Salim Rao

Extreme-value theory applied to boundary estimation

In this paper, the problem of nonparametric estimation of boundaries is investigated from the perspective of extreme value theory. This enables us to revisit the asymptotic theory of the popular three disposal hull estimator in a more general setting, to derive new estimators and to provide useful asymptotic results for the boundary function.

Introduction

In this paper, we consider x ∈ R p + , a vector of production factors (inputs) used to produce a single quantity (output) y ∈ R + . The attainable production set is then defined as P = {(x, y) ∈ R p + × R + | x can produce y}. Assumptions are usually made on this set, such as three disposability of inputs and outputs, meaning that if (x, y) ∈ P, then (x ′ , y ′ ) ∈ P for any (x ′ , y ′ ) such that x ′ ≥ x and y ′ ≤ y. To the extent that the efficiency of a firm is a concern, the boundary of P is of interest. The efficient boundary of P is the locus of optimal production plans (maximal achievable output for a given level of inputs). In our setup, the production boundary is represented by the graph of the production function θ(x) = sup{y | (x, y) ∈ P}. The economic efficiency score of a firm operating at the level (x, y) is then given by the ratio θ(x)/y.

Cazals et al. [START_REF] Cazals | Nonparametric frontier estimation: a robust approach[END_REF] proposed a probabilistic interpretation of the production boundary. Let P be the support of the joint distribution of a random vector (X, Y ) ∈ R p + × R + . The distribution function of (X, Y ) can be denoted F (x, y) and F (•|x) = F (x, •)/F X (x) will be used to denote the conditional distribution function of Y given X ≤ x, with F X (x) = F (x, ∞) > 0. It has been proven in [START_REF] Cazals | Nonparametric frontier estimation: a robust approach[END_REF] that

θ(x) = sup{y ≥ 0 | F (y|x) < 1}
is a monotone non-decreasing function with x. So, for all x ′ ≥ x with respect to the partial order, θ(x ′ ) ≥ θ(x). The graph of θ is the smallest non-decreasing surface which is greater than or equal to the upper boundary of P. Further, it has been shown that under the three disposability assumption, θ ≡ θ, that is, the graph of θ coincides with the production boundary.

Since P is unknown, it must be estimated from a sample of i.i.d. firms X n = {(X i , Y i ) | i = 1, . . . , n}. The three disposal hull (3DH) is

P 3DH = {(x, y) ∈ R p+1 + | y ≤ Y i , x ≥ X i , i = 1, . . . , n}. The resulting 3DH estimator of θ(x) is θ1 (x) = sup{y ≥ 0 | F (y|x) < 1} = max i,Xi≤x Y i ,
where F (y|x) = Fn (x, y)/ FX (x) with Fn (x, y) = (1/n) n i=1 I(X i ≤ x, Y i ≤ y) and FX (x) = Fn (x, ∞). This estimator represents the lowest monotone step function covering all of the data points (X i , Y i ). To summarize, under regularity conditions, the 3DH estimator θ1 (x) is consistent and converges to a Weibull distribution with some unknown parameters.

In this paper, we first analyze the properties of the 3DH estimator from an extreme value theory perspective. First, we provide the necessary and sufficient condition for the 3DH estimator to converge in distribution and we specify the asymptotic distribution with the appropriate rate of convergence. We also provide a limit theorem for moments in a general framework. Second, we show how the unknown parameter ρ x > 0, involved in the necessary and sufficient extreme value conditions, is linked to the dimension p + 1 of the data and to the shape parameter β > -1 of the joint density: in the general setting where p ≥ 1 and β = β x may depend on x, we obtain, under a convenient regularity condition, the general convergence rate n -1/ρx = n -1/(βx+p+1) of the 3DH estimator θ1 (x). Third, we suggest a strongly consistent and asymptotically normal estimator of the unknown parameter ρ x of the asymptotic Weibull distribution of θ1 (x). This also answers the important question of how to estimate the shape parameter β x of the joint density of (X, Y ) when it approaches the boundary of the support P. This branch of statistics has become very active these past ten years, the main contributions to this domain are listed below:

• Theoretical issues: [START_REF] Rao | A review on conditional extreme value analysis[END_REF][START_REF] Rao | Linear aggregation of conditional extreme-value index estimators[END_REF][START_REF] Rao | Nonlinear aggregation of conditional extreme-value index estimators[END_REF][START_REF] Das | Detecting a conditional extreme value model[END_REF][START_REF] Gardes | Functional kernel estimators of large conditional quantiles[END_REF][START_REF] Daouia | Kernel estimators of extreme level curves[END_REF][START_REF] Gardes | Conditional extremes from heavy-tailed distributions: An application to the estimation of extreme rainfall return levels[END_REF][START_REF] Gardes | Functional nonparametric estimation of conditional extreme quantiles[END_REF][START_REF] Gardes | A moving window approach for nonparametric estimation of the conditional tail index[END_REF][START_REF] Beirlant | Local polynomial maximum likelihood estimation for pareto-type distributions[END_REF][START_REF] Ac Davison | Local likelihood smoothing of sample extremes[END_REF][START_REF] Chavez-Demoulin | Generalized additive modelling of sample extremes[END_REF][START_REF] Hall | Nonparametric analysis of temporal trend when fitting parametric models to extreme-value data[END_REF][START_REF] Das | Conditioning on an extreme component: Model consistency with regular variation on cones[END_REF][START_REF] Beirlant | Nonparametric estimation of extreme conditional quantiles[END_REF][START_REF] Abdous | Extreme behaviour for bivariate elliptical distributions[END_REF][START_REF] Wang | Estimation of high conditional quantiles for heavy-tailed distributions[END_REF][START_REF] Wang | Tail index regression[END_REF],

• Quantile regression: [START_REF] Koenker | Quantile regression: An introduction[END_REF][START_REF] Daouia | On kernel smoothing for extremal quantile regression[END_REF][START_REF] Rosen | Extreme percentile regression[END_REF] • Application to finance: [START_REF] Byström | Managing extreme risks in tranquil and volatile markets using conditional extreme value theory[END_REF][START_REF] Bali | A conditional extreme value volatility estimator based on high-frequency returns[END_REF][START_REF] Ghorbel | Predictive performance of conditional extreme value theory in value-at-risk estimation[END_REF][START_REF] Byström | Extreme value theory and extremely large electricity price changes[END_REF][START_REF] Marimoutou | Extreme value theory and value at risk: application to oil market[END_REF][START_REF] Wang | A model of conditional var of high frequency extreme value based on generalized extreme value distribution[END_REF],

• Edge estimation [START_REF] Jacob | Regression and edge estimation[END_REF][START_REF] Jacob | Estimating the edge of a poisson process by orthogonal series[END_REF],

• Frontier estimation [START_REF] Daouia | A γ-moment approach to monotonic boundaries estimation: with applications in econometric and nuclear fields[END_REF][START_REF] Rao | Nonlinear aggregation of frontier estimators[END_REF][START_REF] Rao | Linear aggregation of frontier estimators[END_REF][START_REF] Rao | Frontier estimation as a particular case of conditional extreme value analysis[END_REF][START_REF] Simar | Performance of the bootstrap for dea estimators and iterating the principle[END_REF][START_REF] Simar | Testing restrictions in nonparametric efficiency models[END_REF][START_REF] Simar | Inferences from cross-sectional, stochastic frontier models[END_REF][START_REF] Simar | Of course we can bootstrap dea scores! but does it mean anything? logic trumps wishful thinking[END_REF][START_REF] Daouia | Frontier estimation and extreme value theory[END_REF][START_REF] Daouia | Asymptotic representation theory for nonstandard conditional quantiles[END_REF][START_REF] Daouia | Functional convergence of quantiletype frontiers with application to parametric approximations[END_REF][START_REF] Bouchard | Nonparametric frontier estimation by linear programming[END_REF][START_REF] Aragon | Nonparametric frontier estimation: A conditional quantile-based approach[END_REF][START_REF] Girard | L 1-optimal nonparametric frontier estimation via linear programming[END_REF][START_REF] Bouchard | Some linear programming methods for frontier estimation[END_REF][START_REF] Peng | Bias-corrected estimators for monotone and concave frontier functions[END_REF][START_REF] Daouia | Regularization of nonparametric frontier estimators[END_REF][START_REF] Simar | Detecting outliers in frontier models: A simple approach[END_REF][START_REF] Simar | Aspects of statistical analysis in dea-type frontier models[END_REF][START_REF] Daouia | Robustness and inference in nonparametric partial frontier modeling[END_REF][START_REF] Simar | Stochastic fdh/dea estimators for frontier analysis[END_REF][START_REF] Park | Efficient semiparametric estimation in a stochastic frontier model[END_REF][START_REF] Daouia | Estimating frontier cost models using extremiles[END_REF][START_REF] Greene | Maximum likelihood estimation of econometric frontier functions[END_REF][START_REF] Gijbels | On estimation of monotone and concave frontier functions[END_REF][START_REF] Martins-Filho | A smooth nonparametric conditional quantile frontier estimator[END_REF][START_REF] Daouia | Robust nonparametric frontier estimators: qualitative robustness and influence function[END_REF][START_REF] Daouia | Nonparametric efficiency analysis: A multivariate conditional quantile approach[END_REF][START_REF] Sengupta | Stochastic data envelopment analysis: a new approach[END_REF][START_REF] Berezkin | Hybrid adaptive methods for approximating a nonconvex multidimensional pareto frontier[END_REF][START_REF] Simar | A general methodology for bootstrapping in non-parametric frontier models[END_REF][START_REF] Cazals | Nonparametric frontier estimation: a robust approach[END_REF][START_REF] Kneip | Asymptotics for dea estimators in nonparametric frontier models[END_REF][START_REF] Kumbhakar | Stochastic frontier analysis[END_REF][START_REF] Schmidt | Simple tests of alternative specifications in stochastic frontier models[END_REF][START_REF] Jeong | Linearly interpolated fdh efficiency score for nonconvex frontiers[END_REF][START_REF] Girard | Frontier estimation via kernel regression on high power-transformed data[END_REF][START_REF] Girard | Frontier estimation with local polynomials and high power-transformed data[END_REF][START_REF] Daouia | Nadarayas estimates for large quantiles and free disposal support curves[END_REF][START_REF] Florens | Parametric approximations of nonparametric frontiers[END_REF][START_REF] Simar | Sensitivity analysis of efficiency scores: How to bootstrap in nonparametric frontier models[END_REF][START_REF] Simar | Statistical inference in nonparametric frontier models: The state of the art[END_REF][START_REF] Simar | Statistical inference in nonparametric frontier models: recent developments and perspectives. The measurement of productive efficiency and productivity growth[END_REF][START_REF] Girard | Frontier estimation with kernel regression on high order moments[END_REF],

• Boundary estimation [START_REF] Girard | A note on extreme values and kernel estimators of sample boundaries[END_REF][START_REF] Girard | Smoothed extreme value estimators of non-uniform point processes boundaries with application to star-shaped supports estimation[END_REF][START_REF] Girard | Central limit theorems for smoothed extreme value estimates of poisson point processes boundaries[END_REF][START_REF] Girard | Extreme values and kernel estimates of point processes boundaries[END_REF][START_REF] Girard | Projection estimates of point processes boundaries[END_REF][START_REF] Girard | Extreme values and haar series estimates of point process boundaries[END_REF][START_REF] Hall | Local likelihood tracking of fault lines and boundaries[END_REF][START_REF] Delaigle | Estimation of boundary and discontinuity points in deconvolution problems[END_REF][START_REF] Hall | Estimating a changepoint, boundary, or frontier in the presence of observation error[END_REF][START_REF] Bi | Estimating the self-thinning boundary line as a density-dependent stochastic biomass frontier[END_REF][START_REF] Geffroy | Asymptotic normality of the l 1-error of a boundary estimator[END_REF][START_REF] Girard | On the asymptotic normality of the l1-error for haar series estimates of poisson point processes boundaries[END_REF].

By construction, the 3DH estimator is very non-robust to extremes. Recently, Aragon et al. [START_REF] Aragon | Nonparametric frontier estimation: A conditional quantile-based approach[END_REF] constructed an original estimator of θ(x), which is more robust than θ1 (x), but which keeps the same limiting Weibull distribution as θ1 (x) under the restrictive condition β = 0. In this paper, we provide further insights and generalize their main result. We also suggest attractive estimators of θ(x)

converging to a normal distribution, which appear to be robust to outliers. The paper is organized as follows. Section 2 presents the main results of the paper. Section 3 concludes the paper, with proofs deferred for the Appendix.

Main results

From now on, we assume that x ∈ R p + such that F X (x) > 0 and will denote by θ α (x) and θα (x), respectively, the α-quantiles of the distribution function F (•|x) and its empirical version F (•|x),

θ α (x) = inf{y ≥ 0 | F (y|x) ≥ α} and θα (x) = inf{y ≥ 0 | F (y|x) ≥ α} with α ∈ ]0, 1]. When α ↑ 1,
the conditional quantile θ α (x) tends to θ 1 (x), which coincides with the boundary function θ(x). Likewise, θα (x) tends to the 3DH estimator θ1 (x) of θ(x) as α ↑ 1.

We first derive the following interesting results on the problem of convergence in distribution of suitably normalized maxima b -1 n ( θ1 (x)-θ(x)). We will denote by Γ(•) the gamma function.

-1 n θ1 (x) -θ(x) d -→ G x , (1)
then G x (y) coincides with Ψ ρx (y) = exp{-(-y) ρx } with support ]-∞, 0] for some ρ x > 0.

(ii) There exists b n > 0 such that b -1 n ( θ1 (x)θ(x)) converges in distribution if and only if

lim t→∞ 1-F θ(x)-1/tz | x / 1-F θ(x)-1/t | x = z -ρx
for all z > 0

(2) (regular variation with exponentρ x , notation 1 -F (θ(x) -

1 t | x) ∈ RV -ρx ).
In this case, the norming constants b n can be chosen as b n = θ(x) -

θ 1-(1/nFX (x)) (x). (iii) Given (2), lim n→∞ E{b -1 n (θ(x) -θ1 (x))} k = Γ(1 + kρ -1
x ) for all integers k ≥ 1 and [START_REF] Aragon | Nonparametric frontier estimation: A conditional quantile-based approach[END_REF], this function can be represented as t -ρx L x (t) with L x (•) ∈ RV 0 (L x being slowly varying) and so the extreme value condition (2) holds if and only if we have the following representation:

lim n→∞ P θ1 (x) -E( θ1 (x)) {Var( θ1 (x))} 1/2 ≤ y = Ψ ρx [{Γ(1 + 2ρ -1 x ) -Γ 2 (1 + ρ -1 x )} 1/2 y -Γ(1 + ρ -1 x )]. Since the function t → F X (x)[1 -F (θ(x) -1 t | x)] ∈ RV -ρx (regularly varying in t → ∞) by
F X (x)[1 -F (y|x)] = L x {θ(x) -y} -1 θ(x) -y ρx as y ↑ θ(x). (3)
In the particular case where L x ({θ(x)-y} -1 ) = ℓ x is a strictly positive function in x, it is shown in the next corollary that b n ∼ (nℓ x ) -1/ρx . From now on, a random variable W is said to follow the distribution Weibull(1, ρ x ) if W ρx is exponential with parameter 1.

Corollary 2.1 Given (3) or, equivalently, (2) with L x ({θ(x) -y} -1 ) = ℓ x > 0, we have

(nℓ x ) 1/ρx θ(x) -θ1 (x) d -→ Weibull(1, ρ x ) as n → ∞.
We assume the differentiability of the functions ℓ x , ρ x with ρ x > p and θ(x) in order to ensure the existence of the joint density near its support boundary. We distinguish between three different behaviors of this density at the boundary point (x, θ(x)) ∈ R p+1 based on how the value of ρ x compares to the dimension (p + 1): when ρ x > p + 1, the joint density decays to zero at a speed of power ρ x -(p + 1) of the distance from the boundary; when ρ x = p + 1, the density has a sudden jump at the boundary; when ρ x < p + 1, the density increases toward infinity at a speed of power ρ x -(p + 1) of the distance from the boundary. The case ρ x ≤ p + 1 corresponds to sharp or fault-type boundarys.

As an immediate consequence, we obtain the convergence in distribution of the 3DH estimator with the convergence rate n -1/(β+2) . In the other particular case where the joint density is strictly positive on the boundary, we achieve the best rate of convergence n -1/(p+1) . We answer the important question of how to estimate the shape parameter β x or, equivalently, the regular variation exponent ρ x in [START_REF] Aragon | Nonparametric frontier estimation: A conditional quantile-based approach[END_REF].

As an immediate consequence of Theorem 2.1(iii), we obtain

E{θ(x) -θ1 (x)} k = k{β x + p + 1} -1 {nℓ x } -k/(βx+p+1) Γ(k{β x + p + 1} -1 ) (4) + o n -k/(βx+p+1) .
The result (4) also reflects the well-known curse of dimensionality from which the 3DH estimator θ1 (x) suffers as the number p of inputs-usage increases, as pointed out earlier in the particular case where β x = 0.

By an appropriate choice of α as a function of n, Aragon et al. [START_REF] Aragon | Nonparametric frontier estimation: A conditional quantile-based approach[END_REF] have shown that θα (x) estimates the full boundary θ(x) itself and converges to the same Weibull distribution as the 3DH θ1 (x) . The next theorem provides further insights and generalizes their main result.

Theorem 2.2 (i) If b -1 n ( θ1 (x) -θ(x)) d -→ G x , then for any fixed integer k ≥ 0, b -1 n θ1-k/(n FX (x)) (x) -θ(x) d -→ H x as n → ∞ for the distribution function H x (y) = G x (y) k i=0 (-log G x (y)) i /i!.
(ii) Suppose that the upper bound of the support of

Y is finite. If b -1 n ( θ1 (x) - θ(x)) d -→ G x , then b -1 n ( θαn (x) -θ(x)) d -→ G x for all sequences α n → 1 satisfying nb -1 n (1 -α n ) → 0.
When θ1 (x) converges in distribution, the estimator θαn (x), for α n := 1k/n FX (x) < 1 (that is, k = 1, 2, . . . , in Theorem 2.2(i)), estimates θ(x) itself and also converges in distribution, with the same scaling, but a different limit distribution (here, nb

-1 n (1 -α n ) a.s.
-→ ∞). To recover the same limit distribution as the 3DH estimator, it suffices to require that α n → 1 rapidly so that nb -1 n (1α n ) → 0. This extends the main result of Aragon et al. ( [START_REF] Aragon | Nonparametric frontier estimation: A conditional quantile-based approach[END_REF], Theorem 4.3), where the convergence rate achieves n -1/(p+1) under the restrictive assumption that the density of (X, Y ) is strictly positive on the boundary. Note, also, that the estimate θαn does not envelop all of the data points providing a robust alternative to the 3DH boundary θ1 ; see [START_REF] Daouia | Robust nonparametric frontier estimators: qualitative robustness and influence function[END_REF] for an analysis of its quantitative and qualitative robustness properties.

The important question of how to estimate ρ x from the multivariate random sample X n is very similar to the problem of estimating the so-called extreme value index, which is based on a sample of univariate random variables. An attractive estimation method has been proposed, which can be easily adapted to our conditional approach: let k = k n be a sequence of integers tending to infinity and let k/n → 0 as n → ∞. A Pickands-type estimate of ρ x can be derived as

ρx = log 2 log θ1-(2k-1)/(n FX (x)) (x) -θ1-(4k-1)/(n FX (x)) (x) θ1-(k-1)/(n FX (x)) (x) -θ1-(2k-1)/(n FX (x)) (x) -1
.

The following result is particularly important since it allows the hypothesis ρ x > 0 to be tested and will later be employed to derive asymptotic confidence intervals for θ(x).

Theorem 2.3 (i) If (2) holds, k n → ∞ and k n /n → 0, then ρx p -→ ρ x . (ii) If (2) holds, k n /n → 0 and k n / log log n → ∞, then ρx a.s. -→ ρ x . (iii) Assume that U (t) := θ 1-1/(tFX (x)) (x), t > 1 FX (x)
, has a positive derivative and that there exists a positive function A(•) such that for z > 0, lim t→∞ {(tz) 1+1/ρx U ′ (tz)-t 1+1/ρx U ′ (t)}/A(t) = ± log(z), for either choice of the sign (Π-variation, which will in the sequel be denoted by: ±t

1+1/ρx U ′ (t) ∈ Π(A)). Then, k n (ρ x -ρ x ) d -→ N (0, σ 2 (ρ x )), ( 5
)
with asymptotic variance σ 2 (ρ x ) = ρ 2 x (2 1-2/ρx + 1)/{(2 -1/ρx -1) log 4} 2 , for k n → ∞ satisfying k n = o(n/g -1 (n)), where g -1 is the generalized inverse function of g(t) = t 3+2/ρx {U ′ (t)/A(t)} 2 .
(iv) If, for some κ > 0 and δ > 0, the function

{t ρx-1 F ′ (θ(x) -1 t | x) - δ} ∈ RV -κ , then (5) holds with g(t) = t 3+2/ρx {U ′ (t)/(t 1+1/ρx U ′ (t) - [δF X (x)] -1/ρx (ρ x ) 1/ρx-1 )} 2 .
Note that the second order regular variation conditions (iii) and (iv) of Theorem 2.3 are difficult to check in practice, which makes the theoretical choice of the sequence {k n } a hard problem. In practice, in order to choose a reasonable estimate ρx (k n ) of ρ x , one can construct the plot of ρx , consisting of the points {(k, ρx (k)), 1 ≤ k < n FX (x)/4}, and select a value of ρ x at which the obtained graph looks stable. This technique is known as the Pickands plot in the univariate extreme value literature.

We can also easily adapt the well-known moment estimator for the index of a univariate extreme value distribution to our conditional setup. Define

M (j) n = 1 k k-1 i=0 log θ1-i/(n FX (x)) (x) -log θ1-k/(n FX (x)) (x) j for each j = 1, 2 and k = k n < n.
We can then define the moment-type estimator for the conditional regularvariation exponent ρ x as ρx = -M (1) n + 1 -

1 2 1 -M (1) n 2 /M (2) n -1 -1 . Theorem 2.4 (i) If (2) holds, k n /n → 0 and k n → ∞, then ρx p -→ ρ x . (ii) If (2) holds, k n /n → 0 and k n /(log n) δ → ∞ for some δ > 0, then ρx a.s.
-→ ρ x .

(iii) If ±t 1/ρx {θ(x)-U (t)} ∈ Π(B) for some positive function B, then √ k n (ρ xρ x ) has, asymptotically, a normal distribution with mean zero and variance

ρ x (2 + ρ x )(1 + ρ x ) 2 4 -8 (2 + ρ x ) (3 + ρ x ) + (11 + 5ρ x )(2 + ρ x ) (3 + ρ x )(4 + ρ x ) for k n → ∞ satisfying k n = o(n/g -1 (n)), where g(t) = t 1+2/ρx [{log θ(x)- log U (t)}/B(t)] 2 .
The next theorem enables the construction of confidence intervals for θ(x) and for high quantile-type boundarys θ 1-pn/FX (x) (x) when p n → 0 and np n → ∞.

Theorem 2.5 (i) Suppose that F (•|x) has a positive density F ′ (•|x) such that F ′ (θ(x) -1 t | x) ∈ RV 1-ρx . Then, 2k n θ1-(kn-1)/(n FX (x)) (x) -θ 1-pn/FX (x) (x) θ1-(kn-1)/(n FX (x)) (x) -θ1-(2kn-1)/(n FX (x)) (x) d -→ N (0, V 1 (ρ x )), where V 1 (ρ x ) = ρ -2 x 2 1-2/ρx /(2 -1/ρx -1) 2 , provided that p n → 0, np n → ∞ and k n = [np n ].
(ii) Suppose that the conditions of Theorem 2.3(iii) or (iv) hold, and define

θ * 1 (x) := (2 1/ ρx -1) -1 θ1-(kn-1)/(n FX (x)) (x) -θ1-(2kn-1)/(n FX (x)) (x) + θ1-(kn-1)/(n FX (x)) (x).
Then, putting V 2 (ρ x ) = 3ρ -2 x 2 -1-2/ρx /(2 -1/ρx -1) 6 , we have

2k n θ * 1 (x) -θ(x) θ1-(kn-1)/(n FX (x)) (x) -θ1-(2kn-1)/(n FX (x)) (x) d -→ N (0, V 2 (ρ x )).
(iii) Suppose that the conditions of Theorem 2.3(iii) or (iv) hold, and define

θ * 1 (x) := (2 1/ρx -1) -1 θ1-(kn-1)/(n FX (x)) (x) -θ1-(2kn-1)/(n FX (x)) (x) + θ1-(kn-1)/(n FX (x)) (x).
Then, putting 4 , we have

V 3 (ρ x ) = ρ -2 x 2 -2/ρx /(2 -1/ρx -1)
2k n θ * 1 (x) -θ(x) θ1-(kn-1)/(n FX (x)) (x) -θ1-(2kn-1)/(n FX (x)) (x) d -→ N (0, V 3 (ρ x )), (6) θ1
-(kn-1)/(n FX (x)) (x) -θ1-(2kn-1)/(n FX (x)) (x) n 2k n U ′ n 2k n p -→ ρ x (1 -2 -1/ρx ).
Note that Theorem 2.5(ii) is still valid if the estimate ρx is replaced by the true value ρ x , up to a change of the asymptotic variance. It is easy to see that V 2 (ρ x ) ≥ V 3 (ρ x ) and so the estimator θ * 1 (x) of θ(x) is asymptotically more efficient than θ * 1 (x). We also conclude from (6) that θ * 1 (x) and θ * 1 (x) have the same rate of convergence, namely nU ′ ( n 2kn )/(2k n ) 3/2 . In the particular case where L x ({θ(x) -y} -1 ) = ℓ x in (3), we have U ′ ( n 2kn ) = 1 ρx ( 1 ℓx ) 1/ρx ( 2kn n ) 1+1/ρx . Note, also, that in this particular case, the condition of Theorem 2.5(i) holds, that is,

F ′ (θ(x) -1 t | x) = ℓxρx FX (x) ( 1 t ) ρx-1 ∈ RV 1-ρx .
However, the conditions of Theorem 2.3(iii) and (iv) do not hold since both functions t 1+1/ρx U ′ (t) =

1 ρx ( 1 ℓx ) 1/ρx and t ρx-1 F ′ (θ(x)-1 t | x) = ℓxρx FX (x) are constant in t.
Nevertheless, the conclusions of Theorem 2.3(iii) and (iv) hold in this case for all sequences k n → ∞ satisfying kn n → 0. The same is true for the conclusion of Theorem 2.5(ii).

Theorem 2.6 If the condition of Corollary 2.1 holds, k n → ∞ and k n /n → 0 as n → ∞, then {ρ x k 1/2 n /(k n /nℓ x ) 1/ρx } θ1-(kn-1)/(n FX (x)) (x) + (k n /nℓ x ) 1/ρx -θ(x) d -→ N (0, 1)
as n → ∞.

The optimization of the asymptotic mean-squared error of θ1-(kn-1)/(n FX (x)) (x) is not an appropriate criteria for selecting the optimal k n since the resulting value of k n does not depend on n.

We shall now construct asymptotic confidence intervals for both θ(x) and θ 1-pn/FX (x) (x), using the sums M Theorem 2.7 (i) Under the conditions of Theorem 2.5(i),

k n θ1-kn/(n FX (x)) (x) -θ 1-pn/FX (x) (x) M (1) n θ1-kn/(n FX (x)) (x) d -→ N (0, V 4 (ρ x )),
where

V 4 (ρ x ) = (1 + 1/ρ x ) 2 , provided that p n → 0, np n → ∞ and k n = [np n ].
(ii) Suppose that the conditions of Theorem 2.4(iii) hold and that U (•) has a regularly varying derivative U ′ ∈ RV -ρx . Define the moment estimator θ(x) = θ1-kn/(n FX (x)) (x){1 + M

n (1 + ρx )}. Then,

k n θ(x) -θ(x) M (1)
n (1 + 1/ρ x ) θ1-kn/(n FX (x)) (x) d -→ N (0, V 5 (ρ x )), V 5 (ρ x ) = ρ 2 x ρ x (2 + ρ x ) + ρ x (2 + ρ x ) 4 -8 (2 + ρ x ) (3 + ρ x ) + (11 + 5ρ x )(2 + ρ x ) (3 + ρ x )(4 + ρ x ) - 4ρ x (3 + ρ x )
.

We consider the case where the support boundary is linear. We choose (X, Y ) uniformly distributed over the region D = {(x, y) | 0 ≤ x ≤ 1, 0 ≤ y ≤ x}. In this case (see, for example, [START_REF] Daouia | Robust nonparametric frontier estimators: qualitative robustness and influence function[END_REF]), it is easy to see that θ(x) = x and F X (x)[1 -F (y|x)] = (θ(x)y) 2 for all 0 ≤ y ≤ θ(x). Thus, L x (•) = ℓ x = 1 and ρ x = 2 for all x. Therefore, the conclusions of all Theorems 2.1-2.6 hold.

We now choose a nonlinear monotone upper boundary given by the Cobb-Douglas model Y = X 1/2 exp(-U ), where X is uniform on [0, 1] and U , independent of X, is exponential with parameter λ = 3 (see, for example, [START_REF] Daouia | Robust nonparametric frontier estimators: qualitative robustness and influence function[END_REF]). Here, the boundary function is θ(x) = x 1/2 and the conditional distribution function is F (y|x) = 3x -1 y 2 -2x -3/2 y 3 for 0 < x ≤ 1 and 0 ≤ y ≤ θ(x). It is then easily seen that the extreme value condition (2) or, equivalently, (3) holds with ρ x = 2 and L x (z) = F X (x)[3θ(x) -2 z ]/[θ(x)] 3 for all x ∈ ]0, 1] and z > 0.

Concluding remarks

In our approach, we provide the necessary and sufficient condition for the 3DH estimator θ1 (x) to converge in distribution, we specify its asymptotic distribution with the appropriate convergence rate and provide a limit theorem for moments in a general framework. We also provide further insights and generalize the main result of [START_REF] Aragon | Nonparametric frontier estimation: A conditional quantile-based approach[END_REF] on robust variants of the 3DH estimator, and we provide strongly consistent and asymptotically normal estimators ρx and ρx of the unknown conditional tail index ρ x involved in the limit law of θ1 (x). Moreover, when the joint density of (X, Y ) decreases to zero or increases toward infinity at a speed of power β x > -1 of the distance from the boundary, as is often assumed in the literature, we answer the question of how ρ x is linked to the data dimension p + 1 and to the shape parameter β x . The quantity β x = 0 describes the rate at which the density tends to infinity (in the case β x < 0) or to 0 (in the case β x > 0) at the boundary. When β x = 0, the joint density is strictly positive on the boundary. We establish that ρ x = β x + (p + 1).

We propose new extreme-value-based boundary estimators θ * 1 (x), θ * 1 (x) and θ(x), which are asymptotically normally distributed and provide useful asymptotic confidence bands for the monotone boundary function θ(x). These estimators have the advantage of not being limited to a bi-dimensional support and benefit from their explicit and easy formulations, which is not the case for estimators defined by optimization problems, such as local polynomial estimators. Their asymptotic normality is derived under quite natural and general extreme value conditions, without Lipschitz conditions on the boundary and without recourse to assumptions either on the marginal distribution of X or on the conditional distribution of Y given X = x, as is often the case in both statistical and econometrics literature on boundary estimation. The study of the asymptotic properties of the different estimators considered in the present paper is easily carried out by relating them to a simple dimensionless random sample and then applying standard extreme value theory (for example, [START_REF] Dekkers | On the estimation of extreme-value index and large quantiles estimation[END_REF]).

It should be clear that the monotonicity constraint on the boundary is the main difference with most of the existing approaches in the statistical literature. Indeed, the joint support of a random vector (X, Y ) is often described in the literature as the set {(x, y) | y ≤ θ(x)}, where the graph of θ is interpreted as its upper boundary. As a matter of fact, the function of interest, θ, in our approach is the smallest monotone non-decreasing function which is greater than or equal to the boundary function θ. To our knowledge, only the estimators 3DH and DEA estimate the quantity θ. Of course, θ coincides with θ when the boundary curve is monotone, but the construction of estimators of the end-point θ(x) of the conditional distribution of Y given X = x requires a smoothing procedure, which is not the case when the distribution of Y is conditioned by X ≤ x.

We illustrate how the large-sample theory applies in practice by carrying out some Monte Carlo experiments. Good estimates of θ(x) and ρ x may require a large sample of the order of several thousand. Theoretically selecting the optimal extreme conditional quantiles θα(kn(x)) for estimating θ(x) and/or ρ x is a difficult question that is worthy of future research. Here, we suggest a simple automatic data-driven method that provides a reasonable choice of the sequence {k n (x)} for large samples.

The empirical study reveals that the simultaneous estimation of the tail index and of the boundary function requires large sample sizes to provide sensible results. The moment estimators of ρ x and of θ(x) sometimes provide better estimations than the Pickands estimates and sometimes not. When considering bias and MSE, θ(x) and ρx provide more accurate estimations, but when the sample size is large enough, θ * 1 (x) and ρx significantly improve and even seem to outperform the moment estimators. As far as the inference on ρ x is concerned, ρx also provides quite reliable confidence intervals, but ρx provides more satisfactory results for sufficiently large samples. However, when inference about the boundary function itself is concerned, the moment estimator provides very poor results compared with the Pickands estimator.

On the other hand, the performance of the estimator θ * 1 (x), computed when ρ x is known, is quite remarkable, even compared with the popular 3DH. The confidence intervals for θ(x) are very easy to compute and have quite good coverages. In addition, the results are quite stable with respect to the choice of the 'smoothing' parameter k n (x). As shown in our illustrations, the estimates also have the advantage of being robust to extreme values. This suggests, even if ρ x is unknown, the use of a plug-in version of θ * 1 (x) for making inference on θ(x): here, in a first step, we estimate ρ x (using the moment estimator, unless N x is large enough), then we use the asymptotic results for θ * 1 (x), as if ρ x was known. A sensible practice is not to restrict the first step to one procedure, but rather to check that both Pickands and moment estimators point toward similar conclusions.

Appendix: Proofs

Proof of Theorem 2.1 Let Z x = Y I(X ≤ x) and F x (•) = {1 -F X (x)[1 - F (•|x)]}I(• ≥ 0). It can be easily seen that P(Z x ≤ y) = F x (y) for any y ∈ R. Therefore, {Z x i = Y i I(X i ≤ x), i = 1, .
. . , n} is an i.i.d. sequence of random variables with common distribution function F x . Moreover, it is easy to see that the right end-point of F x coincides with θ(x) and that max i=1,...,n Z x i coincides with θ1 (x). Thus, assertion (i) follows from the Fisher-Tippett theorem. It is well known that the normalized maxima b -1 n ( θ1 (x)θ(x))

d -→ G (that is, F x belongs to the domain of attraction of G = Ψ ρx ) if and only if Fx θ(x) -1/t ∈ RV -ρx , (7)
where Fx = 1 -F x . This necessary and sufficient condition is equivalent to [START_REF] Aragon | Nonparametric frontier estimation: A conditional quantile-based approach[END_REF]. In this case, the norming constant b n can be taken to be equal to θ(x)

-inf{y ≥ 0 | F x (y) ≥ 1 -1 n } = θ(x) -inf{y ≥ 0 | F (y|x) ≥ 1 - 1 nFX (x)
}, which gives assertion (ii). For assertion (iii), since [START_REF] Bi | Estimating the self-thinning boundary line as a density-dependent stochastic biomass frontier[END_REF] holds and

E[|Z x | k ] = F X (x)E(Y k |X ≤ x) ≤ θ(x) k , it is immediate that lim n→∞ E{b -1 n ( θ1 (x) -θ(x))} k = (-1) k Γ(1 + k/ρ x ).
Proof of Corollary 2.1 Following the proof of Theorem 2.1, we can set

b n = θ(x) -F -1 x (1 -1 n ), where F -1 x (t) = inf{y ∈ ]0, θ(x)], F x (y) ≥ t} for all t ∈ ]0, 1]. It follows from (3) that F -1
x (t) = θ(x) -((1t)/ℓ x ) 1/ρx as t ↑ 1 and so b n = (1/nℓ x ) 1/ρx for all n sufficiently large.

For the next proofs, we need the following lemma whose proof is quite easy and is thus omitted.

Lemma 1 Let Z x (1) ≤ • • • ≤ Z x (n)
be the order statistics generated by the random variables Z x 1 , . . . , Z x n :

(i) If FX (x) > 0, then θ1-k/(n FX (x)) (x) = Z x (n-k) for each k ∈ {0, 1, . . . , n FX (x)-1}.

(ii) For any fixed integer k ≥ 0, we have θ1-k/(n FX (x)) (x) = Z x (n-k) as n → ∞, with probability 1.

(iii) For any sequence of integers k n ≥ 0 such that k n /n → 0 as n → ∞, we have

θ1-kn/(n FX (x)) (x) = Z x (n-kn)
as n → ∞, with probability 1.

Proof of Theorem 2.2 (i) Since θ(x) = F -1

x (1) and θ1 (x

) = Z x (n) for all n ≥ 1, we have ( θ1 (x) -θ(x)) = (Z x (n) -F -1 x (1)). Hence, if b -1 n ( θ1 (x) - θ(x)) d -→ G x , then b -1 n (Z x (n) -F -1 x (1)) converges to the same distribution G x . Therefore, b -1 n (Z x (n-k) -F -1 x (1)) d → H
x for any integer k ≥ 0, where

H x (y) = G x (y) k i=0 (-log G(y)) i /i!. Finally, since Z x (n-k) a.s. = θ1-k/(n FX (x)) (x) as n → ∞, in view of Lemma 1(ii), we obtain b -1 n ( θ1-k/(n FX (x)) (x)-F -1 x (1)) d -→ H x . (ii) Writing b -1 n ( θα (x) -θ(x)) = b -1 n ( θα (x) -θ1 (x)) + b -1 n ( θ1 (x) -θ(x)
), it suffices to find an appropriate sequence α = α n → 1 such that b -1 n ( θαn (x) -θ1 (x)) d -→ 0. Aragon et al. [START_REF] Aragon | Nonparametric frontier estimation: A conditional quantile-based approach[END_REF] (see equation [START_REF] Daouia | Kernel estimators of extreme level curves[END_REF]

) showed that | θα (x) -θ1 (x)| ≤ (1 -α)n FX (x)F -1
Y (1), with probability 1, for any α > 0. It thus suffices to choose α = α n → 1 such that nb -1 n (1α n ) → 0.

Proof of Theorem 2.3 (i) Let γ x = -1/ρ x in [START_REF] Bi | Estimating the self-thinning boundary line as a density-dependent stochastic biomass frontier[END_REF]. The Pickands estimate of the exponent of variation γ x < 0 is then given by γx := (log 2) -1 log{(Z x (n-k+1) -Z x (n-2k+1) )/(Z x (n-2k+1) -Z x (n-4k+1) )}. Under (2), Condition [START_REF] Bi | Estimating the self-thinning boundary line as a density-dependent stochastic biomass frontier[END_REF] holds and so there exists b n > 0 such that lim n→∞ P[b -1 n (Z x (n)θ(x)) ≤ y] = Ψ -1/γx (y). Since this limit is unique only up to affine transformations, we have -→ ρ x .

lim n→∞ P c -1 n Z x (n) -d n ≤ y = Ψ -1/γx (-γ x y -1) = exp{-(1 + γ x y) -1/γx } for all y ≤ 0, where c n = -γ x b n and d n = θ(x) -b n . Thus,
(iii) We have U

(t) = inf{y ≥ 0 | 1 1-Fx(y) ≥ t}, which corresponds to the inverse function (1/(1 -F x )) -1 (t). Since ±t 1-γx U ′ (t) ∈ Π(A) with γ x = -1/ρ x < 0, it follows from [29] (see Theorem 2.3) that √ k n (γ x -γ x ) d -→ N (0, σ 2 (γ x )) with σ 2 (γ x ) = γ 2 x (2 2γx+1 + 1)/{2(2 γx -1) log 2} 2 for k n → ∞ sat- isfying k n = o(n/g -1 (n)), where g(t) := t 3-2γx {U ′ (t)/A(t)} 2 . By using the fact that √ k n (ρ x -ρ x ) a.s. = √ k n (-1 γx + 1 γx )
as n → ∞, in view of Lemma 1(iii) and applying the delta method, we conclude that

√ k n (ρ x -ρ x ) d -→ N (0, σ 2 (ρ x )) with asymptotic variance σ 2 (ρ x ) = σ 2 (γ x )/γ 4
x . (iv) Under the regularity condition, we have ±{t -1-1/γx F ′ x (θ(x) -1 t ) -δF X (x)} ∈ RV -κ . The conclusion then follows immediately from Theorem 2.5 of [START_REF] Dekkers | On the estimation of extreme-value index and large quantiles estimation[END_REF] in conjunction with Lemma 1(iii).

Proof of Theorem 2. [START_REF] Beirlant | Nonparametric estimation of extreme conditional quantiles[END_REF] We have, by Lemma 1(iii), that for each j = 1, 2,

M (j) n = (1/k) k-1 i=0 log Z x (n-i) -log Z x (n-k) j
as n → ∞, with probability 1;

(8) -1/ρ x then coincides almost surely, for all n large enough, with the well-known moment estimator γx of the index defined in (7) by γ x = -1/ρ x . Hence, Theorem 2.4(i) and (ii) follow from the weak and strong consistency of γx . Likewise, Theorem 2.4(iii) follows.

Proof of Theorem 2.5 (i) Under the regularity condition, the distribution function F x of Z x has a positive derivative F ′ x (y) = F X (x)F ′ (y|x) for all y > 0 such that F ′ x (θ(x) -1 t ) ∈ RV 1+1/γx . Therefore, according to [START_REF] Dekkers | On the estimation of extreme-value index and large quantiles estimation[END_REF] (see Theorem 3.1),

2k n Z x (n-kn+1) -F -1 x (1 -p n ) Z x (n-kn+1) -Z x (n-2kn+1)
is asymptotically normal with mean zero and variance 2 2γx+1 γ 2 x /(2 γx -1) 2 . We conclude by using the facts that F -1

x (1p n ) = θ 1-pn/FX (x) (x) and

2k n Z x (n-kn+1) -F -1 x (1 -p n ) Z x (n-kn+1) -Z x (n-2kn+1) a.s. = 2k n θ1-(kn-1)/(n FX (x)) (x) -F -1 x (1 -p n ) θ1-(kn-1)/(n FX (x)) (x) -θ1-(2kn-1)/(n FX (x)) (x)
as n → ∞.

(ii) We have θ * 1 (x) is then asymptotically normal with mean zero and variance 3γ 2 x 2 2γx-1 /(2 γx -1) 6 .

F x (y) ≥ 1e -t } := F -1

x (1e -t ). Hence,

Z x (n-kn+1) -F -1 x 1 - k n n d = R -1 x E (n-kn+1) -R -1 x log n k n = E (n-kn+1) -log n k n (R -1 x ) ′ log n k n + 1 2 E (n-kn+1) -log n k n 2 (R -1 x ) ′′ [δ n ],
provided that E (n-kn+1) ∧ log(n/k n ) < δ n < E (n-kn+1) ∨ log(n/k n ). By the regularity condition (3), we have that R -1 x (t) = θ(x) -(e -t /ℓ x ) 1/γx for all t large enough. Therefore, for all n sufficiently large,

{ρ x k 1/2 n /(k n /nℓ x ) 1/ρx } Z x (n-kn+1) -F -1 x (1 -k n /n) d = k 1/2 n E (n-kn+1) -log(n/k n ) -{k 1/2 n /2ρ x } E (n-kn+1) -log(n/k n ) 2 exp{-[δ n -log(n/k n )]/ρ x }.
Since k x (1k n /n)] d -→ N (0, 1) as n → ∞. Since F -1 x (t) = θ(x) -((1t)/ℓ x ) 1/ρx for all t < 1 large enough, we have θ(x) -F -1

x (1 -kn n ) = (k n /nℓ x ) 1/ρx for all n sufficiently large. Thus, {ρ x k

1/2 n /(k n /nℓ x ) 1/ρx } × [Z x
(n-kn+1) + (k n /nℓ x ) 1/ρxθ(x)] d → N (0, 1) as n → ∞. We conclude by using the fact that Z x Proof of Theorem 2.7 (i) As shown in the proof of Theorem 2.5(i), we have F ′

x (θ(x) -1 t ) ∈ RV 1+1/γx . Then, we get

k n Z x (n-kn) -F -1 x (1 -p n ) /M (1) n Z x (n-kn) d -→ N 0, V 4 (-1/γ x ) .
The proof is completed by simply using the fact that F -1

x (1-p n ) = θ 1-pn/(FX (x)) (x) and Z x = -1/ρ x as n → ∞, we have θ(x) a.s.

= Z x

(n-kn) M

n (1 -1/γ x ) + Z x (n-kn) as n → ∞. It is then easy to see from (8) that θ(x) coincides almost surely, for all n large enough, with the end-point estimator x * n of F -1

x (1). It is also easy to check that U (t) = (1/(1 -F x )) -1 (t) and γ x = -1/ρ x < 0. We then have

√ k n {x * n -F -1 x (1)}/M (1)
n Z x (n-kn) (1γx ) d -→ N (0, V 5 (-1/γ x )), which gives the desired convergence in distribution of Theorem 2.7(ii) since F -1

x (1) = θ(x), x * n a.s.

= θ(x), γx a.s.

= -1/ρ x and Z x (n-kn) a.s. = θ1-kn/(n FX (x)) (x) as n → ∞.

Theorem 2 . 1

 21 (i) If there exist b n > 0 and some non-degenerate distribution function G x such that b

- 1 +

 1 Z x (n-kn+1) as n → ∞. Following [29], Theorem 3.2, √ 2k n ( θ * 1 (x)θ(x)) Z x (n-kn+1) -Z x (n-2kn+1)

1 / 2 n 2 n

 122 [E (n-kn+1) -log(n/k n )] d → N (0, 1) and |δ n -log(n/k n )| ≤ |E (n-kn+1)log(n/k n )| p → 0 as n → ∞, we obtain {ρ x k 1//(k n /nℓ x ) 1/ρx }[Z x (n-kn+1) -F -1

  kn-1)/(n FX (x)) (x) as n → ∞.

  (n-kn) a.s. = θ1-kn/(n FX (x)) (x) as n → ∞.(ii) Since Z x (n-kn) a.s. = θ1-kn/(n FX (x)) (x) and γx a.s.

(iii) Let E (1) ≤ • • • ≤ E (n) be the order statistics of i.i.d. exponential variables

.

The first term on the right-hand side tends to zero as established by Dekkers and de Haan ( [START_REF] Dekkers | On the estimation of extreme-value index and large quantiles estimation[END_REF], Proof of Theorem 3.2). The second term converges in distribution to N (0, 1) × 2 γx 1-2 γx , in view of Lemma 3.1 and [START_REF] Dekkers | On the estimation of extreme-value index and large quantiles estimation[END_REF], Corollary 3.1. The third term converges in probability to γx 2 γx -1 by the same Corollary 3.1. This ends the proof of (iii), in conjunction with the fact that

as n → ∞, with probability 1.

Proof of Theorem 2.6 Write Fx (y) := F X (x)[1 -F (y|x)] and F x (y) := 1 -Fx (y) for all y ≥ 0. Let R x (y) := -log{ Fx (y)} for all y ∈ [0, θ(x)[ and let E (n-kn+1) be the statistic of order nk n + 1 generated by n independent standard exponential random variables. Z x (n-kn+1) then has the same distribution as R -1

x [E (n-kn+1) ], where R -1 x (t) := inf{y ≥ 0 | R x (y) ≥ t} = inf{y ≥ 0 |


Samy Skander Bahoura
email: samybahoura@yahoo.fr

ABOUT BREZIS-MERLE PROBLEM WITH HOLDERIAN CONDITION: THE CASE OF THREE BLOW-UP POINTS

set Ω of R 2 :

We assume that :

and,

On the other hand, if we assume that V i s-holderian with 1/2 < s ≤ 1, and,

Ω

V i e u i dy ≤ 32π -ǫ, ǫ > 0 then we have a compactness result, namely: sup

INTRODUCTION AND MAIN RESULTS

We set ∆ = ∂ 11 + ∂ 22 on open set Ω of R 2 with a smooth boundary.

We consider the following problem on Ω ⊂ R 2 :

(P ) -∆u i = V i e ui u i = 0 in Ω ⊂ R 3 , in ∂Ω.
We assume that, Ω e ui dy ≤ C, and,

0 ≤ V i ≤ b < +∞
The previous equation is called, the Prescribed Scalar Curvature equation, in relation with conformal change of metrics. The function V i is the prescribed curvature.

Here, we try to find some a priori estimates for sequences of the previous problem. Equations of this type were studied by many authors, see [START_REF] Bahoura | About Brezis-Merle Problem with holderian condition: the case of one or two blow-up points[END_REF][START_REF] Brezis | A sup+inf inequality for some nonlinear elliptic equations involving exponential nonlinearities[END_REF][START_REF] Brezis | Uniform estimates and blow-up bihavior for solutions of -∆u = V e u in two dimensions[END_REF][START_REF] Chen | A priori Estimates for solutions to Nonlinear Elliptic Equations[END_REF][START_REF] Chen | A sharp sup+inf inequality for a nonlinear elliptic equation in R 2[END_REF][START_REF] Cheng | On the asymptotic behavior of solutions of the conformal Gaussian curvature equations in R 2[END_REF][START_REF] Yy | Harnack Type Inequality: the Method of Moving Planes[END_REF][START_REF] Yy | Blow-up Analysis for Solutions of -∆u = V e u in Dimension Two[END_REF][START_REF] Ma | Convergence for a Liouville equation[END_REF][START_REF] Shafrir | A sup+inf inequality for the equation -∆u = V e u[END_REF][START_REF] Zhang | Blowup solutions of some nonlinear elliptic equations involving exponential nonlinearities[END_REF]. We can see in [START_REF] Brezis | Uniform estimates and blow-up bihavior for solutions of -∆u = V e u in two dimensions[END_REF], different results for the solutions of those type of equations with or without boundaries conditions and, with minimal conditions on V , for example we suppose V i ≥ 0 and V i ∈ L p (Ω) or V i e ui ∈ L p (Ω) with p ∈ [1, +∞].

Among other results, we can see in [START_REF] Brezis | Uniform estimates and blow-up bihavior for solutions of -∆u = V e u in two dimensions[END_REF], the following important Theorem, Theorem A (Brezis-Merle [START_REF] Brezis | Uniform estimates and blow-up bihavior for solutions of -∆u = V e u in two dimensions[END_REF]).If (u i ) i and (V i ) i are two sequences of functions relatively to the previous problem (P ) with, 0 < a ≤ V i ≤ b < +∞, and without the boundary condition, then, for all compact set K of Ω,

1 sup K u i ≤ c = c(a, b, m, K, Ω) if inf Ω u i ≥ m.
A simple consequence of this theorem is that, if we assume u i = 0 on ∂Ω then, the sequence (u i ) i is locally uniformly bounded. We can find in [START_REF] Brezis | Uniform estimates and blow-up bihavior for solutions of -∆u = V e u in two dimensions[END_REF] an interior estimate if we assume a = 0, but we need an assumption on the integral of e ui . We have in [START_REF] Brezis | Uniform estimates and blow-up bihavior for solutions of -∆u = V e u in two dimensions[END_REF]:

Theorem B (Brezis-Merle [START_REF] Brezis | Uniform estimates and blow-up bihavior for solutions of -∆u = V e u in two dimensions[END_REF]).If (u i ) i and (V i ) i are two sequences of functions relatively to the previous problem (P ) with, 0 ≤ V i ≤ b < +∞, and, Ω e ui dy ≤ C, then, for all compact set K of Ω, sup

K u i ≤ c = c(b, C, K, Ω).
If, we assume V with more regularity, we can have another type of estimates, sup + inf. It was proved, by Shafrir, see [START_REF] Shafrir | A sup+inf inequality for the equation -∆u = V e u[END_REF], that, if (u i ) i , (V i ) i are two sequences of functions solutions of the previous equation without assumption on the boundary and, 0 < a ≤ V i ≤ b < +∞, then we have the following interior estimate:

C a b sup K u i + inf Ω u i ≤ c = c(a, b, K, Ω).
We can see in [START_REF] Chen | A sharp sup+inf inequality for a nonlinear elliptic equation in R 2[END_REF], an explicit value of C a b = a b . In his proof, Shafrir has used the Stokes formula and an isoperimetric inequality. For Chen-Lin, they have used the blow-up analysis combined with some geometric type inequality for the integral curvature.

Now, if we suppose (V i ) i uniformly Lipschitzian with A the Lipschitz constant, then, C(a/b) = 1 and c = c(a, b, A, K, Ω), see Brézis-Li-Shafrir [START_REF] Brezis | A sup+inf inequality for some nonlinear elliptic equations involving exponential nonlinearities[END_REF]. This result was extended for Hölderian sequences (V i ) i by Chen-Lin, see [START_REF] Chen | A sharp sup+inf inequality for a nonlinear elliptic equation in R 2[END_REF]. Also, we can see in [START_REF] Yy | Harnack Type Inequality: the Method of Moving Planes[END_REF], an extension of the Brezis-Li-Shafrir to compact Riemann surface without boundary. We can see in [START_REF] Yy | Blow-up Analysis for Solutions of -∆u = V e u in Dimension Two[END_REF] explicit form, (8πm, m ∈ N * exactly), for the numbers in front of the Dirac masses, when the solutions blowup. Here, the notion of isolated blow-up point is used. Also, we can see in [START_REF] Zhang | Blowup solutions of some nonlinear elliptic equations involving exponential nonlinearities[END_REF] refined estimates near the isolated blow-up points and the bubbling behavior of the blow-up sequences.

In [START_REF] Brezis | Uniform estimates and blow-up bihavior for solutions of -∆u = V e u in two dimensions[END_REF], Brezis and Merle proposed the following Problem: Problem (Brezis-Merle [START_REF] Brezis | Uniform estimates and blow-up bihavior for solutions of -∆u = V e u in two dimensions[END_REF]).If (u i ) i and (V i ) i are two sequences of functions relatively to the previous problem (P ) with,

0 ≤ V i → V in C 0 (Ω). Ω e ui dy ≤ C,

Is it possible to prove that:

sup

Ω u i ≤ c = c(C, V, Ω) ?
Here, we assume more regularity on V i , we suppose that

V i ≥ 0 is C s (s-holderian) (1/2 < s ≤ 1) .
We give the answer where bC < 32π.

In the similar way, we have in dimension n ≥ 3, with different methods, some a priori estimates of the type sup × inf for equation of the type:

-∆u + n -2 4(n -1) R g (x)u = V (x)u (n+2)/(n-2) on M.
where R g is the scalar curvature of a riemannian manifold M , and V is a function. The operator ∆ = ∇ i (∇ i ) is the Laplace-Beltrami operator on M . When V ≡ 1 and M compact, the previous equation is the Yamabe equation. T. Aubin and R. Scheon solved the Yamabe problem, see for example [START_REF] Aubin | Some Nonlinear Problems in Riemannian Geometry[END_REF]. If V is not a constant function, the previous equation is called a prescribing curvature equation, we have many existence results see also [START_REF] Aubin | Some Nonlinear Problems in Riemannian Geometry[END_REF], for a detailed summary.

We can see in [START_REF] Brezis | A sup+inf inequality for some nonlinear elliptic equations involving exponential nonlinearities[END_REF], [START_REF] Caffarelli | Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth[END_REF], [START_REF] Gidas | Symmetry and related properties via the maximum principle[END_REF], some results for elliptic equations of this type, and, some application of the method of moving-plane to obtain uniform estimates and estimates of type sup × inf. See also, [START_REF] Siu | The existence of Kahler-Einstein metrics on manifolds with positive anticanonical line bundle and a suitable finite symmetry group[END_REF], [START_REF] Tian | A Harnack type inequality for certain complex Monge-Ampre equations[END_REF] for other sup + inf inequalities.

In [START_REF] Caffarelli | Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth[END_REF], we have a classification result for singular and non-singular solution of the Yamabe equation on open set of R n and on R n . The method used is of moving-plane and some other estimates.

In [START_REF] Gidas | Symmetry and related properties via the maximum principle[END_REF], we have a basic description of the method of moving-plane, and, in [START_REF] Brezis | A sup+inf inequality for some nonlinear elliptic equations involving exponential nonlinearities[END_REF], we have an application of this method, namely; inequality of type sup + inf on a bounded domain of R 2 .

Returning to ourk wrok, we give a compactness result for the Brezis-Merle Problem when the energy is less than 32π -ǫ, ǫ > 0. In fact, we extend the result of the author, see [START_REF] Bahoura | About Brezis-Merle Problem with holderian condition: the case of one or two blow-up points[END_REF]. We argue by contradiction and we use some asymptotic estimates for the blow-up functions. Also, we use a term of the Pohozaev identity to conclude to a contradiction.

Our main result is:

Theorem. Assume that, V i is uniformly s-holderian with 1/2 < s ≤ 1, and, B1 (0)
V i e ui dy ≤ 32π -ǫ, ǫ > 0, then we have:

sup Ω u i ≤ c = c(b, C, A, s, Ω).
for solutions of the problem (P ), here A is the holderian constant of V i .

PROOF OF THE RESULT:

Proof of the theorem: Without loss of generality, we can assume that Ω = B 1 (0) the unit ball centered on the origin.

Here, G is the Green function of the Laplacian with Dirichlet condition on B 1 (0). We have (in complex notation):

G(x, y) = 1 2π log |1 -xy| |x -y| ,
we can write:

u i (x) = B1(0) G(x, y)V i (y)e ui(y) dy,
We assume that we are in the case of one blow-up point. Following the notation of a previous paper, see [START_REF] Bahoura | About Brezis-Merle Problem with holderian condition: the case of one or two blow-up points[END_REF], we have:

max Ω u i = u i (x i ) → +∞, δ i = d(x i , ∂Ω) → 0,
for ǫ > 0 small enough, and |y| = ǫ,

u i (x i + δ i y) ≤ C ǫ ,
and, the sup + inf inequality gives:

u i (x i ) + 4 log δ i ≤ C.
Also, we have the following estimates which imply the smallness for a term of the Pohozaev identity:

||∇u i || L q (B(xi,δiǫ ′ )) = o(1). ∀ 1 ≤ q < 2.
We have; because V i is s-holderian with 1/2 < s ≤ 1, the following term of the Pohozaev identity tends to 0

J i = B(xi,δiǫ ′ ) < x i 1 |∇(u i -u) > (V i -V i (x i ))e ui dy = o(1).
Now, we set:

r i = e -ui(xi)/2 , we write, for |θ| ≤ δ i ǫ ′ r i , 0 < ǫ ′ < 1 4 , u i (x i + r i θ) = Ω G(x i + r i θ, y)V i (y)e ui(y) dx = = Ω-B(xi,2δiǫ ′ ) G(x i , y)V i e ui(y) dy + B(xi,2δiǫ ′ ) G(x i + r i θ, y)V i e ui(y) dy = We write, y = x i + r i θ, with | θ| ≤ 2 δ i r i ǫ ′ , u i (x i + r i θ) = B(0,2 δ i r i ǫ ′ ) 1 2π log |1 -(x i + r i θ)(x i + r i θ)| r i |θ -θ| V i e ui(y) r 2 i dy+ + Ω-B(xi,2δiǫ ′ ) G(x i + r i θ, y)V i e ui(y) dy u i (x i ) = Ω-B(xi,2δiǫ ′ ) G(x i , y)V i e ui(y) dy + B(xi,2δiǫ ′ ) G(x i , y)V i e ui(y) dy
Hence,

u i (x i ) = B(0,2 δ i r i ǫ ′ ) 1 2π log |1 -xi (x i + r i θ)| r i | θ| V i e ui(y) r 2 i dy+ + Ω-B(xi,2δiǫ ′ ) G(x i , y)V i e ui(y) dy
We look to the difference,

v i (θ) = u i (x i + r i θ) -u i (x i ) = B(0,2 δ i r i ǫ ′ ) 1 2π log | θ| |θ -θ| V i e ui(y) r 2 i dy + h 1 + h 2 ,
where,

h 1 (θ) = Ω-B(xi,2δiǫ ′ ) G(x i + r i θ, y)V i e ui(y) dy - Ω-B(xi,2δiǫ ′ ) G(x i , y)V i e ui(y) dy,
and,

h 2 (θ) = B(0,2δiǫ ′ ) 1 2π log |1 -(x i + r i θ)y| |1 -xi y| V i e ui(y) dy.
Remark that, h 1 and h 2 are two harmonic functions, uniformly bounded.

According to the maximum principle, the harmonic function G(x i +r i θ, .) on Ω-B(x i , 2δ i ǫ ′ ) take its maximum on the boundary of B(x i , 2δ i ǫ ′ ), we can compute this maximum:

G(x i +r i θ, y i ) = 1 2π log |1 -(x i + r i θ)y i | |x i + r i θ -y i | ≃ 1 2π log (|1 + |x i |)δ i -δ i (3ǫ ′ + o(1))| δ i ǫ ′ ≤ C ǫ ′ < +∞ with y i = x i + 2δ i θ i ǫ ′ , |θ i | = 1, and |r i θ| ≤ δ i ǫ ′ .
We can remark, for |θ| ≤ δ i ǫ ′ r i , that v i is such that:

v i = h 1 + h 2 + B(0,2 δ i r i ǫ ′ ) 1 2π log | θ| |θ -θ| V i e ui(y) r 2 i dy, v i = h 1 + h 2 + B(0,2 δ i r i ǫ ′ ) 1 2π log | θ| |θ -θ| V i (x i + r i θ)e vi( θ) d θ,
with h 1 and h 2 , the two uniformly bounded harmonic functions.

Remark: In the case of 2 or 3 blow-up points, and if we consider the half ball, we have supplemntary terms, around the 2 other blow-up terms. Note that the Green function of the half ball is quasi-similar to the one of the unit ball and our computations are the same if we consider the half ball.

We assume that, the blow-up limit is 0 and we take:

G(x, y) = 1 2π log |1 -xy| |x -y| - 1 2π log |1 -xy| |x -y| , .
Asymptotic estimates and the case of one, two and three blow-up points :

By the asymptotic estimates of Cheng-Lin see [START_REF] Cheng | On the asymptotic behavior of solutions of the conformal Gaussian curvature equations in R 2[END_REF], we can see that, we have the following uniform estimates at infinity:

Lemma 2.1. ∀ ǫ, ǫ ′ > 0, ∃ k ǫ,ǫ ′ ∈ R + , i ǫ,ǫ ′ ∈ N and C ǫ,ǫ ′ > 0, such that for i ≥ i ǫ,ǫ ′ and k ǫ,ǫ ′ ≤ |θ| ≤ δ i ǫ ′ r i (-4 -ǫ) log |θ| -C ǫ,ǫ ′ ≤ v i (θ) ≤ (-4 + ǫ) log |θ| + C ǫ,ǫ ′ ,
For the proof, we consider the three following sets:

A 1 = { θ, | θ| ≤ k ǫ }, A 2 = { θ, |θ -θ| ≤ |θ| 2 , | θ| ≥ k ǫ },
and,

A 3 = { θ, |θ -θ| ≥ |θ| 2 , | θ| ≥ k ǫ }.
where k ǫ is such that;

8π(1 -ǫ) ≤ B(0,kǫ) V i (x i + r i θ)e vi( θ) d θ = B(xi,kǫe -u i (x i )/2 )
V i e ui(y) dy ≤ 8π(1 + ǫ).

In fact, if we assume that we have one blow-up point:

B(0, δ i 2r i ) V i (x i + r i θ)e vi( θ) d θ = B(xi, δ i
2 )

V i e ui(y) dy → 8π,

To have the uniform bounds C ǫ > 0, we need to bound uniformly the following quantity:

A i = B(0, δ i 2r i ) 1 2π log | θ|V i e ui(y) r 2 i dy = B(0, δ i 2r i ) 1 2π log | θ|V i e vi( θ) d θ.
To obtain this uniform bound, we use the CC.Chen and C.S. Lin computations, see [START_REF] Chen | A sharp sup+inf inequality for a nonlinear elliptic equation in R 2[END_REF], to have the existence of a sequence l i → +∞ such that:

B(0,li) 1 2π log | θ|V i e vi( θ) d θ ≤ C,
and, on the other hand, the computations of YY.Li and I. Shafrir, see [START_REF] Yy | Blow-up Analysis for Solutions of -∆u = V e u in Dimension Two[END_REF], to have, for

l i ≤ | θ| ≤ δ i 2r i : e vi( θ) ≤ C | θ| 2β+2 ,
for some 0 < β < 1.

Finaly,

A i ≤ C.
Remark that, in the estimate of CC.Chen and C.S Lin, see [START_REF] Chen | A sharp sup+inf inequality for a nonlinear elliptic equation in R 2[END_REF], we need the assumption that V i is sholderian with 0 < s ≤ 1.

To explain more the previous lemma, we write:

-2πv i + 2πh 1 + 2πh 2 = - B(0,2 δ i r i ǫ ′ )∩A1 1 2π log | θ| |θ -θ| V i (x i + r i θ)e vi( θ) d θ+, - B(0,2 δ i r i ǫ ′ )∩A2 1 2π log | θ| |θ -θ| V i (x i + r i θ)e vi( θ) d θ+ - B(0,2 δ i r i ǫ ′ )∩A3 1 2π log | θ| |θ -θ| V i (x i + r i θ)e vi( θ) d θ = = -I 1 -I 2 -I 3 .
For I 2 , we have: |θ -θ| ≤ | θ|, hence, -I 2 ≤ 0.

For I 1 , it is easy to see that:

-I 1 ≤ log |θ| A1 V i (x i + r i θ)e vi( θ) d θ + C,
with C a constant independant of x and i. Here we use the estimates of Chen-Lin. Since, |θ -θ| ≤ | θ| + |θ| ≤ | θ||θ| for |θ|, | θ| ≥ 1, we have:

-I 3 ≤ log |θ| A3 V i (x i + r i θ)e vi( θ) d θ,
Thus,

-2πv i + 2πh 1 + 2πh 2 ≤ log |θ| A1∪A3 V i (x i + r i θ)e vi( θ) d θ + C,
Hence,

-2πv i + 2πh 1 + 2πh 2 ≤ (8π + ǫ) log |θ| + C, Thus, v i -h 1 -h 2 ≥ (-4 -ǫ) log |θ| -C.
For the rest of the proof, we use the same argument as in Cheng-Lin, see [START_REF] Cheng | On the asymptotic behavior of solutions of the conformal Gaussian curvature equations in R 2[END_REF].

We write:

v i -h 1 -h 2 = B(0,2 δ i r i ǫ ′ )∩A1 1 2π log | θ| |θ -θ| V i (x i + r i θ)e vi( θ) d θ+, + B(0,2 δ i r i ǫ ′ )∩A2 1 2π log | θ| |θ -θ| V i (x i + r i θ)e vi( θ) d θ+ + B(0,2 δ i r i ǫ ′ )∩A3 1 2π log | θ| |θ -θ| V i (x i + r i θ)e vi( θ) d θ = = I 1 + I 2 + I 3 .
We have:

I 1 ≤ -log |θ| A1 1 2π V i (x i + r i θ)e vi( θ) d θ + C,
with C a constant independant of x and i. Here we use the estimates of Chen-Lin. For I 3 , we have:

I 3 ≤ 1.
For I 2 , we have:

I 2 ≤ 1 2π {| θ-θ|≤|θ| -σ } log 1 |θ -θ| V i (x i + r i θ)e vi( θ) d θ + ǫ 2 log |θ|,
Hence,

v i -h 1 -h 2 ≤ (-4 + ǫ) log |θ| + 1 2π {| θ-θ|≤|θ| -σ } log 1 |θ -θ| V i (x i + r i θ)e vi( θ) d θ.
As in [START_REF] Cheng | On the asymptotic behavior of solutions of the conformal Gaussian curvature equations in R 2[END_REF], we can prove that, (h 1 and h 2 are harmonic and satisfy the mean value theorem):

v i -h 1 -h 2 - {| θ-θ|=r=|θ| -σ } (v i -h 1 -h 2 ) = 1 2π Br (x) log r |θ -θ| V i (x i + r i θ)e vi( θ) d θ {| θ-θ|=r=|θ| -σ } (v i -h 1 -h 2 ) ≤ (-4 + ǫ) log |θ|.
As in the proof of the theorem 1.1 of [START_REF] Cheng | On the asymptotic behavior of solutions of the conformal Gaussian curvature equations in R 2[END_REF], we use the Brezis-Merle estimate and the two previous estimates to prove that for θ large enough, we have:

v i -h 1 -h 2 ≤ (-4 + ǫ) log |θ| + C.
To see this : (We write v i -h 1 -h 2 = k + q, with q harmonic with the same boundary value as v i , we use Brezis-Merle estimate). Note that, h 1 and h 2 are uniformly bounded. Let Ω = B r (θ), where r = 2|θ| -σ we have:

-∆k = V i e k+q in Ω, k = 0 on ∂Ω.
By the Brezis-Merle estimate:

Ω e 2k ≤ C 1 |θ| -2σ .
We use the fact that q is harmonic to have:

q(θ) ≤ Cq(0) + (C -1)(-min Ω q -).
By the previous computations we have:

min Ω q -= min ∂Ω q -= min ∂Ω (v i -h 1 -h 2 ) -≥ (-4 -ǫ) log |θ| -C,
and by the previous mean value estimate, we have:

q(0) = {| θ-θ|=r=|θ| -σ } (v i -h 1 -h 2 ) ≤ (-4 + ǫ) log |θ|.
Thus,

q(θ) ≤ C log |θ|.
Here C is a constant independant of i and σ.

We have by the same computations as in the proof of the theorem 1.1 of [START_REF] Cheng | On the asymptotic behavior of solutions of the conformal Gaussian curvature equations in R 2[END_REF] to conclude that:

{| θ-θ|≤|θ| -σ } e 2vi ≤ |θ| -2σ+2C ,
and by Cauchy-Schwarz inequality, we have:

1 2π {| θ-θ|≤|θ| -σ } log 1 |θ -θ| V i (x i + r i θ)e vi( θ) d θ 2 ≤ C,
and that, for θ and σ large enough:

v i -h 1 -h 2 ≤ (-4 + ǫ) log |θ| + C.
Now, we extend the previous asymptotic estimates to the first derivatives: we have, after derivation under the integral:

∂ j v i = ∂ j h 1 + ∂ j h 2 + B(0,2 δ i r i ǫ ′ ) 1 2π θ j -θj |θ -θ| 2 V i e ui(y) r 2 i dy,
In other words, we have:

∂ j v i = ∂ j h 1 + ∂ j h 2 + B(0,2 δ i r i ǫ ′ ) 1 2π θ j -θj |θ -θ| 2 V i (x i + r i θ)e vi( θ) d θ,
We have the following lemma:

Lemma 2.2. ∀ ǫ, ǫ ′ > 0 ∃ k ǫ,ǫ ′ ∈ R + , i ǫ,ǫ ′ ∈ N, such that, for i ≥ i ǫ,ǫ ′ and k ǫ,ǫ ′ ≤ |θ| ≤ δ i ǫ ′ r i , ∂ j v i (θ) ≃ ∂ j u 0 (θ) ± ǫ |θ| + C r i δ i ,
where u 0 is the solution to:

-∆u 0 = V (0)e u0 , in R 2 .
For the proof, we consider the three following sets:

A 1 = { θ, | θ| ≤ k ǫ }, A 2 = { θ, |θ -θ| ≤ |θ| 2 , | θ| ≥ k ǫ },
and,

A 3 = { θ, |θ -θ| ≥ |θ| 2 , | θ| ≥ k ǫ }.
where k ǫ is such that;

8π(1 -ǫ) ≤ B(0,kǫ) V i (x i + r i θ)e vi( θ) d θ = B(xi,kǫe -u i (x i )/2 )
V i e ui(y) dy ≤ 8π(1 + ǫ).

Remark 1: In the case of 2 or 3 blow-up points, and if we consider the half ball, we have supplemntary terms, around the 2 other blow-up terms. Note that the Green function of the half ball is quasi-similar to the one of the unit ball. In the case of 3 blow-up points, we have the following supplementary term ( x i is the principal blow-up point and y i and t i the 2 other blow-up points):

C 1 r i d(x i , y i ) + C 2 r i d(x i , t i ) .
We assume that, the blow-up limit is 0 and we take:

G(x, y) = 1 2π log |1 -xy| |x -y| - 1 2π log |1 -xy| |x -y| , .
In the previous computations, we have considered the unit ball, but by a conformal transformation , we can have the same estimates on the half ball, with a coefficient of the conformal transformation. We can assume the estimates on the half ball. Now, we consider the following term of the Pohozaev identity

J i = B(xi,δiǫ ′ ) < x i 1 |∇(u i -u) > (V i -V i (x i ))e ui dy,
We want to show that this term tends to 0 as i tends to infinity. We can reduce the problem, after integration by parts, to the following integral:

J ′ i = δ i B(xi,δiǫ ′ ) ∂ 1 u i V i e ui dy = δ i B(xi,δiǫ ′ ) ∂ 1 u i (-∆u i )
But, if we take y = x i + r i θ, with, |θ| ≤ δ i ǫ ′ r i , we have:

J ′ i = δ i r i B(0, δ i r i ǫ ′ ) ∂ 1 v i (-∆v i ) = = δ i r i ∂B(0, δ i r i ǫ ′ ) (∂ 1 v i ) 2 2 ν 1 - (∂ 2 v i ) 2 2 ν 1 + (∂ 1 v i )(∂ 2 v i )ν 2 dσ i ,
Thus, if we use the uniform asymptotic estimates, we can see that, we reduce the computation to the Pohozaev identity for the limit blow-up function (which equal to 0), plus terms in ǫ|θ| and |θ|. First, we tend i to infinity, after ǫ to 0 and finaly , we tend ǫ ′ to 0 .

With this method we can have a compactness result for 3 blow-ups points. First, we can see the case of 3 exteriors blow-up points, then by the previous formulation we have a compactness result, it is the case for one of the following cases ( if we set δ i , δ ′ i and δ ′′ i for the radii of each exterior blow-up) :

d(x i , y i ) δ i → +∞ and d(x i , t i ) δ i → +∞, or, d(y i , x i ) δ ′ i → +∞ and d(y i , t i ) δ ′ i → +∞, or, d(t i , x i ) δ ′′ i → +∞ and d(t i , y i ) δ ′′ i → +∞,
or, the case when the distance to two exterior blow-up points is of order the radii. In this last case, we divide the region in 3 parts and use the Pohozaev identity directly. In fact, we are reduced to the case of two blow-up points.

In fact, in the case of 3 exterior blow-up points. By the previous formulation around each exterior blow-up point we look to the one of the 3 first cases. For example, assume the first case. Then we work around the first blow-up. In fact we have, for 3 blow-up points :

Lemma 2.3. ∀ ǫ > 0, ǫ ′ > 0 ∃ k ǫ,ǫ ′ ∈ R + , i ǫ,ǫ ′ ∈ N and C ǫ,ǫ ′ > 0, such that, for i ≥ i ǫ,ǫ ′ and k ǫ,ǫ ′ ≤ |θ| ≤ δ i ǫ ′ r i , (-4 -ǫ) log |θ| -C ǫ,ǫ ′ ≤ v i (θ) ≤ (-4 + ǫ) log |θ| + C ǫ,ǫ ′ , and, ∂ j v i ≃ ∂ j u 0 (θ) ± ǫ |θ| + C r i δ i 2 |θ| + m × r i δ i + C 1 r i d(x i , y i ) + C 2 r i d(x i , t i )
Proof of the compactness : By using the lemma we can see that we have the compactness, because: (to understand this, it is sufficient to do the computations for the half ball directly by using the Green function of the half ball directly).

We have after using the previous term of the Pohozaev identity:

o(1) = J ′ i = m ′ + C 1 o(1) + C 2 o(1), 0 = lim ǫ ′ lim ǫ lim i J ′ i = m ′ , which contradict the fact that m ′ > 0.
Proof of the second estimate of the lemma:

For the proof, we consider the three following sets:

A 1 = { θ, | θ| ≤ k ǫ }, A 2 = { θ, |θ -θ| ≤ |θ| 2 , | θ| ≥ k ǫ },
and,

A 3 = { θ, |θ -θ| ≥ |θ| 2 , | θ| ≥ k ǫ }.
where k ǫ (large enough), is such that;

8π(1 -ǫ) ≤ B(0,kǫ) V i (x i + r i θ)e vi( θ) d θ = B(xi,kǫe -u i (x i )/2 )
V i e ui(y) dy ≤ 8π(1 + ǫ).

We write:

∂ j v i -∂ j h 1 -∂ j h 2 = B(0,2 δ i r i ǫ ′ ) 1 2π
θ j -θj |θ -θ| 2 V i (x i + r i θ)e vi( θ) d θ,

∂ j v i -∂ j h 1 -∂ j h 2 = A1 1 2π
θ j -θj |θ -θ| 2 V i (x i +r i θ)e vi( θ) d θ+ A2∪A3 1 2π

θ j -θj |θ -θ| 2 V i (x i +r i θ)e vi( θ) d θ Using the estimates of v i , we obtain:

∂ j v i -∂ j h 1 -∂ j h 2 = o(1) |θ| + A1 1 2π θ j -θj |θ -θ| 2 V 0 e u0( θ) d θ+ A2∪A3 1 2π
θ j -θj |θ -θ| 2 V i (x i +r i θ)e vi( θ) d θ Thus,

∂ j v i -∂ j h 1 -∂ j h 2 = ∂ j u 0 + o(1) |θ| + A2∪A3 1 2π
θ j -θj |θ -θ| 2 V i (x i + r i θ)e vi( θ) d θ Finaly,

∂ j v i -∂ j h 1 -∂ j h 2 -∂ j u 0 = o(1) |θ| + A2∪A3 1 2π
θ j -θj |θ -θ| 2 V i (x i + r i θ)e vi( θ) d θ.

For A 2 and A 3 , we have: for θ large enough and ǫ 0 small enough.

| A3 1
Finaly, we have:

|∂ j v i -∂ j h 1 -∂ j h 2 -∂ j u 0 | ≤ ǫ |θ| ,
for θ large enough. Now, it is easy to see from the definition of h 1 and h 2 that:

|∂ j h 1 -∂ j h 2 -m r i δ i | ≤ C 1 r i d(x i , y i ) + C 2 r i d(x i , t i ) Thus, |∂ j v i -∂ j u 0 -m r i δ i | ≤ C 1 r i d(x i , y i )
+ C 2 r i d(x i , t i ) for θ large enough.

  2π θ j -θj |θ -θ| 2 V i (x i + r i θ)e vi( θ) d θ| ≤ For θ ∈ A 2 , | θ| ≥ |θ| 2 and, | θ| ≥ |θ -θ| 2, and we use the estimate of v i to have:θj |θ -θ| 2 V i (x i + r i θ)e vi( θ) d θ| ≤ C|θ| |θ| 4-ǫ0 ≤

	because,	1 2π {| θ|≥kǫ} θ j ǫ 1 |θ| V i (x i + r i θ)e vi( θ) d θ ≤ {| θ|≥kǫ} V | A2 1 2π |θ| ,	ǫ |θ|	,

i (x i + r i θ)e vi( θ) d θ → 0,

for k ǫ large enough.