A compactness result for a system with weight and boundary singularity.

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A compactness result for a system with weight and boundary singularity.

Samy Skander Bahoura

To cite this version:

Samy Skander Bahoura. A compactness result for a system with weight and boundary singularity..

2019. �hal-01989191�

A COMPACTNESS RESULT FOR A SYSTEM WITH WEIGHT AND BOUNDARY SINGULARITY.

SAMY SKANDER BAHOURA

ABSTRACT. We give blow-up behavior for solutions to an elliptic system with Dirichlet condition, and, weight and boundary singularity. Also, we have a compactness result for this elliptic system with regular H¨olderian weight and boundary singularity and Lipschitz condition.

Mathematics Subject Classification: 35J60 35B45 35B50

Keywords: blow-up, boundary, system, Dirichlet condition, a priori estimate, analytic domain, regular weight, boundary singularity, Lipschitz condition.

1. I

M

R

We set ∆ = ∂

+ ∂

on open analytic domain Ω of R

.

We consider the following equation:

(P)



 

 



 

 



−∆u = |x|

V e

in Ω ⊂ R

,

−∆v = W e

in Ω ⊂ R

,

u = 0 in ∂Ω,

v = 0 in ∂Ω.

Here, we assume that:

0 ≤ V ≤ b

< +∞, e

∈ L

(Ω) and u ∈ W

(Ω),

0 ≤ W ≤ b

< +∞, |x|

e

∈ L

(Ω) and v ∈ W

(Ω),

and,

0 ∈ ∂Ω, β ≥ 0.

1

When u = v and β = 0, the above system is reduced to an equation which was studied by many authors, with or without the boundary condition, also for Riemann surfaces, see [1-17], one can find some existence and compactness results, also for a system.

Among other results, we can see in [6] the following important Theorems (β = 0):,

Theorem A.(Brezis-Merle [6]).Consider the case of one equation; if (u

)

= (v

)

and (V

)

= (W

)

are two sequences of functions relatively to the problem (P ) with, 0 < a ≤ V

≤ b < +∞, then, for all compact set K of Ω,

sup

K

u

≤ c = c(a, b, K, Ω).

Theorem B (Brezis-Merle [6]).Consider the case of one equation and assume that (u

)

and (V

)

are two sequences of functions relatively to the previous problem (P ) with, 0 ≤ V

≤ b < +∞, and,

Z

Ω

e

dy ≤ C, then, for all compact set K of Ω,

sup

K

u

≤ c = c(b, C, K, Ω).

Next, we call energy the following quantity:

E = Z

Ω

e

dy.

The boundedness of the energy is a necessary condition to work on the problem (P ) as showed in [6], by the following counterexample (β = 0):

Theorem C (Brezis-Merle [6]).Consider the case of one equation, then there are two sequences (u

)

and (V

)

of the problem (P ) with, 0 ≤ V

≤ b < +∞, and,

Z

Ω

e

dy ≤ C, and

sup

Ω

u

→ +∞.

2

When β = 0, the above system have many properties in the constant and the Lipschitzian cases. Indeed we have (when β = 0):

In [12], Dupaigne-Farina-Sirakov proved (by an existence result of Montenegro, see [16]) that the so- lutions of the above system when V and W are constants can be extremal and this condition imply the boundedness of the energy and directly the compactness. Note that in [11], if we assume (in particular) that

∇ log V and ∇ log W and V > a > 0 or W > a

> 0 and V, W are nonegative and uniformly bounded then the energy is bounded and we have a compactness result.

Note that in the case of one equation (and β = 0), we can prove by using the Pohozaev identity that if +∞ > b ≥ V ≥ a > 0, ∇V is uniformely Lipschitzian that the energy is bounded when Ω is starshaped.

In [15] Ma-Wei, using the moving-plane method showed that this fact is true for all domain Ω with the same assumptions on V . In [11] De Figueiredo-do O-Ruf extend this fact to a system by using the moving-plane method for a system.

Theorem C, shows that we have not a global compactness to the previous problem with one equation, perhaps we need more information on V to conclude to the boundedness of the solutions. When ∇ log V is Lipschitz function and β = 0, Chen-Li and Ma-Wei see [7] and [15], showed that we have a compactness on all the open set. The proof is via the moving plane-Method of Serrin and Gidas-Ni-Nirenberg. Note that in [11], we have the same result for this system when ∇ log V and ∇ log W are uniformly bounded. We will see below that for a system we also have a compactness result when V and W are Lipschitzian and β ≥ 0.

Now consider the case of one equation. In this case our equation have nice properties.

If we assume V with more regularity, we can have another type of estimates, a sup + inf type inequalities.

It was proved by Shafrir see [17], that, if (u

)

, (V

)

are two sequences of functions solutions of the previous equation without assumption on the boundary and, 0 < a ≤ V

≤ b < +∞, then we have the following interior estimate:

C a b

sup

K

u

+ inf

Ω

u

≤ c = c(a, b, K, Ω).

Now, if we suppose (V

)

uniformly Lipschitzian with A the Lipschitz constant, then, C(a/b) = 1 and c = c(a, b, A, K, Ω), see [5].

Here we are interested by the case of a system of this type of equation. First, we give the behavior of the blow-up points on the boundary, with weight and boundary singularity, and in the second time we have a proof of compactness of the solutions to Gelfand-Liouville type system with weight and boundary singularity and Lipschitz condition.

Here, we write an extention of Brezis-Merle Problem (see [6]) to a system:

Problem. Suppose that V

→ V and W

→ W in C

( ¯ Ω), with, 0 ≤ V

and 0 ≤ W

. Also, we consider two sequences of solutions (u

), (v

) of (P ) relatively to (V

), (W

) such that,

Z

Ω

e

dx ≤ C

, Z

Ω

|x|

e

dx ≤ C

,

3

is it possible to have:

||u

||

≤ C

= C

(β, C

, C

, Ω)?

and,

||v

||

≤ C

= C

(β, C

, C

, Ω)?

In this paper we give a caracterization of the behavior of the blow-up points on the boundary and also a proof of the compactness theorem when V

and W

are uniformly Lipschitzian and β ≥ 0. For the behavior of the blow-up points on the boundary, the following condition are enough,

0 ≤ V

≤ b

, 0 ≤ W

≤ b

,

The conditions V

→ V and W

→ W in C

( ¯ Ω) are not necessary.

But for the proof of the compactness for the system, we assume that:

||∇V

||

≤ A

, ||∇W

||

≤ A

, β ≥ 0.

Our main result are:

Theorem 1.1. Assume that max

u

→ +∞ and max

v

→ +∞ Where (u

) and (v

) are solutions of the probleme (P ) with (β ≥ 0), and:

0 ≤ V

≤ b

, and Z

Ω

e

dx ≤ C

, ∀ i, and,

0 ≤ W

≤ b

, and Z

Ω

|x|

e

dx ≤ C

, ∀ i,

then; after passing to a subsequence, there is a finction u, there is a number N ∈ N and N points x

, x

, . . . , x

∈ ∂Ω, such that,

Z

∂Ω

∂

u

ϕ → Z

∂Ω

∂

uϕ +

N

X

j=1

α

ϕ(x

), α

≥ 4π, for any ϕ ∈ C

(∂Ω), and,

u

→ u in C

( ¯ Ω − {x

, . . . , x

}).

4

Z

∂Ω

∂

v

ϕ → Z

∂Ω

∂

vϕ +

N

X

j=1

β

ϕ(x

), β

≥ 4π, for any ϕ ∈ C

(∂Ω), and,

v

→ v in C

( ¯ Ω − {x

, . . . , x

}).

In the following theorem, we have a proof for the global a priori estimate which concern the problem (P).

Theorem 1.2. Assume that (u

), (v

) are solutions of (P) relatively to (V

), (W

) with the following conditions:

x

= 0 ∈ ∂Ω, β ≥ 0, and,

0 ≤ V

≤ b

, ||∇V

||

≤ A

, and Z

Ω

e

≤ C

,

0 ≤ W

≤ b

, ||∇W

||

≤ A

, and Z

Ω

|x|

e

≤ C

, We have,

||u

||

≤ C

(b

, b

, β, A

, A

, C

, C

, Ω), and,

||v

||

≤ C

(b

, b

, β, A

, A

, C

, C

, Ω),

2. P

Proof of theorem 1.1:

We have:

u

, v

∈ W

(Ω).

Since e

∈ L

(Ω) by the corollary 1 of Brezis-Merle’s paper (see [6]) we have e

∈ L

(Ω) for all k > 2 and the elliptic estimates of Agmon and the Sobolev embedding (see [1]) imply that:

5

u

∈ W

(Ω) ∩ C

( ¯ Ω).

And, We have:

v

, u

∈ W

(Ω).

Since |x|

e

∈ L

(Ω) by the corollary 1 of Brezis-Merle’s paper (see [6]) we have e

∈ L

(Ω) for all k > 2 and the elliptic estimates of Agmon and the Sobolev embedding (see [1]) imply that:

v

∈ W

(Ω) ∩ C

( ¯ Ω).

Since |x|

V

e

and W

e

are bounded in L

(Ω), we can extract from those two sequences two subse- quences which converge to two nonegative measures µ

and µ

. (This procedure is similar to the procedure of Brezis-Merle, we apply corollary 4 of Brezis-Merle paper, see [6]).

If µ

(x

) < 4π, by a Brezis-Merle estimate for the first equation, we have e

∈ L

around x

, by the elliptic estimates, for the second equation, we have v

∈ W

⊂ L

around x

, and , returning to the first equation, we have u

∈ L

around x

.

If µ

(x

) < 4π, then u

and v

are also locally bounded around x

.

Thus, we take a look to the case when, µ

(x

) ≥ 4π and µ

(x

) ≥ 4π. By our hypothesis, those points x

are finite.

We will see that inside Ω no such points exist. By contradiction, assume that, we have µ

(x

) ≥ 4π. Let us consider a ball B

(x

) which contain only x

as nonregular point. Thus, on ∂B

(x

), the two sequence u

and v

are uniformly bounded. Let us consider:

( −∆z

= |x|

V

e

in B

(x

) ⊂ R

, z

= 0 in ∂B

(x

).

By the maximum principle we have:

z

≤ u

and z

→ z almost everywhere on this ball, and thus, Z

e

≤ Z

e

≤ C, and,

6

Z

e

≤ C.

but, z is a solution in W

(B

(x

)), 1 ≤ q < 2, of the following equation:

( −∆z = µ

in B

(x

) ⊂ R

, z = 0 in ∂B

(x

).

with, µ

≥ 4π and thus, µ

≥ 4πδ

and then, by the maximum principle in W

(B

(x

)):

z ≥ −2 log |x − x

| + C

thus,

Z

e

= +∞,

which is a contradiction. Thus, there is no nonregular points inside Ω

Thus, we consider the case where we have nonregular points on the boundary, we use two estimates:

Z

∂Ω

∂

u

dσ ≤ C

, Z

∂Ω

∂

v

dσ ≤ C

,

and,

||∇u

||

≤ C

, ||∇v

||

≤ C

, ∀ i and 1 < q < 2.

We have the same computations, as in the case of one equation.

We consider a points x

∈ ∂Ω such that:

µ

(x

) < 4π.

We consider a test function on the boundary η we extend η by a harmonic function on Ω, we write the equation:

−∆((u

− u)η) = |x|

(V

e

− V e

)η+ < ∇(u

− u)|∇η >= f

with,

7

Z

|f

| ≤ 4π − ǫ + o(1) < 4π − 2ǫ < 4π,

−∆((v

− v)η) = (W

e

− W e

)η+ < ∇(v

− v)|∇η >= g

,

with,

Z

|g

| ≤ 4π − ǫ + o(1) < 4π − 2ǫ < 4π,

By the Brezis-Merle estimate, we have uniformly, e

∈ L

around x

, by the elliptic estimates, for the second equation, we have v

∈ W

⊂ L

around x

, and , returning to the first equation, we have u

∈ L

around x

.

We have the same thing if we assume:

µ

(x

) < 4π.

Thus, if µ

(x

) < 4π or µ

(x

) < 4π, we have for R > 0 small enough:

(u

, v

) ∈ L

(B

(x

) ∩ Ω). ¯

By our hypothesis the set of the points such that:

µ

(x

) ≥ 4π, µ

(x

) ≥ 4π,

is finite, and, outside this set u

and v

are locally uniformly bounded. By the elliptic estimates, we have the C

convergence to u and v on each compact set of Ω ¯ − {x

, . . . x

}.

Indeed,

By the Stokes formula we have,

Z

∂Ω

∂

u

dσ ≤ C,

We use the weak convergence in the space of Radon measures to have the existence of a nonnegative Radon measure µ

such that,

Z

∂Ω

∂

u

ϕdσ → µ

(ϕ), ∀ ϕ ∈ C

(∂Ω).

8

We take an x

∈ ∂Ω such that, µ

(x

) < 4π. For ǫ > 0 small enough set I

= B(x

, ǫ) ∩ ∂Ω on the unt disk or one can assume it as an interval. We choose a function η

such that,



 

 



 

 



η

≡ 1, on I

, 0 < ǫ < δ/2, η

≡ 0, outside I

,

0 ≤ η

≤ 1,

||∇η

||

≤ C

(Ω, x

)

ǫ .

We take a η ˜

such that,

( −∆˜ η

= 0 in Ω ⊂ R

,

˜

η

= η

in ∂Ω.

Remark: We use the following steps in the construction of η ˜

: We take a cutoff function η

in B (0, 2) or B(x

, 2):

1- We set η

(x) = η

(|x − x

|/ǫ) in the case of the unit disk it is sufficient.

2- Or, in the general case: we use a chart (f, Ω) ˜ with f (0) = x

and we take µ

(x) = η

(f (|x|/ǫ)) to have connected sets I

and we take η

(y) = µ

(f

(y)). Because f, f

are Lipschitz, |f(x) − x

| ≤ k

|x| ≤ 1 for |x| ≤ 1/k

and |f(x) − x

| ≥ k

|x| ≥ 2 for |x| ≥ 2/k

> 1/k

, the support of η is in I

.



 

 



 

 



η

≡ 1, on f(I

), 0 < ǫ < δ/2, η

≡ 0, outside f (I

),

0 ≤ η

≤ 1,

||∇η

||

1)ǫ)

≤ C

(Ω, x

)

ǫ .

3- Also, we can take: µ

(x) = η

(|x|/ǫ) and η

(y) = µ

(f

(y)), we extend it by 0 outside f (B

(0)).

We have f (B

(0)) = D

(x

), f (B

(0)) = D

(x

) and f(B

) = D

(x

) with f and f

smooth diffeo- morphism.



 

 



 

 



η

≡ 1, on a the connected set J

= f (I

), 0 < ǫ < δ/2, η

≡ 0, outside J

= f (I

),

0 ≤ η

≤ 1,

||∇η

||

≤ C

(Ω, x

)

ǫ .

And, H

(J

) ≤ C

H

(I

) = C

4ǫ, since f is Lipschitz. Here H

is the Hausdorff measure.

We solve the Dirichlet Problem:

9

( −∆¯ η

= −∆η

in Ω ⊂ R

,

¯

η

= 0 in ∂Ω.

and finaly we set η ˜

= −¯ η

+ η

. Also, by the maximum principle and the elliptic estimates we have :

||∇˜ η

||

≤ C(||η

||

+ ||∇η

||

+ ||∆η

||

) ≤ C

ǫ

, with C

depends on Ω.

We use the following estimate, see [8],

||∇v

||

≤ C

, ||∇u

||

≤ C

, ∀ i and 1 < q < 2.

We deduce from the last estimate that, (v

) converge weakly in W

(Ω), almost everywhere to a function v ≥ 0 and R

Ω

|x|

e

< +∞ (by Fatou lemma). Also, V

weakly converge to a nonnegative function V in L

.

We deduce from the last estimate that, (u

) converge weakly in W

(Ω), almost everywhere to a function u ≥ 0 and R

Ω

e

< +∞ (by Fatou lemma). Also, W

weakly converge to a nonnegative function W in L

. The function u, v are in W

(Ω) solutions of :

( −∆u = |x|

V e

∈ L

(Ω) in Ω ⊂ R

,

u = 0 in ∂Ω.

And,

( −∆v = W e

∈ L

(Ω) in Ω ⊂ R

,

v = 0 in ∂Ω.

According to the corollary 1 of Brezis-Merle’s result, see [6], we have e

∈ L

(Ω), k > 1. By the elliptic estimates, we have v ∈ C

( ¯ Ω).

According to the corollary 1 of Brezis-Merle’s result, see [6], we have e

∈ L

(Ω), k > 1. By the elliptic estimates, we have u ∈ C

( ¯ Ω).

For two vectors f and g we denote by f · g the inner product of f and g.

We can write:

−∆((u

− u)˜ η

) = |x|

(V

e

− V e

)˜ η

− 2∇(u

− u) · ∇˜ η

. (1)

10

−∆((v

− v)˜ η

) = (W

e

− W e

)˜ η

− 2∇(v

− v) · ∇˜ η

.

We use the interior esimate of Brezis-Merle, see [6],

Step 1: Estimate of the integral of the first term of the right hand side of (1).

We use the Green formula between η ˜

and u, we obtain, Z

Ω

|x|

V e

η ˜

dx = Z

∂Ω

∂

uη

≤ C

ǫ||∂

u||

= Cǫ (2) We have,

( −∆u

= |x|

V

e

in Ω ⊂ R

,

u

= 0 in ∂Ω.

We use the Green formula between u

and η ˜

to have:

Z

Ω

|x|

V

e

η ˜

dx = Z

∂Ω

∂

u

η

dσ → µ

(η

) ≤ µ

(J

) ≤ 4π − ǫ

, ǫ

> 0 (3) From (2) and (3) we have for all ǫ > 0 there is i

= i

(ǫ) such that, for i ≥ i

,

Z

Ω

||x|

(V

e

− V e

)˜ η

|dx ≤ 4π − ǫ

+ Cǫ (4)

Step 2: Estimate of integral of the second term of the right hand side of (1).

Let Σ

= {x ∈ Ω, d(x, ∂Ω) = ǫ

} and Ω

= {x ∈ Ω, d(x, ∂Ω) ≥ ǫ

}, ǫ > 0. Then, for ǫ small enough, Σ

is hypersurface.

The measure of Ω − Ω

is k

ǫ

≤ meas(Ω − Ω

) = µ

(Ω − Ω

) ≤ k

ǫ

. Remark: for the unit ball B(0, ¯ 1), our new manifold is B(0, ¯ 1 − ǫ

).

( Proof of this fact; let’s consider d(x, ∂Ω) = d(x, z

), z

∈ ∂Ω, this imply that (d(x, z

))

≤ (d(x, z))

for all z ∈ ∂Ω which it is equivalent to (z − z

) · (2x − z − z

) ≤ 0 for all z ∈ ∂Ω, let’s consider a chart around z

and γ(t) a curve in ∂Ω, we have;

(γ(t) − γ(t

) · (2x − γ(t) − γ(t

)) ≤ 0 if we divide by (t − t

) (with the sign and tend t to t

), we have γ

(t

) · (x − γ(t

)) = 0, this imply that x = z

− sν

where ν

is the outward normal of ∂Ω at z

))

11

With this fact, we can say that S = {x, d(x, ∂Ω) ≤ ǫ} = {x = z

− sν

, z

∈ ∂Ω, −ǫ ≤ s ≤ ǫ}. It is sufficient to work on ∂Ω. Let’s consider a charts (z, D = B(z, 4ǫ

), γ

) with z ∈ ∂Ω such that ∪

B(z, ǫ

) is cover of ∂Ω . One can extract a finite cover (B (z

, ǫ

)), k = 1, ..., m, by the area formula the measure of S ∩ B (z

, ǫ

) is less than a kǫ (a ǫ-rectangle). For the reverse inequality, it is sufficient to consider one chart around one point of the boundary.

We write, Z

Ω

|∇(u

− u) · ∇ η ˜

|dx = Z

Ω_ǫ3

|∇(u

− u) · ∇˜ η

|dx + Z

Ω−Ω_ǫ3

|∇(u

− u) · ∇˜ η

|dx. (5)

Step 2.1: Estimate of R

Ω−Ω_ǫ3

|∇(u

− u) · ∇˜ η

|dx.

First, we know from the elliptic estimates that ||∇˜ η

||

≤ C

/ǫ

, C

depends on Ω

We know that (|∇u

|)

is bounded in L

, 1 < q < 2, we can extract from this sequence a subsequence which converge weakly to h ∈ L

. But, we know that we have locally the uniform convergence to |∇u| (by Brezis-Merle’s theorem), then, h = |∇u| a.e. Let q

be the conjugate of q.

We have, ∀f ∈ L

(Ω)

Z

Ω

|∇u

|f dx → Z

Ω

|∇u|f dx If we take f = 1

ǫ3

, we have:

for ǫ > 0 ∃ i

= i

(ǫ) ∈ N , i ≥ i

, Z

Ω−Ω_ǫ3

|∇u

| ≤ Z

Ω−Ω_ǫ3

|∇u| + ǫ

.

Then, for i ≥ i

(ǫ), Z

Ω−Ω_ǫ3

|∇u

| ≤ meas(Ω − Ω

)||∇u||

+ ǫ

= ǫ

(k

||∇u||

+ 1).

Thus, we obtain,

Z

Ω−Ω_ǫ3

|∇(u

− u) · ∇˜ η

|dx ≤ ǫC

(2k

||∇u||

+ 1) (6) The constant C

does not depend on ǫ but on Ω.

Step 2.2: Estimate of R

Ω_ǫ3

|∇(u

− u) · ∇˜ η

|dx.

12

We know that, Ω

⊂⊂ Ω, and ( because of Brezis-Merle’s interior estimates) u

→ u in C

(Ω

). We have,

||∇(u

− u)||

ǫ3)

≤ ǫ

, for i ≥ i

= i

(ǫ).

We write, Z

Ω_ǫ3

|∇(u

− u) · ∇˜ η

|dx ≤ ||∇(u

− u)||

ǫ3)

||∇˜ η

||

≤ C

ǫ for i ≥ i

,

For ǫ > 0, we have for i ∈ N , i ≥ max{i

, i

, i

}, Z

Ω

|∇(u

− u) · ∇˜ η

|dx ≤ ǫC

(2k

||∇u||

+ 2) (7) From (4) and (7), we have, for ǫ > 0, there is i

= i

(ǫ) ∈ N , i

= max{i

, i

, i

} such that,

Z

Ω

| − ∆[(u

− u)˜ η

]|dx ≤ 4π − ǫ

+ ǫ2C

(2k

||∇u||

+ 2 + C) (8) We choose ǫ > 0 small enough to have a good estimate of (1).

Indeed, we have:

( −∆[(u

− u)˜ η

] = g

in Ω ⊂ R

, (u

− u)˜ η

= 0 in ∂Ω.

with ||g

||

≤ 4π − ǫ

2 .

We can use Theorem 1 of [6] to conclude that there are q ≥ q > ˜ 1 such that:

Z

Vǫ(x0)

e

dx ≤ Z

Ω

e

dx ≤ C(ǫ, Ω).

where, V

(x

) is a neighberhood of x

in Ω. Here we have used that in a neighborhood of ¯ x

by the elliptic estimates, 1 − Cǫ ≤ η ˜

≤ 1.

Thus, for each x

∈ ∂Ω − {¯ x

, . . . , x ¯

} there is ǫ

> 0, q

> 1 such that:

Z

B(x0,ǫx0)

e

dx ≤ C, ∀ i. (9)

Now, we consider a cutoff function η ∈ C

( R

) such that

13

η ≡ 1 on B (x

, ǫ

/2) and η ≡ 0 on R

− B(x

, 2ǫ

/3).

We write

−∆(v

η) = W

e

η − 2∇v

· ∇η − v

∆η.

Because, by Poincar´e and Gagliardo-Nirenberg-Sobolev inequalities:

||v

||

≤ c

||∇v

||

≤ C

, 1 ≤ q < 2,

with, q

= 2q/(2 − q) > 2 > 1.

By the elliptic estimates, (v

)

is uniformly bounded in L

(V

(x

)). Finaly, we have, for some ǫ > 0 small enough,

||v

||

≤ c

∀ i.

Now, we consider a cutoff function η ∈ C

( R

) such that

η ≡ 1 on B (x

, ǫ

/2) and η ≡ 0 on R

− B(x

, 2ǫ

/3).

We write

−∆(u

η) = |x|

V

e

η − 2∇u

· ∇η − u

∆η.

By the elliptic estimates, (u

)

is uniformly bounded in L

(V

(x

)) and also in C

norm.

If we repeat this procedure another time, we have a boundedness of (u

)

and (v

)

in the C

norm, because they are bounded in W

⊂ W

norms with 2q/(2 − q) = q

> 2.

We have the same computations and conclusion if we consider a regular point x

for the measure µ

. We have proved that, there is a finite number of points x ¯

, . . . , x ¯

such that the squence (u

)

and (v

)

are locally uniformly bounded (in C

, θ > 0) in Ω ¯ − {¯ x

, . . . , x ¯

}.

Proof of theorem 1.2:

Without loss of generality, we can assume that 0 = x

is a blow-up point. Since the boundary is an analytic curve γ

(t), there is a neighborhood of 0 such that the curve γ

can be extend to a holomorphic map such that γ

(0) 6= 0 (series) and by the inverse mapping one can assume that this map is univalent around 0. In the case when the boundary is a simple Jordan curve the domain is simply connected. In the case that the domains has a finite number of holes it is conformally equivalent to a disk with a finite number of disks removed. Here we consider a general domain. Without loss of generality one can assume that γ

(B

) ⊂ Ω

14

and also γ

(B

) ⊂ ( ¯ Ω)

and γ

(−1, 1) ⊂ ∂Ω and γ

is univalent. This means that (B

, γ

) is a local chart around 0 for Ω and γ

univalent. (This fact holds if we assume that we have an analytic domain, (below a graph of an analytic function), we have necessary the condition ∂ Ω = ¯ ∂Ω and the graph is analytic, in this case γ

(t) = (t, ϕ(t)) with ϕ real analytic and an example of this fact is the unit disk around the point (0, 1) for example).

By this conformal transformation, we can assume that Ω = B

, the half ball, and ∂

B

is the exterior part, a part which not contain 0 and on which u

converge in the C

norm to u. Let us consider B

, the half ball with radius ǫ > 0. Also, one can consider a C

domain (a rectangle between two half disks) and by charts its image is a C

domain) We know that:

u

∈ C

( ¯ Ω).

Thus we can use integrations by parts (Stokes formula). The Pohozaev identity applied around the blow- up 0:

Z

B_ǫ⁺

∆u

< x|∇v

> dx = − Z

B_ǫ⁺

∆v

< x|∇u

> dx + Z

∂⁺B⁺_ǫ

g(∇u

, ∇v

)dσ, (10)

Thus, Z

Bǫ⁺

|x|

V

e

< x|∇v

> dx = − Z

B⁺ǫ

W

e

< x|∇u

> dx − Z

∂⁺Bǫ⁺

g(∇u

, ∇v

)dσ, (11) After integration by parts, we obtain:

Z

B⁺ǫ

2V

(1 + β)|x|

e

dx + Z

Bǫ⁺

< x|∇V

> |x|

e

dx + Z

∂B⁺ǫ

< ν|x > |x|

V

e

dσ+

+ Z

Bǫ⁺

W

e

dx + Z

Bǫ⁺

< x|∇W

> e

dx + Z

∂B⁺ǫ

< ν|x > W

e

dσ =

= − Z

∂⁺B⁺ǫ

g(∇u

, ∇v

)dσ,

Also, for u and v, we have:

Z

Bǫ⁺

2V (1 + β)|x|

e

dx + Z

B⁺ǫ

< x|∇V > |x|

e

dx + Z

∂B⁺ǫ

< ν|x > |x|

V e

dσ+

+ Z

B⁺ǫ

W e

dx + Z

B⁺ǫ

< x|∇W > e

dx + Z

∂B⁺ǫ

< ν|x > W e

dσ =

15

= − Z

∂⁺B⁺ǫ

g(∇u, ∇v)dσ,

If, we take the difference, we obtain:

Z

γ1(Bǫ⁺)

|x|

V

e

dx + Z

γ1(B⁺ǫ)

W

e

dx = o(ǫ) + o(1)

But, Z

γ1(B⁺_ǫ)

|x|

V

e

dx + Z

γ1(B_ǫ⁺)

W

e

dx = Z

∂γ1(B⁺_ǫ)

∂

u

dσ + Z

∂γ1(B_ǫ⁺)

∂

v

dσ

and,

Z

∂γ1(B⁺ǫ)

∂

u

dσ + Z

∂γ1(B⁺ǫ)

∂

v

dσ → α

+ β

> 0 (12) a contradiction.

R

[1] T. Aubin. Some Nonlinear Problems in Riemannian Geometry. Springer-Verlag, 1998.

[2] C. Bandle. Isoperimetric Inequalities and Applications. Pitman, 1980.

[3] Bartolucci, D. A ”sup+Cinf” inequality for Liouville-type equations with singular potentials. Math. Nachr. 284 (2011), no.

13, 1639-1651.

[4] Bartolucci, D. A ‘sup+Cinf’ inequality for the equation−∆u=V e^u/|x|^2α. Proc. Roy. Soc. Edinburgh Sect. A 140 (2010), no. 6, 1119-1139

[5] H. Brezis, YY. Li and I. Shafrir. A sup+inf inequality for some nonlinear elliptic equations involving exponential nonlineari- ties. J.Funct.Anal.115 (1993) 344-358.

[6] H. Brezis, F. Merle. Uniform estimates and Blow-up behavior for solutions of−∆u=V(x)e^uin two dimension. Commun.

in Partial Differential Equations, 16 (8 and 9), 1223-1253(1991).

[7] W. Chen, C. Li. A priori estimates for solutions to nonlinear elliptic equations. Arch. Rational. Mech. Anal. 122 (1993) 145-157.

[8] H. Brezis, W. A. Strauss. Semi-linear second-order elliptic equations in L1. J. Math. Soc. Japan 25 (1973), 565-590.

[9] C-C. Chen, C-S. Lin. A sharp sup+inf inequality for a nonlinear elliptic equation inR². Commun. Anal. Geom. 6, No.1, 1-19 (1998).

[10] D.G. De Figueiredo, P.L. Lions, R.D. Nussbaum, A priori Estimates and Existence of Positive Solutions of Semilinear Elliptic Equations, J. Math. Pures et Appl., vol 61, 1982, pp.41-63.

[11] D.G. De Figueiredo. J. M. do O. B. Ruf. Semilinear elliptic systems with exponential nonlinearities in two dimensions. Adv.

Nonlinear Stud. 6 (2006), no. 2, 199-213.

[12] Dupaigne, L. Farina, A. Sirakov, B. Regularity of the extremal solutions for the Liouville system. Geometric partial differential equations, 139-144, CRM Series, 15, Ed. Norm., Pisa, 2013.

[13] YY. Li, I. Shafrir. Blow-up analysis for solutions of−∆u=V e^uin dimension two. Indiana. Math. J. Vol 3, no 4. (1994).

1255-1270.

[14] YY. Li. Harnack Type Inequality: the method of moving planes. Commun. Math. Phys. 200,421-444 (1999).

[15] L. Ma, J-C. Wei. Convergence for a Liouville equation. Comment. Math. Helv. 76 (2001) 506-514.

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[16] Montenegro. M. Minimal solutions for a class of ellptic systems. Bull. London. Math. Soc. 37 (2005), no. 3, 405-416.

[17] I. Shafrir. A sup+inf inequality for the equation−∆u=V e^u. C. R. Acad.Sci. Paris S´er. I Math. 315 (1992), no. 2, 159-164.

DEPARTEMENT DEMATHEMATIQUES, UNIVERSITEPIERRE ETMARIECURIE, 2PLACEJUSSIEU, 75005, PARIS, FRANCE. E-mail address:samybahoura@yahoo.fr, samybahoura@gmail.com

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