scalar curvature equation in dimension 4 and 5

Samy Skander Bahoura

To cite this version:

Samy Skander Bahoura. sup*inf inequalities for scalar curvature equation in dimension 4 and 5. 2014.

�hal-00923289�

SUP × INF INEQUALITIES FOR THE SCALAR CURVATURE EQUATION IN DIMENSIONS 4 AND 5.

SAMY SKANDER BAHOURA

ABSTRACT. We consider the following problem on bounded open setΩofRⁿ: (

−∆u=V u

n+2

n−2 in Ω⊂Rⁿ, n= 4,5,

u >0 in Ω.

We assume that :

V ∈C^1,β(Ω), 0< β≤1 0< a≤V ≤b <+∞,

|∇V| ≤AinΩ,

|∇^1+βV| ≤BinΩ.

then, we have asup×infinequality for the solutions of the previous equation, namely:

(sup

K

u)^β×inf

Ω u≤c=c(a, b, A, B, β, K,Ω), forn= 4, and,

(sup

K

u)^1/3×inf

Ω u≤c=c(a, b, A, B, K,Ω), forn= 5, andβ= 1.

1. I

M

R

We work on Ω ⊂⊂ R

and we consider the following equation:

( −∆u = V u

in Ω ⊂ R

, n = 4, 5,

u > 0 in Ω. (E)

with,



 

 

 

 

V ∈ C

(Ω),

0 < a ≤ V ≤ b < +∞ in Ω,

|∇V | ≤ A in Ω,

|∇

V | ≤ B in Ω.

(C

)

Without loss of genarality, we suppose Ω = B

(0) the unit ball of R

. The corresponding equation in two dimensions on open set Ω of R

, is:

−∆u = V (x)e

, (E

)

The equation (E

) was studied by many authors and we can find very important result about a priori estimates in [8], [9], [12], [16], and [19]. In particular in [9] we have the following interior estimate:

sup

K

u ≤ c = c(inf

Ω

V, ||V ||

, inf

Ω

u, K, Ω).

And, precisely, in [8], [12], [16], and [19], we have:

C sup

K

u + inf

Ω

u ≤ c = c(inf

Ω

V, ||V ||

, K, Ω), and,

1

sup

K

u + inf

Ω

u ≤ c = c(inf

Ω

V, ||V ||

, K, Ω).

where K is a compact subset of Ω, C is a positive constant which depends on inf

V sup

V , and, α ∈ (0, 1].

For n ≥ 3 we have the following general equation on a riemannian manifold:

−∆u + hu = V (x)u

, u > 0. (E

)

Where h, V are two continuous functions. In the case c

h = R

the scalar curvature, we call V the prescribed scalar curvature. Here c

is a universal constant.

The equation (E

) was studied a lot, when M = Ω ⊂ R

or M = S

see for example, [2-4], [11], [15]. In this case we have a sup × inf inequality.

In the case V ≡ 1 and M compact, the equation (E

) is Yamabe equation. T.Aubin and R.Schoen proved the existence of solution in this case, see for example [1] and [14] for a complete and detailed summary.

When M is a compact Riemannian manifold, there exist some compactness result for equation (E

) see [18]. Li and Zhu see [18], proved that the energy is bounded and if we suppose M not diffeormorfic to the three sphere, the solutions are uniformly bounded. To have this result they use the positive mass theorem.

Now, if we suppose M Riemannian manifold (not necessarily compact) and V ≡ 1, Li and Zhang [17] proved that the product sup × inf is bounded. On other handm see [3], [5] and [6] for other Harnack type inequalities, and, see [3] and [7] about some caracterisation of the solutions of this equation (E

) in this case (V ≡ 1).

Here we extend a result of [11] on an open set of R

, n = 4, 5. In fact we consider the prescribed scalar curvature equation on an open set of R

, n = 4, 5, and, we prove a sup × inf inequality on compact set of the domain when the derivative of the prescribed scalar curvature is β-holderian, β > 0.

Our proof is an extension of Chen-Lin result in dimension 4 and 5, see [11] , and, the moving- plane method is used to have this estimate. We refer to Gidas-Ni-Nirenberg for the moving-plane method, see [13]. Also, we can see in [10], one of the application of this method.

We have the following result in dimension 4, which is the consequence of the work of Chen- Lin.

Theorem A. For all a, b, m, A, B > 0, and for all compact K of Ω, there exists a positive constant c = c(a, b, A, B, K, Ω) such that:

sup

K

u × inf

Ω

≤ c,

where u is solution of (E) with V , C

satisfying (C

) for β = 1.

Here, we give an inequality of type sup × inf for the equation (E) in dimension 4 and with general conditions on the prescribed scalar curvature, exactly we take a C

condition. In fact we extend the result of Chen-Lin in dimension 4.

Here we prove:

Theorem 1.1. . For all a, b, A, B > 0, 1 ≥ β > 0, and for all compact K of Ω, there exists a positive constant c = c(a, b, A, B, β, K, Ω) such that:

(sup

K

u)

× inf

Ω

u ≤ c, where u is solution of (E) with V satisfying (C

).

2

We have the following result in dimension 5, which is the consequence of the work of Chen- Lin.

Theorem B. For all a, b, m, A, B > 0, and for all compact K of Ω, there exists a positive constant c = c(a, b, m, A, B, K, Ω) such that:

sup

K

u ≤ c, if inf

Ω

u ≥ m,

where u is solution of (E) with V satisfying (C

) = (C

) for β = 1.

Here, we give an inequality of type sup × inf for the equation (E) in dimension 5 and with general conditions on the prescribed scalar curvature, exactly we take a C

condition (β = 1 in (C

)). In fact we extend the result of Chen-Lin in dimension 5.

Here we prove:

Theorem 1.2. . For all a, b, A, B > 0, and for all compact K of Ω, there exists a positive constant c = c(a, b, A, B, K, Ω) such that:

(sup

K

u)

× inf

Ω

u ≤ c, where u is solution of (E) with V satisfying (C

) for β = 1.

2. T

-

.

In this section we will formulate a modified version of the method of moving-plane for use later. Let Ω an open set and Ω

the complement of Ω. We consider a solution u of the following equation:

( ∆u + f (x, u) = 0,

u > 0, (E

)

where f (x, u) is nonegative, Holder continuous in x, C

in u, and defined on Ω ¯ × (0, +∞).

Let e be a unit vector in R

. For λ < 0, we let

T

= {x ∈ R

, hx, ei = λ}, Σ

= {x ∈ R

, hx, ei > λ}, and x

= x + (2λ − 2hx, ei)e to denote the reflexion point of x with respect to T

, where h., .i is the standard inner product of R

. Define:

λ

≡ sup{λ < 0, Ω

⊂ Σ

},

Σ

= Σ

− Ω

for λ ≤ λ

, and Σ ¯

the closure of Σ

. Let u

(x) = u(x

) and w

(x) = u(x) − u

(x) for x ∈ Σ

. Then we have, for any arbitrary function b

(x),

∆w

(x) + b

(x)w

(x) = Q(x, b

(x)), where,

Q(x, b

(x)) = f (x

, u

) − f (x, u) + b

(x)w

(x).

The hypothesis (∗) is said to be satisfied if there are two families of functions b

(x) and h

(x) defined in Σ

, for λ ∈ (−∞, λ

) such that, the following assertions holds:

0 ≤ b

(x) ≤ c(x)|x|

,

where c(x) is independant of λ and tends to zero as |x| tends to +∞, h

(x) ∈ C

(Σ

∩ Ω),

and satisfies:

( ∆h

(x) ≥ Q(x, b

(x)) in Σ

∩ Ω h

(x) > 0 in Σ

∩ Ω

in the distributional sense and,

3

h

(x) = 0 on T

and h

(x) = O(|x|

), as |x| → +∞ for some constant t

> 0,

h

(x) + ǫ < w

(x),

in a neighborhood of ∂Ω, where ǫ is a positive constant independant of x.



 

 

h

(x) and ∇

h

are continuous with respect to both variables x and λ, and for any compact set of Ω, w

(x) > h

(x) holds when −λ is sufficiently large.

We have the following lemma:

Lemma 2.1. Let u be a solution of (E

). Suppose that u(x) ≥ C > 0 in a neighborhood of ∂Ω and u(x) = O(|x|

) at +∞ for some positive t

. Assume there exist b

(x) and h

(x) such that the hypothesis (∗) is satisfied for λ ≤ λ

. Then w

(x) > 0 in Σ

, and h∇u, ei > 0 on T

for λ ∈ (−∞, λ

).

For the proof see Chen and Lin, [10].

Remark 2.2. If we know that w

− h

> 0 for some λ = λ

< λ

and b

and h

satisfy the hypothesis (∗) for λ

≤ λ ≤ λ

, then the conclusion of the lemma 2.1 holds.

3. P

: Proof of the theorem 1, n = 4 :

To prove the theorem, we argue by contradiction and we assume that the (sup)

× inf tends to infinity.

Step 1: blow-up analysis We want to prove that:

R ˜

( sup

BR˜(0)

u)

× inf

B_{3 ˜R}(0)

u ≤ c = c(a, b, A, B, β), If it is not the case, we have:

R ˜

( sup

BRi˜ (0)

u

)

× inf

B_{3 ˜Ri}(0)

u

= i

→ +∞, For positive solutions u

> 0 of the equation (E) and R ˜

→ 0.

Thus,

1

i R ˜

( sup

BRi˜ (0)

u

)

→ +∞, and,

1

i R ˜

[ sup

BRi˜ (0)

u

]

→ +∞, Let a

such that:

u

(a

) = max

BRi˜ (0)

u

, We set,

s

(x) = ( ˜ R

− |x − a

|)

u

(x), we have,

s

(¯ x

) = max

BRi˜ (ai)

s

≥ s

(a

) = ˜ R

sup

B_Ri(0)

u

→ +∞,

4

we set,

R

= 1

2 ( ˜ R

− |¯ x

− a

|), We have, for |x − x ¯

| ≤ R

i ,

R ˜

− |x − a

| ≥ R ˜

− |¯ x

− a

| − |x − a

| ≥ 2R

− R

= R

Thus,

u

(x)

u

(¯ x

) ≤ β

≤ 2

. with β

→ 1.

We set,

M

= u

(¯ x

), v

(y) = u

(¯ x

+ M

y)

u

(¯ x

) , |y| ≤ 1

i R

M

= 2L

. And,

1 i

R

M

× inf

B_{3 ˜Ri}(0)

u

→ +∞,

By the elliptic estimates, v

converge on each compact set of R

to a function U

> 0 solution of :

( −∆U

= V (0)U

in R

, U

(0) = 1 = max

U

.

For simplicity, we assume that 0 < V (0) = n(n − 2) = 8. By a result of Caffarelli-Gidas- Spruck, see [10], we have:

U

(y) = (1 + |y|

)

. We set,

v

(y) = v

(y + e),

where v

is the blow-up function. Then, v

has a local maximum near −e.

U

(y) = U

(y + e).

We want to prove that:

{0≤|y|≤r}

min v

≤ (1 + ǫ)U

(r).

for 0 ≤ r ≤ L

, with L

= 1

2i R

M

.

We assume that it is not true, then, there is a sequence of number r

∈ (0, L

) and ǫ > 0, such that:

{0≤|y|≤r

min

v

≥ (1 + ǫ)U

(r

).

We have:

r

→ +∞.

Thus , we have for r

∈ (0, L

) :

{0≤|y|≤r

min

v

≥ (1 + ǫ)U

(r

).

Also, we can find a sequence of number l

→ +∞ such that:

5

||v

− U

||

→ 0.

Thus,

{0≤|y|≤l

min

v

≥ (1 − ǫ/2)U

(l

).

Step 2 : The Kelvin transform and the Moving-plane method

Step 2.1: a linear equation perturbed by a term, and, the auxiliary function Step 2.1.1: D

= |∇V

(x

)| → 0.

We have the same estimate as in the paper of Chen-Lin. We argue by contradiction. We consider r

∈ (0, L

) where L

is the number of the blow-up analysis.

L

= 1

2i R

M

.

We use the assumption that the sup times inf is not bounded to prove w

> h

in Σ

= {y, y

> λ}, and on the boundary.

The function v

has a local maximum near −e and converge to U

(y) = U

(y + e) on each compact set of R

. U

has a maximum at −e.

We argue by contradiction and we suppose that:

D

= |∇V

(x

)| 6→ 0.

Then, without loss of generality we can assume that:

∇V

(x

) → e = (1, 0, ...0).

Where x

is :

x

= ¯ x

+ M

e, with x ¯

is the local maximum in the blow-up analysis.

As in the paper of Chen-Lin, we use the Kelvin transform twice and we set (we take the same notations):

I

(y) =

|y|

|y|²

− δe |

− δe|

, v

(y) = v

(I

(y))

|y|

|y − e/δ|

, and,

V

(y) = V

(x

+ M

I

(y)).

U

(y) = U

(I

(y))

|y|

|y − e/δ|

.

Then, U

has a local maximum near e

→ −e when δ → 0. The function v

has a local maximum near −e.

We want to prove by the application of the maximum principle and the Hopf lemma that near e

we have not a local maximum, which is a contradiction.

We set on Σ

= Σ

− {y, |y −

| ≤

} ≃ Σ

− {y, |I

(y)| ≥ r

}:

h

(y) = − Z

Σλ

G

(y, η)Q

(η)dη.

with,

6

Q

(η) = (V

(η) − V

(η

))(v

(η

))

. And, by the same estimates, we have for η ∈ A

= {η, |η| ≤ R = ǫ

/δ},

V

(η) − V

(η

) ≥ M

(η

− λ) + o(1)M

|η

|, and, we have for η ∈ A

= Σ

− A

:

|V

(η) − V

(η

)| ≤ CM

(|I

(η)| + |I

(η

)|), And, we have for some λ

≤ −2 and C

> 0:

w

(y) = v

(y) − v

(y

) ≥ C

y

− λ

(1 + |y|)

, for y

> λ

.

Because , by the maximum principle:

{li≤|I

min

v

= min{ min

{|Iδ(y)|=li}

, v

min

{|Iδ(y)|=ri}

v

} ≥ (1 − ǫ)U

( e δ )

≥ (1 + c

δ − ǫ)U

(( e

δ )

) ≥ (1 + c

δ − 2ǫ)v

(y

), and for |I

(y)| ≤ l

we use the C

convergence of v

to U

.

Thus,

w

(y) > 2ǫ > 0,

By the same estimates as in Chen-Lin paper (we apply the lemma 2.1 of the second section), and by our hypothesis on v

, we have:

0 < h

(y) = O(1)M

(y

− λ)(1 + |y|)

< 2ǫ < w

(y).

also, we have the same etimate on the boundary, |I

(η)| = r

or |y − e/δ| = c

r

: Step 2.1.1: |∇V

(x

)|

[u

(x

)] ≤ C

Here, also, we argue by contradiction. We use the same computation as in Chen-Lin paper, we choose the same h

, except the fact that here we use the computation with M

in front the regular part of h

.

Here also, we consider r

∈ (0, L

) where L

is the number of the blow-up analysis.

L

= 1

2i R

M

. We argue by contradiction and we suppose that:

M

D

→ +∞.

Then, without loss of generality we can assume that:

∇V

(x

)

|∇V

(x

)| → e = (1, 0, ...0).

We use the Kelvin transform twice and around this point and around 0.

h

(y) = ǫr

G

(y, e δ ) −

Z

Σλ

G

(y, η)Q

(η)dη.

with,

Q

(η) = (V

(η) − V

(η

))(v

(η

))

. And, by the same estimates, we have for η ∈ A

V

(η) − V

(η

) ≥ M

D

(η

− λ) + o(1)M

|η

|,

7

and, we have for η ∈ A

, |I

(η)| ≤ c

M

D

,

|V

(η) − V

(η

)| ≤ CM

D

(|I

(η)| + |I

(η

)|), and for M

D

≤ |I

(η)| ≤ r

,

|V

(η) − V

(η

)| ≤ M

D

|I

(η)| + M

|I

(η)|

, By the same estimates, we have for |I

(η)| ≤ r

or |y − e/δ| ≥ c

r

: h

(y) ≃ ǫr

G

(y, e

δ )+c

M

D

(y

− λ)

|y|

+o(1)M

D

(y

− λ)

|y|

+o(1)M

G

(y, e δ ).

with c

> 0.

And, we have for some λ

≤ −2 and C

> 0:

v

(y) − v

(y

) ≥ C

y

− λ

(1 + |y|)

, for y

> λ

.

By the same estimates as in Chen-Lin paper (we apply the lemma 2.1 of the second section), and by our hypothesis on v

, we have:

0 < h

(y) < 2ǫ < w

(y).

also, we have the same etimate on the boundary, |I

(η)| = r

or |y − e/δ| = c

r

Step 2.2 conclusion : a linear equation perturbed by a term, and, the auxiliary function Here also, we use the computations of Chen-Lin, and, we take the same auxiliary function h

(which correspond to this step), except the fact that here in front the regular part of this function we have M

.

Here also, we consider r

∈ (0, L

) where L

is the number of the blow-up analysis.

L

= 1

2i R

M

. We set,

v

(z) = v

(z + e),

where v

is the blow-up function. Then, v

has a local maximum near −e.

U

(z) = U

(z + e).

We have, for |y| ≥ L

, L

= 1 2 R

M

,

¯

v

(y) = 1

|y|

v

y

|y|

.

|V

(¯ x

+ M

y

|y|

) − V

(¯ x

)| ≤ M

(1 + |y|

).

x

= ¯ x

+ M

e,

Then, for simplicity, we can assume that, v ¯

has a local maximum near e

= (−1/2, 0, ...0).

Also, we have:

|V

(x

+ M

y

|y|

) − V

(x

+ M

y

|y

|

)| ≤ M

(1 + |y|

).

h

(y) ≃ ǫr

G

(y, 0) − Z

Σ^′_λ

G

(y, η)Q

(η)dη.

where, Σ

= Σ

− {η, |η| ≤ r

}, and,

8

Q

(η) =

V

(x

+ M

y

|y|

) − V

(x

+ M

y

|y

|

)

(v

(y

))

. we have by the same computations that:

Z

Σ^′_λ

G

(y, η)Q

(η)dη ≤ CM

G

(y, 0) << ǫr

G

(y, 0).

By the same estimates as in Chen-Lin paper (we apply the lemma 2.1 of the second section), and by our hypothesis on v

, we have:

0 < h

(y) < 2ǫ < w

(y).

also, we have the same estimate on the boundary, |y| = 1 r

. Proof of the theorem 2, n = 5:

To prove the theorem, we argue by contradiction and we assume that the (sup)

× inf tends to infinity.

Step 1: blow-up analysis We want to prove that:

R ˜

( sup

BR˜(0)

u)

× inf

B_{3 ˜R}(0)

u ≤ c = c(a, b, A, B), If it is not the case, we have:

R ˜

( sup

BRi˜ (0)

u

)

× inf

B_{3 ˜Ri}(0)

u

= i

→ +∞, For positive solutions u

> 0 of the equation (E) and R ˜

→ 0.

Thus,

1

i R ˜

( sup

BRi˜ (0)

u

)

→ +∞, and,

1

i R ˜

[ sup

BRi˜ (0)

u

]

→ +∞, Let a

such that:

u

(a

) = max

BRi˜ (0)

u

, We set,

s

(x) = ( ˜ R

− |x − a

|)

u

(x), we have,

s

(¯ x

) = max

BRi˜ (ai)

s

≥ s

(a

) = ˜ R

sup

B_Ri(0)

u

→ +∞, we set,

R

= 1

2 ( ˜ R

− |¯ x

− a

|), We have, for |x − x ¯

| ≤ R

i ,

R ˜

− |x − a

| ≥ R ˜

− |¯ x

− a

| − |x − a

| ≥ 2R

− R

= R

Thus,

9

u

(x)

u

(¯ x

) ≤ β

≤ 2

. with β

→ 1.

We set,

M

= u

(¯ x

), v

(y) = u

(¯ x

+ M

y)

u

(¯ x

) , |y| ≤ 1

i R

M

= 2L

. And,

1

i

R

M

× inf

B_{3 ˜Ri}(0)

u

→ +∞,

By the elliptic estimates, v

converge on each compact set of R

to a function U

> 0 solution of :

( −∆U

= V (0)U

in R

, U

(0) = 1 = max

U

.

For simplicity, we assume that 0 < V (0) = n(n − 2) = 15. By a result of Caffarelli-Gidas- Spruck, see [10], we have:

U

(y) = (1 + |y|

)

. We set,

v

(y) = v

(y + e),

where v

is the blow-up function. Then, v

has a local maximum near −e.

U

(y) = U

(y + e).

We want to prove that:

{0≤|y|≤r}

min v

≤ (1 + ǫ)U

(r).

for 0 ≤ r ≤ L

, with L

= 1

2i R

M

.

We assume that it is not true, then, there is a sequence of number r

∈ (0, L

) and ǫ > 0, such that:

{0≤|y|≤r

min

v

≥ (1 + ǫ)U

(r

).

We have:

r

→ +∞.

Thus , we have for r

∈ (0, L

) :

{0≤|y|≤r

min

v

≥ (1 + ǫ)U

(r

).

Also, we can find a sequence of number l

→ +∞ such that:

||v

− U

||

→ 0.

Thus,

{0≤|y|≤l

min

v

≥ (1 − ǫ/2)U

(l

).

Step 2 : The Kelvin transform and the Moving-plane method

Step 2.1: a linear equation perturbed by a term, and, the auxiliary function

10

Step 2.1.1: D

= |∇V

(x

)| → 0.

We have the same estimate as in the paper of Chen-Lin. We argue by contradiction. We consider r

∈ (0, L

) where L

is the number of the blow-up analysis.

L

= 1

2i R

M

.

We use the assumption that the sup times inf is not bounded to prove w

> h

in Σ

= {y, y

> λ}, and on the boundary.

The function v

has a local maximum near −e and converge to U

(y) = U

(y + e) on each compact set of R

. U

has a maximum at −e.

We argue by contradiction and we suppose that:

D

= |∇V

(x

)| 6→ 0.

Then, without loss of generality we can assume that:

∇V

(x

) → e = (1, 0, ...0).

Where x

is :

x

= ¯ x

+ M

e, with x ¯

is the local maximum in the blow-up analysis.

As in the paper of Chen-Lin, we use the Kelvin transform twice and we set (we take the same notations):

I

(y) =

|y|

|y|²

− δe |

− δe|

, v

(y) = v

(I

(y))

|y|

|y − e/δ|

, and,

V

(y) = V

(x

+ M

I

(y)).

U

(y) = U

(I

(y))

|y|

|y − e/δ|

.

Then, U

has a local maximum near e

→ −e when δ → 0. The function v

has a local maximum near −e.

We want to prove by the application of the maximum principle and the Hopf lemma that near e

we have not a local maximum, which is a contradiction.

We set on Σ

= Σ

− {y, |y −

| ≤

i

} ≃ Σ

− {y, |I

(y)| ≥ r

}:

h

(y) = − Z

Σλ

G

(y, η)Q

(η)dη.

with,

Q

(η) = (V

(η) − V

(η

))(v

(η

))

. And, by the same estimates, we have for η ∈ A

= {η, |η| ≤ R = ǫ

/δ},

V

(η) − V

(η

) ≥ M

(η

− λ) + o(1)M

|η

|, and, we have for η ∈ A

= Σ

− A

:

|V

(η) − V

(η

)| ≤ CM

(|I

(η)| + |I

(η

)|), And, we have for some λ

≤ −2 and C

> 0:

11

v

(y) − v

(y

) ≥ C

y

− λ

(1 + |y|)

, for y

> λ

.

By the same estimates, and by our hypothesis on v

, we have, for c

> 0:

0 < h

(y) < 2ǫ < w

(y).

also, we have the same etimate on the boundary, |I

(η)| = r

or |y − e/δ| = c

r

Step 2.1.1: |∇V

(x

)|[u

(x

)]

≤ C

Here, also, we argue by contradiction. We use the same computation as in Chen-Lin paper, we take α = 2 and we choose the same h

, except the fact that here we use the computation with M

in front the regular part of h

.

Here also, we consider r

∈ (0, L

) where L

is the number of the blow-up analysis.

L

= 1

2i R

M

. We argue by contradiction and we suppose that:

M

D

→ +∞.

Then, without loss of generality we can assume that:

∇V

(x

)

|∇V

(x

)| → e = (1, 0, ...0).

We use the Kelvin transform twice and around this point and around 0.

h

(y) = ǫr

G

(y, e δ ) −

Z

Σλ

G

(y, η)Q

(η)dη.

with,

Q

(η) = (V

(η) − V

(η

))(v

(η

))

. And, by the same estimates, we have for η ∈ A

V

(η) − V

(η

) ≥ M

D

(η − λ) + o(1)M

|η

|, and, we have for η ∈ A

, |I

(η)| ≤ c

M

D

,

|V

(η) − V

(η

)| ≤ CM

D

(|I

(η)| + |I

(η

)|), and for M

D

≤ |I

(η)| ≤ r

,

|V

(η) − V

(η

)| ≤ M

D

|I

(η)| + M

|I

(η)|

, By the same estimates, we have for |I

(η)| ≤ r

or |y − e/δ| ≥ c

r

: h

(y) ≃ ǫr

G

(y, e

δ )+c

M

D

(y

− λ)

|y|

+o(1)M

D

(y

− λ)

|y|

+o(1)M

G

(y, e δ ).

with c

> 0.

And, we have for some λ

≤ −2 and C

> 0:

v

(y) − v

(y

) ≥ C

y

− λ

(1 + |y|)

, for y

> λ

.

By the same estimates as in Chen-Lin paper (we apply the lemma 2.1 of the second section), and by our hypothesis on v

, we have:

0 < h

(y) < 2ǫ < w

(y).

12

also, we have the same etimate on the boundary, |I

(η)| = r

or |y − e/δ| = c

r

: Step 2.2 conclusion : a linear equation perturbed by a term, and, the auxiliary function Here also, we use the computations of Chen-Lin, and, we take the same auxiliary function h

(which correspond to this step), except the fact that here in front the regular part of this function we have M

.

Here also, we consider r

∈ (0, L

) where L

is the number of the blow-up analysis.

L

= 1

2i R

M

. We set,

v

(z) = v

(z + e),

where v

is the blow-up function. Then, v

has a local maximum near −e.

U

(z) = U

(z + e).

We have, for |y| ≥ L

, L

= 1

2 R

M

,

¯

v

(y) = 1

|y|

v

y

|y|

.

|V

(¯ x

+ M

y

|y|

) − V

(¯ x

)| ≤ M

(1 + |y|

).

x

= ¯ x

+ M

e,

Then, for simplicity, we can assume that, v ¯

has a local maximum near e

= (−1/2, 0, ...0).

Also, we have:

|V

(x

+ M

y

|y|

) − V

(x

+ M

y

|y

|

)| ≤ M

(1 + |y|

).

h

(y) ≃ ǫr

G

(y, 0) − Z

Σ^′_λ

G

(y, η)Q

(η)dη.

where, Σ

= Σ

− {η, |η| ≤ r

}, and, Q

(η) =

V

(x

+ M

y

|y|

) − V

(x

+ M

y

|y

|

)

(v

(y

))

. we have by the same computations that:

Z

Σ^′_λ

G

(y, η)Q

(η)dη ≤ CM

G

(y, 0) << ǫr

G

(y, 0).

By the same estimates as in Chen-Lin paper (we apply the lemma 2.1 of the second section), and by our hypothesis on v

, we have:

0 < h

(y) < 2ǫ < w

(y).

also, we have the same estimate on the boundary, |η| = 1 r

.

13

R

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DEPARTEMENT DEMATHEMATIQUES, UNIVERSITEPIERRE ETMARIECURIE, 2PLACEJUSSIEU, 75005, PARIS, FRANCE.

E-mail address:samybahoura@yahoo.fr

14