Olivier Rey

The question of interior blow-up points for an elliptic Neumann problem : the critical case

In contrast with the subcritical case, we prove that for any bounded domain Ω in R 3 , the Neumann elliptic problem with critical nonlinearity -∆u + µu = u 5 , u > 0 in Ω ; ∂u/∂ν = 0 on ∂Ω has no solution blowing up at only interior points as µ goes to infinity.

Introduction and Results

Wondering about the mechanisms of pattern formation in biology, Turing [START_REF] Tenenblat | Turing -The chemical basis of morphogenesis[END_REF] made the very important discovery that contrary to the intuition, which associates diffusion phenomena to a smoothing of initial data, spatial concentration structures may result from the interaction of two substances with different diffusion rates. Since that time many biological patterns have be explained in such a way. Models describing the evolution of the two involved substances concentrations, as those proposed by Keller and Segel, or Gierer and Meinhardt, consist in a system of two coupled nonlinear parabolic equations. Under some further assumptions, finding stationary solutions to the system reduces to solving a single nonlinear elliptic equation with Neumann boundary conditions [START_REF] Lin | Takagi -Large amplitude stationary solutions to a chemotaxis system[END_REF] (P µ )

-∆u + µu = u p , u > 0 in Ω ∂u ∂ν = 0 on ∂Ω where p > 1, µ > 0 are fixed parameters, and Ω is a smooth bounded domain in R n . Note that setting v = µ -1/(p-1) u, d 2 = 1/µ, problem (P µ ) is equivalent to

(P ′ d ) -d 2 ∆v + v = v p , v > 0 in Ω ∂v ∂ν = 0 on ∂Ω.
Since the works of Lin, Ni and Takagi [START_REF] Lin | Ni -On the Diffusion Coefficient of a Semilinear Neumann Problem[END_REF][START_REF] Lin | Takagi -Large amplitude stationary solutions to a chemotaxis system[END_REF][START_REF] Ni | Takagi -On the shape of least-energy solutions to a semi-linear problem Neumann problem[END_REF][START_REF] Ni | Takagi -Locating the peaks of least-energy solutions to a semi-linear Neumann problem[END_REF], many papers have been devoted to the study of (P ′ d ), under the assumption that p is subcritical, i.e. n = 2, or n ≥ 3 and p < (n + 2)/(n -2). A natural question is to know whether the results which hold for subcritical exponents are true, or not, for critical or supercritical exponents. The study of the critical case made similarities and differences appear with respect to the subcritical case. For example, it was proved that when p is subcritical, the only solution to (P µ ) for small µ is constant [START_REF] Lin | Takagi -Large amplitude stationary solutions to a chemotaxis system[END_REF], and the same holds when n = 3 and p is critical [START_REF] Zhu | Uniqueness results through a priori estimates, I. A three dimensional Neumann problem[END_REF][START_REF] Wei | Winter -Stationary solutions for the Cahn-Hilliard equation[END_REF]. However, if n = 4, 5, 6, Ω is a ball and p is critical, (P µ ) has at least one nonconstant radial solution for small µ [START_REF] Adimurthi | Existence and nonexistence of positive radial solutions of Neumann problems with critical Sobolev exponents[END_REF].

For large µ, it is known that in both subcritical and critical cases, (P µ ) has solutions which concentrate at some points of the domain as µ goes to infinity (alternatively, d goes to zero in (P ′ d )). The next question is to characterize such concentration points. In both cases, the least energy solutions have, for large µ, exactly one maximum point which lies on the boundary of the domain, and which goes, as µ goes to infinity, to a maximum point of the mean curvature of the boundary [START_REF] Ni | Takagi -On the shape of least-energy solutions to a semi-linear problem Neumann problem[END_REF][START_REF] Ni | Takagi -Locating the peaks of least-energy solutions to a semi-linear Neumann problem[END_REF] [START_REF] Adimurthi | Interaction between the geometry of the boundary and positive solutions of a semilinear Neumann problem with critical nonlinearity[END_REF][START_REF] Adimurthi | Characterization of concentration points and L ∞ -estimates for solutions of a semilinear Neumann problem involving the critical Sobolev exponent[END_REF][START_REF] Ni | Takagi -Singular behavior of least-energy solutions of a semi-linear Neumann problem involving critical Sobolev exponents[END_REF][START_REF] Wang | Remarks on a nonlinear Neumann problem with critical exponent[END_REF][START_REF] Rey | An elliptic Neumann problem with critical nonlinearity in three dimensional domains[END_REF].

For subcritical exponents, higher energy solutions exist which blow up at one or several points of the boundary [START_REF] Gui | Multi-peak solutions for a semilinear Neumann problem[END_REF][START_REF] Li | On a singularly perturbed equation with Neumann boundary condition[END_REF][START_REF] Wei | Winter -Stationary solutions for the Cahn-Hilliard equation[END_REF][START_REF] Dancer | Multipeak solutions for a singularly perturbed Neumann problem[END_REF] as µ goes to infinity. Solutions also exist which blow up at one or several points in the interior of the domain [START_REF] Wei | On the interior spike layer solutions of singularly perturbed semilinear Neumann problems[END_REF][START_REF] Grossi | Pistoia -On the effect of critical points of distance function in superlinear elliptic problems[END_REF][START_REF] Cerami | Multiplicity of multiple interior peaks solutions for some singularly perturbed Neumann problems[END_REF][START_REF] Gui | Estimates for boundary bubbling solutions to an elliptic Neumann problem[END_REF][START_REF] Kowalczyk | Multiple spike layers in the shadow Gierer-Meinhardt system : existence of equilibria and quasi-invariant manifold[END_REF][START_REF] Grossi | Existence of multipeak solutions for a semilinear elliptic problem via nonsmooth critical point theory[END_REF][START_REF] Yan | On the number of interior multipeak solutions for singularly perturbed Neumann problems[END_REF][START_REF] Bates | Fusco -Equilibria with many nuclei for the Cahn-Hilliard equation[END_REF]. In particular, (P µ ) has single interior spike solutions which blow up at a local maximum point of the distance function d(x, ∂Ω), x ∈ Ω. (Solutions have also been built which blow up at interior and boundary points at the same time [START_REF] Gui | Multiple mixed boundary and interior peak solutions for some singularly perturbed Neumann problems[END_REF].) For critical exponent, all the existence results concern solutions which blow up at one or several points of the boundary as µ goes to infinity [1-6, 12, 38, 40, 43-47, 33, 22, 24-25]. Hence the question : do solutions blowing up at interior points exist ?

The only known result is partial and negative [START_REF] Cao | Existence and nonexistence of interiorpeaked solution for a nonlinear Neumann problem[END_REF] : for n ≥ 5 and critical p, (P µ ) has no solution u µ such that

u µ -α n k i=1 U λ i µ ,y i µ H 1 (Ω) → 0 as µ → ∞ with α n = (n(n -2)) (n-2)/4 , k ∈ N * U λ,y (x) = λ (n-2)/2 (1 + λ 2 |x -y| 2 ) (n-2)/2 λ ∈ R * + , x, y ∈ R n (1.1)
and λ i µ → ∞, y i µ → y i in Ω as µ → ∞, y i = y j if i = j. Such a result could have also been derived from the arguments in [START_REF] Rey | Boundary effect for an elliptic Neumann problem with critical nonlinearity[END_REF] and, in the case n = 3 and k = 1, from [START_REF] Rey | An elliptic Neumann problem with critical nonlinearity in three dimensional domains[END_REF] (see the final remark at the end of Section 3). A recent paper shows that the case k = 1 cannot happen in any dimension [START_REF] Ghoussoub | On a singularly perturbed Neumann problem with the critical exponent[END_REF].

Our aim in this paper is to consider the question of interior blow-up points, for n = 3 and critical p, without any assumption neither on the number of those points, nor on the distance between them, which may be zero.

(u µ ) µ≥µ0 being a sequence of nonconstant solutions to (P µ ), there are several and equivalent ways to define blow-up points of (u µ ). For example, y ∈ Ω will be said to be a blow-up point of (u µ ) if and only if

lim inf r→0 lim sup µ→∞ B(y,r)∩Ω |∇u µ | 2 or B(y,r)∩Ω u 6 µ > 0.
Our main result is :

Theorem 1 Let (u µ ) µ≥µ0 a sequence, bounded in H 1 (Ω), of solutions to (P µ ).
There exists at least one blow-up point which lies on the boundary of Ω.

We notice that, in contrast with the subcritical case, the existence of solutions with a finite number of blow-up points all lying in the interior of the domain, is excluded. We emphasize that the main difficulty in this work is to eliminate the possibility of multiple interior peaks without a priori assumption on the location of those peaks, which may be very close from each other, or even centered at the same point.

The next section is devoted to an a priori analysis of the solutions to (P µ ) as µ goes to infinity. This analysis, mixing together energy-dependent and energy-independent estimates, provides us with informations about the shape of solutions, which allow us to prove, through variational methods, the theorem in Section 3.

Blow-up analysis 2.1 Energy-independent estimates

We begin with an energy-independent description of the nonconstant solutions to (P µ ) as µ goes to infinity. We have the following proposition : Proposition 2.1 Let (u µ ) µ≥µ0 be a sequence of nonconstant solutions to (P µ ).

Let ε > 0, R > 1. For µ large enough, u µ has N µ ∈ N * local maximum points x i µ ∈ Ω, 1 ≤ i ≤ N µ , such that : (i) µ 1/4 u µ (x i µ ) < ε (ii) 1 u µ (x i µ ) u µ ( x u 2 µ (x i µ ) + x i µ ) - 1 (1 + |x| 2 3 ) 1/2 C 2 (B(0,2R)∩Ω i µ ) < ε with Ω i µ = u 2 µ (x i µ )(Ω -x i µ ) (iii) x i µ ∈ ∂Ω or u 2 µ (x i µ )d(x i µ , ∂Ω) > 1 ε (iv) B(x i µ , r i µ ) ∩ B(x j µ , r j µ ) = ∅ for i = j, r i µ = R u 2 µ (x i µ ) (v) d x, {x i µ , 1 ≤ i ≤ N µ } 1/2 u µ (x) ≤ C(ε, R).
(i) says that the maxima increase faster than µ 1/4 as µ goes to infinity; (ii) describes the shape of u µ in a neighbourhood of a maximum point x i µ ; (iii) shows that either the maximum points x i µ are on the boundary, or are not too close from the boundary with respect to the heighth of the maximum; (iv) shows that these maximum points are not too close from each other with respect to their heighths, and (v) provides us with a global bound for u µ in Ω.

Such a proposition relies on arguments initially developped by Schoen [START_REF] Schoen | On the number of constant scalar curvature metrics in a conformal class[END_REF], in the context of the Yamabe problem. The proof, which follows the same scheme as in [START_REF] Li | Yamabe type equations on three dimensional Riemannian manifolds[END_REF]Proposition 5.1], with the convenient additional arguments, is given in Appendix A.

Let now x µ be any point in Ω. We set

v µ (y) = 1 µ 1/4 u µ ( y µ 1/2 + x µ ) y ∈ Ω µ = µ 1/2 (Ω -x µ ) (2.1)
which satisfies

-∆v µ + v µ = v 5 µ , v µ > 0 in Ω µ ; ∂v µ ∂ν = 0 on ∂Ω µ . (2.2) Proposition 2.1 is equivalent to : Proposition 2.2 Let ε > 0, R > 1. For µ large enough, v µ has N µ ∈ N * local maximum points y i µ = µ 1/2 (x i µ -x µ ) ∈ Ω, 1 ≤ i ≤ N µ , such that : (i) v µ (y i µ ) > 1 ε (ii) 1 v µ (y i µ ) v µ ( y v 2 µ (y i µ ) + y i µ ) - 1 (1 + |y| 2 3 ) 1/2 C 2 (B(0,2R)∩Ω i µ ) < ε with Ω i µ = v 2 µ (y i µ )(Ω µ -y i µ ) = u 2 µ (x i µ )(Ω -x i µ ) (iii) y i µ ∈ ∂Ω or v 2 µ (y i µ )d(y i µ , ∂Ω) > 1 ε (iv) B(y i µ , s i µ ) ∩ B(y j µ , s j µ ) = ∅ for i = j, s i µ = R v 2 µ (x i µ ) (v) d y, {y i µ , 1 ≤ i ≤ N µ } 1/2 v µ (y) ≤ C(ε, R).
The interest of considering v µ instead of u µ is that v µ solves a nonlinear elliptic equation (2.2) whose linear part has constant, hence bounded coefficients. This fact allows us to use techniques and results of Li, concerning the scalar curvature problem [START_REF] Li | Prescribing scalar curvature on S n and related problems, Part. I[END_REF], Li and Zhu concerning Yamabe type equations [START_REF] Li | Yamabe type equations on three dimensional Riemannian manifolds[END_REF].

In view of Proposition 2.2, we set :

Definition 1 ȳ ∈ R 3 is called an isolated blow-up point of (v µ ) if there exist r > 0, C ∈ R and a sequence (y µ ) in Ωµ , converging to ȳ, such that y µ is a local maximum of v µ , v µ (y µ ) → ∞ and v µ (y) ≤ C |y -y µ | 1/2 y ∈ B(y µ , r) ∩ Ω µ .
ȳ is called a simple isolated blow-up point if moreover there exists r 0 > 0 such that, for µ large enough, r 1/2 vµ (r) has exactly one critical point in (0, r 0 ), with

vµ (r) = 1 |∂B(y µ , r) ∩ Ω µ | ∂B(yµ,r)∩Ωµ v µ 0 < r < r.
Then, we can state : In view of our further needs, we consider the points y i µ defined in Proposition 2.2, and we prove the following proposition, which is of crucial interest in the sequel : Proposition 2.4 Let y i µ be as in Proposition 2.2.

Proposition 2.3 (
(i) Assume that there exists ρ > 0 such that, for large µ, d(y i µ , ∂Ω µ ) ≥ ρ, 1 ≤ i ≤ N µ . Then, there exists δ > 0 such that, for µ large enough

|y i µ -y j µ | ≥ δ ∀i, j i = j. (2.5) (ii) If Ω is convex, (2.5) holds with ρ = 0.
This proposition still follows from [START_REF] Li | Prescribing scalar curvature on S n and related problems, Part. I[END_REF][START_REF] Li | Yamabe type equations on three dimensional Riemannian manifolds[END_REF][START_REF] Zhu | Uniqueness results through a priori estimates, I. A three dimensional Neumann problem[END_REF]. Let us sketch the argument for (i). Assuming that the proposition is false, we may suppose, without loss of generality

δ µ = |y 1 µ -y 2 µ | = min i =j |y i µ -y j µ | → 0 as µ → ∞. (2.6)
We set

w µ (z) = δ 1/2 µ v µ (δ µ z + y 1 µ ) z ∈ Ωµ = (Ω µ -y 1 µ )/δ µ . w µ satisfies -∆w µ + δ 2 µ w µ = w 5 µ , w µ > 0 in Ωµ ; ∂w µ ∂ν = 0 on ∂ Ωµ (2.7)
and, denoting

z i µ = (y i µ -y 1 µ )/δ µ , 1 ≤ i ≤ N µ , we know that w µ (0), w µ (z 2 µ ) > R 1/2 because of Prop. 2.2(iv) (2.8) |z i µ -z j µ | ≥ 1 i = j because of (2.6) (2.9) d y, {z i µ , 1 ≤ i ≤ N µ } 1/2 w µ (y) ≤ C because of Prop. 2.2(v). (2.10)
Up to a subsequence, we may assume that z 2 µ → z 2 as µ → ∞, |z 2 | = 1. We claim that w µ (0) and w µ (z 2 µ ) go to infinity as µ goes to infinity. Indeed, let us assume that w µ (z 2 µ ) stays bounded. Prop. 2.2(ii) then implies that w µ stays bounded in a fixed neighbourhood of z 2 . Taking also account of (2.9-10), we see that w µ stays bounded in any ball B(z 2 , r), 0 < r < 1. If w µ (0) goes to infinity, 0 is an isolated blow-up point, hence a simple isolated blow-up point, and an inequality as (2.4) holds for w µ , i.e.

w µ (0)w µ (z) ≤ C |z| z ∈ B(0, r ′ 0 ) ∩ Ωµ (2.11)
Consequently, at a small and fixed distance of 0, w µ goes to zero. Therefore, Harnack inequality applied to (2.7) shows that w µ goes uniformly to zero in B(z 2 , r), 0 < r < 1, in contradiction with (2.8).

If w µ (0) stays bounded, either there is some z i µ , i > 2, such that |z i µ | stays bounded and w µ (z i µ ) goes to infinity, and we can repeat the previous argument with z i µ instead of 0, whence again a contradiction; or w µ stays bounded in any ball centered in 0. Then, elliptic theory shows that, up to a subsequence, w µ goes in C 2 loc (R 3 ) to a limit w which satisfies -∆w = w 5 , w ≥ 0 in R 3 , w ≡ 0, ∇w(0) = ∇w(z 2 ) = 0. According to [START_REF] Caffarelli | Spruck -Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth[END_REF], such a w does not exist, hence again a contradiction. Therefore, 0 and z 2 are two simple isolated blow-up points.

Up to a reindexation and passing to a subsequence, we may assume that for i ≥ 2, either

z i µ → z i , or |z i µ | → ∞ as µ → ∞. Because of (2.9), |z i -z j | ≥ 1 if i = j.
If z is a blow-up point for w µ , z = z i for some index i, because of (2.10). Let S be the set of these blow-up points, which are isolated and simple (note that d(z i µ , ∂ Ωµ ) ≥ ρ/δ µ → ∞, and the equivalent of Prop. 2.3(i) holds for the solutions of (2.7), as [START_REF] Zhu | Uniqueness results through a priori estimates, I. A three dimensional Neumann problem[END_REF] shows). We consider

ξ µ (z) = w µ (0)w µ (z) which satisfies -∆ξ µ + δ 2 µ ξ µ = w 4 µ ξ µ , ξ µ > 0 in Ωµ ; ∂ξ µ ∂ν = 0 on ∂ Ωµ . (2.12)
From (2.10-11) and Harnack inequality applied to (2.7), we know that w µ goes uniformly to zero in any compact set [START_REF] Brezis | Coron -Convergence of solutions of H-systems or how to blow bubbles[END_REF] and Harnack inequality applied to (2.12), we know that ξ µ stays uniformly bounded in any compact set K ⊂ R 3 \ S. Then, elliptic theory ensures that, along some subsequence, ξ µ converges in C 2 loc (R 3 \ S) to a limit ξ, which is a positive regular harmonic function in R 3 \ S. Therefore, we can write

K ⊂ R 3 \ S (note that K ⊂ Ωµ for µ large enough, since d(0, ∂ Ωµ ) ≥ ρ/δ µ ). From (2.
ξ(z) = a |z| + b |z -z 2 | + h(z)
with a ≥ 0, b ≥ 0 and h is regular positive harmonic function in R 3 \ (S -{0, z 2 }). 0 being a simple isolated blow-up point of (w µ ), r → r 1/2 ξµ (r) has a unique critical point in (0, r ′ 0 ), and Prop. 2.2(ii) shows that this function has a maximum point which goes to zero as µ goes to infinity. Therefore, r → r 1/2 ξ(r) is nonincreasing in (0, r ′ 0 ). Then, either ξ ≡ 0, or a > 0. Integrating (2.12) on B r ′ 0 , we find

∂B r ′ 0 ∂ξ µ ∂ν + δ 2 µ B r ′ 0 ξ µ = w µ (0) B r ′ 0 ξ 5 µ .
From Prop. 2.2(ii), we have

w µ (0) B r ′ 0 ξ 5 µ ≥ w µ (0) 2 B(0,2R/w 2 µ (0)) w 5 µ (0) (1 + w 4 µ (0)|y| 2 3 ) 5/2 dy ≥ 1 2 B(0,2R) dy (1 + |y| 2 3 ) 5/2 ≥ τ
with τ a strictly positive constant. On the other hand, (2.11) implies that

δ 2 µ B r ′ 0 ξ µ ≤ Cδ 2 µ B r ′ 0 dy |y| = o(1)
as µ goes to infinity. Moreover, if ξ ≡ 0, we have also

∂B r ′ 0 ∂ξ µ ∂ν = o(1)
hence a contradiction. Consequently, a > 0. In the same way b > 0. The classical Pohozaev identity for (2.12) provides us with the equality

-δ 2 µ Bσ(0) ξ 2 µ = ∂Bσ(0) 1 2 ξ µ ∂ξ µ ∂ν - σ 2 |∇ξ µ | 2 + σ( ∂ξ µ ∂ν ) 2 - 1 2 ∂Bσ(0) δ 2 µ 2 ξ 2 µ - 1 6w 4 µ (0) ξ 6 µ
for any small σ > 0. As δ µ goes to zero and w µ (0) goes to infinity, (2.11) implies that the left hand side, and the last integral on the right hand side, go to zero as µ goes to infinity. In the same time, a straightforward computation shows that lim σ→0 lim µ→∞ ∂Bσ(0)

1 2 ξ µ ∂ξ µ ∂ν - σ 2 |∇ξ µ | 2 + σ( ∂ξ µ ∂ν ) 2 → -2πa(b + h(0)) < 0 whence a contradiction.
(ii), which is not necessary for our further purposes, may be proved in the same way, using the analysis of boundary blow-up points performed in [START_REF] Zhu | Uniqueness results through a priori estimates, I. A three dimensional Neumann problem[END_REF].

Remark. If (u µ ) is assumed to be bounded in H 1 (Ω), N µ is also bounded, since u µ H 1 (Ω) ≥ N µ τ
where τ > 0 is some fixed constant, as Prop. 2.1(ii) shows. Then, up to a reindexation and passing to a subsequence, we may assume that for µ large enough

N µ = k 1 + k 2 = k ∈ N * d(y i µ , ∂Ω µ ) ≥ ρ for some ρ > 0 1 ≤ i ≤ k 1 d(y i µ , ∂Ω µ ) → 0 as µ → ∞ k 1 + 1 ≤ i ≤ k Setting δ µ = min 1≤i,j≤k 1 i =j |y i µ -y j µ |
, the same arguments as previously show the existence of δ > 0 such that |y i µ -

y j µ | > δ, i = j, 1 ≤ i, j ≤ k 1 .

Energy-dependent estimates

We turn now to an energy-dependent blow-up analysis of (u µ ), whose comparison with the previous results will provide us with the informations that we need to prove the theorem in the next section. First, as in [START_REF] Rey | An elliptic Neumann problem with critical nonlinearity in three dimensional domains[END_REF], we define for λ ∈ R * + and a ∈ R 3 the function

V µ,λ,a (x) = U λ,a (x) -ϕ µ,λ,a (x) x ∈ R 3 (2.13)
where U λ,a is given by (1.3), i.e. U λ,a = λ 1/2 (1 + λ 2 |x -a| 2 ) -1/2 , and

ϕ µ,λ,a (x) = 1 -e -µ 1/2 |x-a| λ 1/2 |x -a| . (2.14) V µ,λ,a , which satisfies in R 3 -∆(3 1/4 V µ,λ,a ) + µ(3 1/4 V µ,λ,a ) = (3 1/4 V µ,λ,a ) 5 + µ3 1/4 (U λ,a - 1 λ 1/2 |x -a| ) (2.15
) is an improved approximate solution to (P µ ) with respect to 3 1/4 U λ,a , as µ 1/2 /λ goes to zero -see [START_REF] Rey | An elliptic Neumann problem with critical nonlinearity in three dimensional domains[END_REF]. For µ 1/2 /λ small, ϕ µ,λ,a acts as a perturbation of

U λ,a in H 1 (Ω), since ϕ µ,λ,a 2
H 1 (Ω) = O(µ 1/2 /λ) (2.16)
as integral estimates show [START_REF] Rey | An elliptic Neumann problem with critical nonlinearity in three dimensional domains[END_REF]. Now, we can state :

Proposition 2.5 Let (u µ ) µ≥µ0 be a sequence of nonconstant solutions to (P µ ), bounded in H 1 (Ω). There exist k ∈ N * , (λ i µ ) and (a i µ ) sequences in R * + and Ω respectively such that, for some subsequence

u µ -3 1/4 k i=1 V µ,λ i µ ,a i µ H 1 (Ω) → 0 as µ → ∞ (2.17) with µ 1/2 /λ i µ → 0 (2.18) λ i µ d(a i µ , ∂Ω) → 0 or ∞ (2.19) λ i µ /λ j µ + λ j µ /λ i µ + λ i µ λ j µ |a i µ -a j µ | 2 → ∞ if i = j (2.20)
as µ → ∞.

Note that Proposition 2.5 holds for Palais-Smale sequences as well, whereas the previous one applies to exact solutions of the equation only. Such an analysis is performed for the first time in [START_REF] Bahri | Coron -On a nonlinear elliptic equation involving the critical Sobolev exponent : the effect of the topology of the domain[END_REF], and [START_REF] Brezis | Coron -Convergence of solutions of H-systems or how to blow bubbles[END_REF]. A proof of it, following Bahri's arguments, is given in Appendix B.

In view of Proposition 2.5 we may assume, extracting some subsequence, that each of the sequences (a i µ ) converges to a limit a i ∈ Ω. It is easily seen that the a i 's are exactly the blow-up points of this subsequence.

The shape of solutions with only interior blow-up points

Let us assume that all the a i 's which occur in Proposition 2.5 lie in the interior of Ω. In order to prove the theorem, we have to prove that such a case cannot occur.

Comparing Proposition 2.5 and Proposition 2.2, Prop. 2.2(ii) implies that to each x i µ corresponds some a j(i) µ

such that, ε being small

λ j(i) µ /u 2 µ (x i µ ) is close to 1 ; λ j(i) µ |a j(i) µ -x i µ | is close to 0 (2.21) with j(i 1 ) = j(i 2 ) if i 1 = i 2 .
Conversely, Prop. 2.2(v) implies that to each a j µ corresponds some

x i(j) µ such that λ j µ |a j µ -x i(j) µ | is bounded. We claim that for µ large enough, i(j 1 ) = i(j 2 ) if j 1 = j 2 .
Otherwise, up to subsequences and reindexations, we may assume that j = 1, . . . , p are the p indices, p ≥ 2, such that i(j) = 1. We consider v µ defined by (2.1), with x µ = x 1 µ . Through the change of variable

x = y µ 1/2 + x 1 µ x = x 1
µ is sent to y = 0, and x = a i µ is sent to y = µ 1/2 (a i µx 1 µ ). We know that 0 is an isolated blow-up point of (v µ ). Since x 1 µ goes to a 1 = . . . = a p which lies in the interior of Ω, d(0, Ω µ ) → ∞, and Proposition 2.3 implies that 0 is a simple isolated blow-up point.

On the other hand, we notice that the λ j µ 's are not of the same order as µ goes to infinity, that is

λ i µ /λ j µ + λ j µ /λ j µ → ∞ 0 ≤ i < j ≤ p. Otherwise, λ j µ |a j µ -x 1 µ | being bounded, λ i µ λ j µ |a i µ -a j µ | 2
would also be bounded, and (2.20) could not be satisfied.

Finally, we notice that the boundedness of λ j µ |a j µx 1 µ | implies, through (2.18), that µ 1/2 (a j µx 1 µ ) goes to 0 as µ goes to infinity. It follows that r → r 1/2 vµ , r = |yx 1 µ |, has several maximum points in any fixed interval (0, r 0 ), for µ large enough. This contradicts the fact that 0 is a simple blow-up point.

Once we know that there is a correspondance one to one between the x i µ 's and the a j µ 's, we infer from (2.21) and Proposition 2.4 that there exists γ > 0 such that

|a i µ -a j µ | > γ µ 1/2 i = j
for µ large enough. As a consequence, we know that a sequence (u µ ), bounded in H 1 (Ω), of solutions to (P µ ) whose all blow-up points lie in the interior of Ω writes as

u µ = 3 1/4 k i=1 V µ,λ i µ ,a i µ + v µ k ∈ N * (2.22) with µ 1/2 λ i µ → 0 ; a i µ → a i ∈ Ω |a i µ -a j µ | > γ µ 1/2 if i = j (2.23)
and

v µ → 0 in H 1 (Ω) (2.24)
as µ goes to infinity.

We are going to prove, in the next section, that such a u µ cannot solves (P µ ) for large µ -hence the theorem.

Proof of the theorem

We adopt in this section a variational approach of the problem. We define the functional

J µ (u) = 1 2 Ω (|∇u| 2 + µu 2 ) - 1 6 Ω u 6 u ∈ H 1 (Ω) (3.1)
whose strictly positive critical points are exactly the solutions to (P µ ).

A parametrization of the variational problem

This subsection is devoted to a parametrization of the variational problem in a neighbourhood of the eventual solutions to (P µ ) defined by (2.22-24). k ∈ N * and ρ > 0 being fixed, for ε > 0 we set

V ε,µ = u ∈ H 1 (Ω)/∃(λ i ) 1≤i≤k ∈ (R * + ) k , µ 1/2 /λ i < ε, ∃(a i ) 1≤i≤k ∈ (Ω ρ ) k , |a i -a j | > γ 4µ 1/2 if i = j, s.t. |∇(u -3 1/4 k i=1 V µ,λi,ai )| 2 < ε with Ω ρ = x ∈ Ω, d(x, ∂Ω) > ρ . Defining also B ε,µ = (α, λ, a) ∈ R k × (R * + ) k × (Ω ρ-ε ) k s.t. 3 1/4 /2 < α i < 2.3 1/4 , µ 1/2 /λ i < ε, |a i -a j | > γ -ε 4µ 1/2 if i = j
we have :

Lemma 3.1 There exist µ 0 > 0, ε 0 > 0 such that for any µ ≥ µ 0 , any ε, 0 < ε ≤ ε 0 , and any u ∈ V ε,µ , the infimum

inf (α,λ,y)∈B4ǫ,µ |∇(u -3 1/4 k i=1 V µ,λi
E λ,a,µ = v ∈ H 1 (Ω)/ Ω ∇v.∇V i µ = Ω ∇v.∇ ∂V i µ ∂λi = Ω ∇v.∇ ∂V i µ ∂(ai) l = 0 1 ≤ i ≤ k, 1 ≤ l ≤ 3 (3.2) with V i µ = V µ,λi,ai
, for sake of simplicity. For µ ≥ µ 0 , Lemma 3.1 induces a map Φ from the open subset V ε0,µ of H 1 (Ω) to the manifold

M µ = (α, λ, a, v) ∈ R k × (R * + ) k × Ω k × H 1 (Ω) s.t. (α, λ, a) ∈ B 2ε0,µ , v ∈ E λ,a,µ , |∇v| 2 < ε
where (α(u), λ(u), a(u)) is the unique point in B 2ε0,µ at which the infimum of |∇(u -

3 1/4 k i=1 V µ,λi,ai | 2 is achieved, and v(u) = u - k i=1 α i (u)V µ,λi(u),ai(u) .
This map is open, and induces a diffeomorphism between V ε0,µ and its image, which contains

N µ = (α, λ, a, v) ∈ R k × (R * + ) k × Ω k ρ × H 1 (Ω) s.t.|α i -3 1/4 | < η 0 , µ 1/2 /λ i < η 0 , |a i -a j | > γ 2µ 1/2 if i = j, v ∈ E λ,a,µ and |∇v| 2 < ε (3.
3) for some η 0 > 0 small enough. Setting

K µ : N µ → R (α, λ, a, v) → J µ ( k i=1 α i V µ,λi,ai + v) (3.4) we know that (α, λ, a, v) ∈ N µ is a critical point of K µ if and only if u = k i=1 α i V µ,λi,ai + v is a critical point of J µ .
Let us notice that, for µ large enough, u µ given by (2. [START_REF] Gui | Ghoussoub -Multi-peak solutions for a Neumann problem involving the critical Sobolev exponent[END_REF][START_REF] Gui | Estimates for boundary bubbling solutions to an elliptic Neumann problem[END_REF][START_REF] Gui | Multiple interior peak solutions for some singularly perturbed Neumann problems[END_REF] is in V ε0,µ . Moreover, setting

Φ(u µ ) = ( αµ , λµ , ãµ , ṽµ ) it follows from [8,38, Lemma A.1] that αi µ → 3 1/4 µ 1/2 / λµ → 0 ãi µ → a i |ã i µ -ãj µ | > γ 2µ 1/2 if i = j and ṽµ → 0 in H 1 (Ω).
In particular, Φ(u µ ) ∈ N µ for µ large enough. We are going to show that (α µ , λµ , ãµ , ṽµ ) cannot be a critical point of K µ for µ large enough, whence the theorem.

The v-derivative of K µ

In this subsection, we estimate the H 1 -norm of v µ as µ goes to infinity. In view of (3.3-4), expanding K µ with respect to v in a neighbourhood of v = 0, we find

K µ (α, λ, a, v) = K µ (α, λ, a, 0) + f α,λ,a,µ (v) + Q α,λ,a,µ (v) + R α,λ,a,µ (v) (3.5) with f α,λ,a,µ (v) = µ Ω ( k i=1 α i V i µ )v - Ω ( k i=1 α i V i µ ) 5 v (3.6) Q α,λ,a,µ (v) = 1 2 Ω (|∇u| 2 + µu 2 ) - 5 2 Ω ( k i=1 α i V i µ ) 4 v 2 (3.7) R α,λ,a,µ (v) = O( v 3 H 1 (Ω) ). (3.8)
Moreover, choosing some η 0 > 0 sufficiently small, there exist κ > 0, κ ′ > 0 such that for µ large enough and any (α, λ, a, v)

∈ N µ κ Ω (|∇u| 2 + µu 2 ) ≤ Q α,λ,a,µ (v) ≤ κ ′ Ω (|∇u| 2 + µu 2 ).
(3.9)

The second inequality is a direct consequence of Hölder inequality, Sobolev embedding theorem and estimate (C.3) in appendix. The coercivity property follows from [40, Lemma 3.2] in the case k = 1. The result extends to the case k > 1 using the arguments of [7, Prop. 3.1], which are valid provided that

λ i λ j |a i -a j | 2 , i = j, is large enough. But (α, λ, a, v) ∈ N µ implies that λ i λ j |a i -a j | 2 > γ 2 /4η 2 0
, whence the desired result choosing η 0 small enough. On the other hand, we claim that there exists C > 0 such that, for µ large enough and any (α, λ, a, v) ∈ N µ , we have

|f α,λ,a,µ (v)| ≤ 1 µ 1/4 |λ| 1/2 + µ 1/2 |λ| 1 2 Ω (|∇u| 2 + µu 2 ) 1/2 (3.10) with |λ| = ( k i=1 λ 2 i ) 1/2 .
Let us assume that the claim is true. Then, we deduce from (3.8-10) and the implicit functions theorem the following proposition : Proposition 3.1 There exist η 1 > 0, η 2 > 0 such that, for µ large enough, there exists a smooth map

ѵ = (α, λ, a) ∈ R k × (R * + ) k × Ω k ρ s.t.|α i -3 1/4 | < η 1 , µ 1/2 /λ i < η 1 , |a i -a j | > γ 2µ 1/2 if i = j → E λ,a,µ (α, λ, a) → vµ (α, λ, a) such that vµ (α, λ, a) is the unique point v ∈ E λ,a,µ , |∇v| 2 2 +µ|v| 2 2 < η 2 , satisfying ∂K µ ∂v (α, λ, a, vµ (α, λ, a)) = 0 in T (α,λ,a,vµ) N µ . (3.11)
Moreover, there exists C > 0 such that

Ω |∇v µ | 2 + µ Ω v2 µ ≤ C 1 µ 1/2 |λ| + µ |λ| 2 .
(3.12)

We notice that (3.11) means that there exist (A,

B, C) ∈ R k × R k × (R 3 ) k such that, for w ∈ H 1 (Ω) ∂K µ ∂v (α, λ, a, vµ ).w = k i=1 A i Ω ∇V i µ .∇w + B i Ω ∇ ∂V i µ ∂λ i .∇w + 3 l=1 C il Ω ∇ ∂V i µ ∂(a i ) l .∇w .
Taking respectively w = V i µ ,

∂V i µ ∂λi , ∂V i µ ∂(ai) l , 1 ≤ i ≤ k, 1 ≤ l ≤ 3
, we see that the A i , B i , C il 's solve a linear system which is nearly diagonal since, using the integral estimates in [START_REF] Bahri | Critical points at infinity in some variational problems[END_REF][START_REF] Rey | An elliptic Neumann problem with critical nonlinearity in three dimensional domains[END_REF] and Appendix C, we have

Ω ∇V i µ .∇V j µ = 3π 2 4 δ ij + O µ 1/2 λ 1/2 i λ 1/2 j Ω ∇V i µ .∇ ∂V j µ ∂λ j = O µ 1/2 λ 1/2 i λ 3/2 j Ω ∇V i µ .∇ ∂V j µ ∂(a j ) l = O µ 1/2 λ 1/2 j λ 1/2 i Ω ∇ ∂V i µ ∂λ i .∇ ∂V j µ ∂λ j = 15π 2 64
δ ij λ 2 i + O µ 1/2 λ 3/2 i λ 3/2 j Ω ∇ ∂V i µ ∂λ i .∇ ∂V j µ ∂(a j ) l = O µ 1/2 λ 1/2 j λ 3/2 i Ω ∇ ∂V i µ ∂(a i ) l .∇ ∂V j µ ∂(a j ) m = 15π 2 64 λ 2 i δ ij δ lm + O µ 1/2 λ 1/2 j λ 3/2 i . (3.13)
On the other hand

∂K µ ∂v (α, λ, a, vµ ).V i µ = 1 α i ∂K µ ∂α i (α, λ, a, vµ ) = O |α i -3 1/4 | + µ 1/2 |λ| ∂K µ ∂v (α, λ, a, vµ ). V i µ ∂λ i = 1 α i ∂K µ ∂λ i (α, λ, a, vµ ) = O µ 1/2 λ i |λ| ∂K µ ∂v (α, λ, a, vµ ). V i µ ∂(a i ) l = 1 α i ∂K µ ∂(a i ) l (α, λ, a, vµ ) = O µ 1/2 λ i |λ|
as it follows from Proposition C.1 in appendix. Then, solving the linear system provides us with the estimates

A i = O |α i -3 1/4 | + µ 1/2 |λ| B i = O µ 1/2 λ i |λ| C il = O µ 1/2 λ i |λ|                    (3.14)
Before ending this subsection, we prove claim (3.10). From the proof of Lemma 3.1 in [START_REF] Rey | An elliptic Neumann problem with critical nonlinearity in three dimensional domains[END_REF], we know that there exists C > 0 such that, for µ large enough and any (α, λ, a, v)

∈ N µ µ Ω α i V i µ v - Ω (α i V i µ ) 5 v ≤ C 1 µ 1/4 |λ| 1/2 + µ 1/2 |λ| 1 2 Ω (|∇u| 2 + µu 2 ) 1/2 (3.15) 1 ≤ i ≤ k.
On the other hand, if i = j, the Hölder inequality and the Sobolev embedding theorem yield From the definition (3.1) of E λ,a,µ , it follows that the partial derivative of vµ with respect to α i is also in E λ,a,µ . Therefore, we deduce from (3.11) that

Ω (V i µ ) 4 V j µ v ≤ C v H 1 (Ω) Ω |V i µ | 24/5 |V j µ | 6/5
∂ Kµ ∂α i (α, λ, a) = ∂K µ ∂α i (α, λ, a, vµ (α, λ, a))
that is, according to Proposition C.1 in appendix

∂ Kµ ∂α i (α, λ, a) = π 2 4 α i (3 -α 4 i ) + O( µ 1/2 |λ| ).
In the same way, we have

∂ 2 Kµ ∂α 2 i (α, λ, a) = ∂ 2 K µ ∂α 2 i (α, λ, a, vµ (α, λ, a))
whence, according to (C.22)

∂ 2 Kµ ∂α 2 i (α, λ, a) = π 2 4 (3 -5α 4 i ) + O( µ 1/2 |λ| ).
Consequently, we obtain : Proposition 3.2 For µ large enough, there exists a smooth map

ѵ = (λ, a) ∈ (R * + ) k × Ω k ρ s.t. µ 1/2 /λ i < η 1 |a i -a j | > γ 2µ 1/2 if i = j → R k + (λ, a) → ᾱµ (λ, a) such that ᾱµ (λ, a) is the unique point α ∈ R k + , |α i -3 1/4 | < η 1 , satisfying ∂ Kµ ∂α (ᾱ µ (λ, a), λ, a) = 0.
Moreover, ᾱµ (λ, a) satisfies

ᾱi µ (λ, a) = 3 1/4 + O( µ 1/2 |λ| ).
(3.17)

The λ-derivative of K µ

This last subsection will provide us with the contradiction which proves the theorem. For (λ, a) ∈ ѵ , we set Kµ (λ, a) = Kµ ᾱµ (λ, a), λ, a, vµ (ᾱ µ (λ, a), λ, a) .

From (3.11) we know that

∂ Kµ ∂λ i (λ, a) = ∂ Kµ ∂λ i (ᾱ µ , λ, a, vµ ) + k j=1 A j Ω ∇V j µ .∇ ∂v µ ∂λ i + B j Ω ∇ ∂V j µ ∂λ j .∇ ∂v µ ∂λ i + 3 l=1 C jl Ω ∇ ∂V j µ ∂(a i ) l .∇ ∂v µ ∂λ i .
Since vµ ∈ E λ,a,µ , we have

Ω ∇V j µ .∇ ∂v µ ∂λ i = - Ω ∇ ∂V j µ ∂λ i .∇v µ = 0 Ω ∇ ∂V j µ ∂λ j .∇ ∂v µ ∂λ i = - Ω ∇ ∂ 2 V j µ ∂λ j ∂λ i .∇v µ = O δ ij ∇ ∂ 2 V i µ ∂λ 2 i 2 |∇v µ | 2 Ω ∇ ∂V j µ ∂(a j ) l .∇ ∂v µ ∂λ i - Ω ∇ ∂ 2 V j µ ∂(a j ) l ∂λ i .∇v µ = O δ ij ∇ ∂ 2 V i µ ∂(a i ) l ∂λ i 2 |∇v µ | 2 .
Then, (3.12,14) and estimates (C.8) yield

∂ Kµ ∂λ i (λ, a) = ∂ Kµ ∂λ i (ᾱ µ , λ, a, vµ ) + O 1 λ i ( µ 1/4 |λ| 3/2 + µ |λ| 2 )
whence, according to (3.17) and (C.20)

∂ Kµ ∂λ i (λ, a) = -2π µ 1/2 λ 2 i + o µ 1/2 λ 3/2 i |λ| 1/2 . (3.18)
On the other hand, Propositions 3.1 and 3.2 show that, necessarily ṽµ = vµ (α µ , λµ , ãµ ) αµ = ᾱµ ( λµ , ãµ )

for µ large enough where, according to Subsection 3.1, ( αµ , λµ , ãµ , ṽµ ) = Φ(u µ ). Moreover, u µ being a critical point of J µ , ( λµ , ãµ ) satisfies

∂ Kµ ∂λ i ( λµ , ãµ ) = 0 1 ≤ i ≤ k. (3.19)
However, up to a subsequence and a reindexation, we may assume, without loss of generality, that λ1 µ = min 1≤i≤k λi µ for µ large enough. Then, (3.18) (3.19). This completes the proof of the theorem.

implies that ∂ Kµ ∂λ 1 ( λµ , ãµ ) = -2π µ 1/2 ( λ1 µ ) 2 + o µ 1/2 ( λ1 µ ) 2 in contradiction with

Remarks.

(1) If a i were on the boundary of Ω, some additional term would occur in (3.18), involving the mean curvature of the frontier at a i , and (3.19) would not lead to a contradiction -see [START_REF] Rey | An elliptic Neumann problem with critical nonlinearity in three dimensional domains[END_REF].

(2) If, instead of (P µ ), we consider the subcritical problem -∆u + µu = u 5-εµ , u > 0 in Ω ; ∂u ∂ν = 0 on ∂Ω ε µ > 0, ε µ ln µ → 0 as µ → ∞, and the corresponding modified functional, an additional term would occur in (3.18) which, up to a strictly positive constant, would be equal to ε µ /λ i -see [START_REF] Bahri | Rey -On a variational problem with lack of compactness : the topological effect of the critical point at infinity[END_REF][START_REF] Rey | The topological impact of critical points at infinity ia a variational problem with lack of compactness : the dimension 3[END_REF]. Then, the derivative of Kµ with respect to λ i would vanish for some λ i ∼ µ 1/2 /ε µ . Therefore, the obstruction to the fulfillment of (3.19) disappears in the subcritical case, in accordance with the known results.

(3) In [START_REF] Cao | Existence and nonexistence of interiorpeaked solution for a nonlinear Neumann problem[END_REF] are considered problems as

-∆u + µu = u (n+2)/(n-2) + a(x)u q , u > 0 in Ω ; ∂u ∂ν = 0 on ∂Ω Ω ∈ R n , n ≥ 3, 1 < q < (n + 2)/(n -2)
. We notice that the additional term with respect to (P µ ) would introduce in (3.18) a quantity as a(a i )/λ

(q+3)/2 i if q < 2, a(a i )(ln λ i )/λ 5/2 i if q = 2, a(a i )/λ (7-q)/2 i if q > 2.
Then, we see that (3.19) could be satisfied for a(a i ) > 0 and q > 3. This agrees with Theorem 1.1 in [START_REF] Cao | Existence and nonexistence of interiorpeaked solution for a nonlinear Neumann problem[END_REF].

(4) In the special case k = 1, the result could be easily derived from [START_REF] Rey | An elliptic Neumann problem with critical nonlinearity in three dimensional domains[END_REF]. Namely, considering the functional

I µ (u) = Ω (|∇u| 2 + µu 2 ) Ω u 6 1/3
[40] provides us with the following expansion for the equivalent of Kµ , that we denote in the same way :

∂ Kµ ∂λ (λ, a) = 2π 1/3 -2 µ 1/2 λ 2 + H(y) λ 2 (ln λ 2µ 1/2 -γ - 1 2
)

+ O 1 λ 2 µ 1/2 + µ λ 3 + µ 1/2 λ 3 ln λ µ 1/2
for a concentration point a on the boundary. If a lies in the interior of the domain, similar and easier computations would have given

∂ Kµ ∂λ (λ, a) = -4π 1/3 µ 1/2 λ 2 + +O 1 λ 2 µ 1/2 µ λ 3
which cannot vanish for large λ.

For n ≥ 5, computations in [START_REF] Rey | Boundary effect for an elliptic Neumann problem with critical nonlinearity[END_REF] provide us with the expansion

∂ Kµ ∂λ i (λ, a) = C n H(a i ) λ 2 i -C ′ n µ λ 3 i
+ lower order terms (with C n and C ′ n strictly positive constant) when the concentrations points are on the boundary. For interior points we would have

∂ Kµ ∂λ i (λ, a) = -2C ′ n µ λ 3 i + lower order terms.
Again, these quantities cannot vanish for large λ i 's, whence the equivalent of the Theorem under the assumption that the concentration points stay far from each other -as in [START_REF] Cao | Existence and nonexistence of interiorpeaked solution for a nonlinear Neumann problem[END_REF].

Treating the case n = 4 in the same way would require, in order to obtain convenient expansions, to consider suitable approximate solutions, as we did in the case n = 3.

Let us assume that µu -4 µ (x µ ) goes to infinity as µ goes to infinity. w µ defined as

w µ = v µ ( x α µ ) x ∈ α µ Ωµ with α µ = µ 1/2 u 2 µ (x µ ) satisfies -∆w µ + w µ = 1 α 2 µ w 5 µ , w µ > 0 in α µ Ωµ ; ∂w µ ∂ν = 0 on ∂(α µ Ωµ ) and w µ (x) ≤ √ 2 ∀x ∈ B(0, α µ R µ ).
Elliptic theory shows that, up to a subsequence and a space rotation, w µ converges in C 2 loc (R 3 T ) to a limit w which satisfies

-∆w + w = 0 , 0 ≤ w ≤ √ 2 in R 3 T ; ∂w ∂ν = 0 on ∂R 3 T with R 3 T = x = (x 1 , x 2 , x 3 ) ∈ R 3 , x 3 > -T 0 ≤ T ≤ ∞.
This implies w ≡ 0, in contradiction with w(0) = 1. Therefore, µu -4 µ (x µ ) is bounded.

Up to a subsequence, we can assume

µu -4 µ (x µ ) → θ as µ → ∞ 0 ≤ θ < ∞
and, up to a space rotation, v µ converges in C 2 loc (R 3 S ) to a limit v which satisfies

-∆v + θv = v 5 , 0 ≤ v ≤ √ 2 in R 3 S ; ∂w ∂ν = 0 on ∂R 3 S .
Note that if S < ∞, v may be continued by reflection in a bounded solution of the equation in whole R 3 . As a consequence, if θ > 0, v is constant, i.e. v ≡ 0 or v = θ 1/4 . Since v(0) = 1, θ = 1 and v ≡ 1. This implies that u µ is constant for µ large enough, in contradiction with the initial assumption. Before proving this fact, let us complete the proof of the lemma. If θ = 0, we know that v writes as -see [START_REF] Caffarelli | Spruck -Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth[END_REF] In view of (B.3-4)

v(x) = λ 1/2 (1 + λ 2 |x-x0| 2 3 ) 1/2 for some x 0 ∈ R 3 , λ ∈ R * + . v ≤ √ 2 and v(0) = 1 imply that λ ≤ √ 2 λ 1/2 (1 + λ 2 |x 0 | 2 3 ) -1/2 = 1 whence 1 ≤ λ ≤ √ 2 |x 0 | 2 = 3(λ -1)/λ 2 ≤ 3/4. Noticing that inf v∈H 1 (Ωµ) |∇v| 2 2 + ε 2 µ |v| 2 2 |v| 2 6 = inf u∈H 1 (Ω)
S∩Ωµ ũ6 µ ≤ (aµ+εµb+εµB(0,1))∩Ω u 6 µ ≤ Q µ (ε µ ) < A 3/2
so that, for any smooth

ζ such that S = suppζ ⊂ b + B(0, 1), b ∈ R 3 Ωµ (ζ ũµ ) 6 → 0 as µ → ∞ (B.7)
in contradiction with (B.3).

Let us assume now that µε 2 µ is bounded. In particular, ε µ goes to zero and, up to a subsequence and a space rotation, Ω µ goes to R 3 T , with

R 3 T = x = (x 1 , x 2 , x 3 ) ∈ R 3 , x 3 > -T T = lim µ→∞ d(a µ , ∂Ω)/ε µ . Noticing that |ũ µ | L 6 (Ωµ) = |u µ | L 6 (Ω) and |∇ũ µ | L 2 (Ωµ) = |∇u µ | L 2 (Ω)
, we may assume that there exist ũ ∈ L 6 (R 3 T ), ṽ ∈ L 2 (R 3 T ) such that for any compact set

K ∈ R 3 T (K ∈ Ω µ for µ large enough) ũµ ⇀ ũ in L 6 (K) ∇ũ µ ⇀ ṽ in L 2 (K). Moreover, for any ϕ ∈ C ∞ 0 (R 3 T ) Ωµ ∇ũ µ .∇ϕ → R 3 T ṽ.∇ϕ and Ωµ ∇ũ µ .∇ϕ → - R 3
T ũ∆ϕ whence ṽ = ∇ũ.

Assuming that case (ii) occurs, ũ satisfies

-∆ũ + αũ = ũ5 in R 3 T . (B.8)
Moreover, ũ ≥ 0, since, along some subsequence, ũµ → ũ almost everywhere. If T = ∞, R 3 T = R 3 , and the only solutions to (B.8) are ũ ≡ 0 or ũ ≡ α 1/4 . ũ ≡ α 1/4 is excluded, since ũ ∈ L 6 (R 3 ). If ũ ≡ 0, ũµ goes to zero in L 2 (K), for any compact set K. Then, using this result instead of (B.5), the same argument as previously leads to (B.7), that is a contradiction.

If T < ∞, we first notice that the normal derivative of ũ vanishes on ∂R 3 T . Indeed, from (B.4) we know that for any ϕ ∈ C ∞ 0 (R 3 )

Ωµ ∇ũ µ .∇ϕ + µε 2 µ Ωµ ũµ ϕ = Ωµ ũ5 µ ϕ
from which we easily deduce that

R 3 T ∇ũ.∇ϕ + α R 3 T ũϕ = R 3 T ũ5 ϕ whence ∂ ũ ∂ν = 0 on ∂R 3 T .
Then, we can continue ũ by reflection with respect to x 3 = -T , and the continuation, still denoted ũ, satisfies (B.8) in whole R 3 , whence ũ ≡ 0 or ũ ≡ α 1/4 . ũ ≡ α 1/4 is again impossible, since ũ ∈ L 6 (R 3 ). ũ ≡ 0 implies, as previously, that ũµ goes to zero in L 2 (K), for any compact set

K ∈ R 3 T . If B(0, 1) ∈ R 3
T (i.e. T > 1), we can repeat the argument which leads to (B.7), hence a contradiction. If 0 ≤ T ≤ 1, ũµ may be continued in B(0, 2) \ Ω µ in such a way that, ũ′ µ denoting the continuation of ũµ in Ω µ ∪ B(0, 2)

ũ′ µ H 1 (B(0,2)) ≤ C ũµ H 1 (B(0,2)∩Ωµ) ≤ C ′
C and C ′ independent of µ. Then, along some subsequence

ũ′ µ ⇀ ũ′ in H 1 (B(0, 2)) ũ′ µ → ũ′ in L 2 (B(0, 2))
with ũ′ = ũ = 0 in B(0, 2) ∩ R 3 T . Consequently

Ωµ∩B(0,2) ũ2 µ ≤ 2 Ωµ∩B(0,2) (ũ µ -ũ′ µ ) 2 + 2
Ωµ∩B(0,2)

ũ′ 2 µ → 0 as µ → ∞
and we are still able to prove (B.7), hence again a contradiction.

As announced, (iii) is the only possible case, and ũ satisfies

-∆ũ = ũ5 , ũ ≥ 0 in R 3 T ; ∂ ũ ∂ν = 0 on ∂R 3 T (B.9) with ∇ũ ∈ L 2 (R 3 T ), ũ ∈ L 6 (R 3
T ). If ũ ≡ 0, reasoning as previously provides us with a contradiction. Therefore, there exist λ ∈ R * + and a ∈ ∂R 3 such that, according to [START_REF] Caffarelli | Spruck -Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth[END_REF] ũ

(x) = 3 1/4 λ 1/2 (1 + λ 2 |x -a| 2 ) 1/2
and either T = ∞, or T < ∞ and a ∈ ∂R 3

T . Then, we set

u (1) µ (x) = u µ (x) -3 1/4 V µ,λ/εµ,aµ+εµa (x) x ∈ Ω (B.10)

C Estimates

In this last part, we collect the integral estimates which are used in Section 3. First, we recall that in [START_REF] Rey | An elliptic Neumann problem with critical nonlinearity in three dimensional domains[END_REF] the following is proved, for V = V µ,λ,a

Ω |∇V | 2 = 3π 2 4 -6π µ 1/2 λ + O( 1 λµ 1/2 ) (C.1) Ω V 2 = 2π λµ 1/2 + O( 1 λµ 3/2 ) (C.2) Ω V 6 = π 2 4 -8π µ 1/2 λ + O( µ 1/2 λ 2 ) (C.3)
as a does not approach the boundary of Ω, and µ 1/2 /λ goes to infinity. Actually, a is assumed in [START_REF] Rey | An elliptic Neumann problem with critical nonlinearity in three dimensional domains[END_REF] to be on ∂Ω. The above results follow from the same computations, made easier by the fact that a boundary effect has not to be considered. In the same way, we have

Ω ∇V.∇ ∂V ∂λ = 3π µ 1/2 λ 2 + O( 1 λ 2 µ 1/2 ) Ω ∇V.∇ ∂V ∂a j = O(µ 1/2 ) (C.4) Ω V ∂V ∂λ = - π λ 2 µ 1/2 + O( 1 λ 2 µ 3/2 ) Ω V ∂V ∂a j = O( 1 µ 1/2 ) (C.5) Ω V 5 ∂V ∂λ = 4πµ 1/2 3λ 2 + O( µ 1/2 λ 3 ) Ω V 5 ∂V ∂a j = O(µ 1/2 ) (C.6)
and

Ω ∇ ∂V ∂λ 2 = 15π 2 64λ 2 + O( µ 1/2 λ 3 ) Ω ∇ ∂V ∂a j 2 = 15π 2 64 λ 2 + O(λµ 1/2 ) (C.7) Ω ∇ ∂ 2 V ∂λ 2 2 = O( 1 λ 2 ) Ω ∇ ∂ 2 V ∂λ∂a j 2 = O(1). (C.8)
In view of (3.13), we have also to estimate integrals in which both V i and V j occur, i = j. Let us prove, for example

Ω ∇V i .∇V j = O( µ 1/2 λ 1/2 i λ 1/2 j ) (C.9)
as µ 1/2 /λ i , µ 1/2 /λ j go to infinity, a i and a j do not approach the boundary of Ω and |a ia j | > γ/2µ 1/2 . From (2.15) we deduce that

Ω ∇V i .∇V j = ∂Ω ∂V i ∂ν V j + Ω 3U 5 i - µe -µ 1/2 |x-ai| λ 1/2 |x -a i | V j . (C.10)
It follows from the definition (2.13-14) of V that, a i and a j staying far from ∂Ω

V j = o( 1
λ 1/2 j ) |∇V i | = o( µ 1/2 λ 1/2 i ) on ∂Ω whence ∂Ω ∂V i ∂ν V j = o( µ 1/2 λ 1/2 i λ 1/2 j
).

(C.11)

Turning now to the integral on Ω, we notice that

V j = O( µ 1/2 λ 1/2 j ) outside of B j = B(a j , γ/4µ 1/2 ). (C.12)
On the other hand, taking

R ′ > 0 such that Ω ⊂ B(0, R ′ ) Ω U 5 i ≤ 4π λ 1/2 i 2R ′ λi 0 r 2 (1 + r 2 ) 5/2 dr = O( 1 λ 1/2 i ) (C.13)
and

Ω µe -µ 1/2 |x-ai| λ 1/2 |x -a i | dx ≤ 4π λ 1/2 i 2R ′ 0 µre -µ 1/2 r dr = O( 1
λ 1/2 i ). Therefore Ω\Bj 3U 5 i - µe -µ 1/2 |x-ai| λ 1/2 |x -a i | V j = O( µ 1/2 λ 1/2 i λ 1/2 j ). (C.14)
Finally, we notice that

U 5 i = O( µ 5/2 λ 5/2 i ) µe -µ 1/2 |x-ai| λ 1/2 |x -a i | = O( µ 3/2 λ 1/2 i ) in B j and Bj |V j | ≤ 4π λ 5/2 j γλ j 4µ 1/2 0 1 (1 + r 2 ) 1/2 - 1 -e -(µ 1/2 /λj )r r r 2 dr ≤ 4π λ 5/2 j   γλ j 4µ 1/2 0 (r - r 2 (1 + r 2 ) 1/2 )dr + γλ j 4µ 1/2 0 re -(µ 1/2 /λj )r dr   . Consequently Bj |V j | = O ln(λ j /µ 1/2 ) λ 5/2 j + 1 λ 1/2 j µ (C.15) and Bj 3U 5 i - µe -µ 1/2 |x-ai| λ 1/2 |x -a i | V j = O( µ 1/2 λ 1/2 i λ 1/2 j
).

This estimate, joined to (C. [START_REF] Bates | Fusco -Equilibria with many nuclei for the Cahn-Hilliard equation[END_REF][START_REF] Brezis | Coron -Convergence of solutions of H-systems or how to blow bubbles[END_REF][START_REF] Cerami | Multiplicity of multiple interior peaks solutions for some singularly perturbed Neumann problems[END_REF], yield (C.9).

The other quantities in (3.13), with i = j, may be estimated in the same way. Similar computations also yield

Ω |V i | 24/5 |V j | 6/5 5/6 = O( µ 1/2 λ 1/2 i λ 1/2 j
).

(C.16) Indeed, we have

Ω |V i | 24/5 ≤ 4π λ 3/5 i 2R ′ λi 0 1 (1 + r 2 ) 1/2 - 1 -e -(µ 1/2 /λi)r r 24/5
r 2 dr and the integral on the right hand side goes to a finite limit as µ 1/2 /λ i goes to zero, whence

Ω |V i | 24/5 5/6 = O( 1
λ 1/2 i )
and, taking account of (C.12)

Ω\Bj |V i | 24/5 |V j | 6/5 5/6 = O( µ 1/2 λ 1/2 i λ 1/2 j
).

(C.17)

On the other hand ) and (C.12) applied to V i in B j provides us with

Bj |V j | 6/5 ≤ 4π λ 5/2 j γλ j 4µ 1/2 0 1 (1 + r 2 ) 1/2 - 1 -e -(µ 1/2 /λj )r r 6
Bj |V i | 24/5 |V j | 6/5 5/6 = O( µ 1/2 λ 1/2 i λ 1/2 j • µ 3/4 λ 3/2 i
).

(C.18) (C.17-18) prove (C.16).

We state now :

In view of (C.1-3), we have

α i Ω |∇V i | 2 + µα i Ω V 2 i -α 5 i Ω V 6 i = π 2 4 α i (3 -α 4 i ) + O( µ 1/2 λ i ).
We recall that, according to (C.9), we have also

Ω ∇V i .∇V j = O( µ 1/2 λ 1/2 i λ 1/2 j
).

Similar computations show that

Ω V 5 i V j = O( µ 1/2 λ 1/2 i λ 1/2 j )
and

Ω ( j α j V j ) 5 V i - j α 5 j V 5 j V i = o( µ 1/2 λ 1/2 i |λ| 1/2
).

Lastly, from (C.12,15) we deduce that

µ Ω V i V j = O( µ 1/2 λ 1/2 i λ 1/2 j
).

Collecting thse results, (C.19) is established.

(C. [START_REF] Gui | Multi-peak solutions for a semilinear Neumann problem[END_REF][START_REF] Gui | Ghoussoub -Multi-peak solutions for a Neumann problem involving the critical Sobolev exponent[END_REF] follow from the same kind of integral estimates. The only result which requires to be more careful is (C.20). From (3.3-4), we have

1 α i ∂K µ ∂λ i (α, λ, a, v) = Ω ∇( j α j V j ).∇ ∂V i ∂λ i + µ Ω ( j α j V j + v) ∂V i ∂λ i - Ω ( j α j V j + v) 5 ∂V i ∂λ i
for (α, λ, a, v) ∈ N µ . The same arguments as in the proof of Lemma 3.1 in [START_REF] Rey | An elliptic Neumann problem with critical nonlinearity in three dimensional domains[END_REF] show that ).

µ Ω ∂V i ∂λ i v -α 4 i Ω V 4 i ∂V i ∂λ i v ≤ C µ 1/2 λ 3/2 i 1 |λ| 1/2 + 1 µ 3/4 Ω (|∇v| 2 + µv 2 ) 1/2 . We have also, if i = j Ω V 4 i ∂V i ∂λ i v ≤ C v
Lastly, using Hölder inequality, Sobolev embedding theorem and integral estimates as the previous one, we see that the terms in which v occurs with an exponent larger or equal to 2 have a contribution to the integral which is dominated by v 2 H 1 (Ω) /λ i . Therefore, similarly to (C.23) we obtain, using (3.12)

1 α i ∂K µ ∂α i (α, λ, a, vµ ) = 1 α i ∂K µ ∂α i (α, λ, a, 0) + O µ 1/2 λ 3/2 i |λ| 3/2 + 1 µ 1/2 λ 3/2 i |λ| 1/2
for any (α, λ, a) ∈ ѵ . We notice that, according to (C.4-6)

Ω ∇V i .∇ ∂V i ∂λ i + µ Ω V i ∂V i ∂λ i -α 4 i Ω V 5 i ∂V i ∂λ i = π(2 - 4 3 α 4 ) µ 1/2 λ 2 i + O 1 λ 2 i µ 1/2 l |3 -α 4 l | .
In addition, using (2.15), we have

Ω ∇(α j V j ).∇ ∂V i ∂λ i + µ Ω α j V j ∂V i ∂λ i - Ω (α j V j ) 5 ∂V i ∂λ i = α j ∂Ω ∂V j ∂ν ∂V i ∂λ i + Ω 3U 5 j -α 4 j V 5 j + µ(U j - 1 λ 1/2 j |x -a j | ) ∂V i ∂λ i .
As a i and a j do not approach the boundary of Ω, the definition of V i , V j shows that ∂V j ∂ν = o( µ ).

On the other hand, similarly to (C.2), we have

Ω ∂V i ∂λ i 2 = O( 1 λ 3 i µ 1/2
) and ).

Ω U j - 1 λ 1/2 j |x -a j | 2 ≤ 4π λ 2 j 2R ′ λj 0 1 (1 + r 2 ) 1/2 -
Lastly, we write Ω (3U 5 jα 4 j V 5 j )

∂V i ∂λ i = 3

Ω (U 5 j -V 5 j )

∂V i ∂λ i + (3 -α 4 j ) Ω V 5 j ∂V i ∂λ i .
Proceeding as in the proof of (C.10), we find

Ω V 5 j ∂V i ∂λ i = O( µ 1/2 λ 1/2 i λ 1/2 j
).

Concerning the last integral that we have to estimate, setting as previously B l = B(a l , γ/4µ ).

As we have also

∂V i ∂λ i = O( µ 1/2 λ 3/2 j
) outside of B i U 5 j -V

  [START_REF] Cerami | Bifurcation and multiplicity results for nonlinear elliptic problems involving critical Sobolev exponents[END_REF][START_REF] Cerami | Multiplicity of multiple interior peaks solutions for some singularly perturbed Neumann problems[END_REF] and estimate (C.16) in Appendix show that (3.10) is satisfied.

3. 3

 3 The α-derivative of K µ For (α, λ, a) ∈ ѵ , we set Kµ (α, λ, a) = K µ (α, λ, a, vµ (α, λ, a)).

5 γ/ 4 0t 4 / 5 e

 5445 5 e -(6µ 1/2 /5λj )r dr = ( λ j µ 1/2 ) 9/-6t/5 dt = O( λ

  in the proof of (C.16), we findΩ |V j | 24/5 ∂V i ∂λ i

  .20) is proved.

  ,ai | 2 is achieved at only one point, which lies in B 2ε,µ . Such a lemma is proved, for k = 1, in [40, App. A]. The result extends easily to the case k > 1, proceeding as in [8, Prop. 7][38, App. A]. Then, for λ ∈ (R * + ) k and a ∈ Ω k , we set

  1/2 ), we have, similarly to (C.15)

	Bi	∂V i ∂λ i	= O	ln(λ i /µ 1/2 ) 7/2 λ i

APPENDIX

A Proof of Proposition 2.1

The proof relies on the following lemma : Lemma A.1 Let (u µ ) µ≥µ0 be a sequence of nonconstant solutions to (P µ ). Let ε > 0, R > 1. There exists a constant C 0 = C 0 (R) such that, for µ large enough and any compact set

Proof of Proposition 2.1. Lemma A.1, applied with K = ∅, provides us with x µ = x 1 µ such that (i) (ii) (iii) of Proposition 3.1 are satisfied. Then, we set

.

If max x∈ Ω\K d(x, K) 1/2 u µ (x) < C 0 , there is nothing more to prove. If not, Lemma A.1 provides us with a new x µ = x 2 µ such that (i) (ii) (iii) of Proposition 2.1 are again satisfied, and B(x

µ ) . The process must stop after a finite number N µ of steps since, because of (ii)

with τ a constant which does not depend on i, µ. (Note that if (u µ ) is assumed to be bounded in H 1 (Ω), N µ is also bounded as µ goes to infinity.) We have max

Therefore, (v) is satisfied and the proof of Proposition 2.1 is complete.

Proof of Lemma A.1. Arguing by contradiction, we assume that there exist a sequence (u µ ) of nonconstant solutions (P µ ) and a sequence (K µ ) of compact sets such that

and there is no x µ , as specified in the lemma. Let xµ

By assumption, R µ → ∞ as µ → ∞. Since

On the other hand, v µ satisfies

Moreover, if S < ∞, necessarily x 0 ∈ ∂R 3 S . From the shape of v and the convergence of v µ to v in C 2 loc (R 3 S ), we deduce the existence, for large µ, of a local maximum point z µ of v µ , which goes to x 0 as µ goes to infinity. Then

+ xµ is a local maximum point of u µ , and

→ ∞, we know that for large µ, x µ ∈ Ω \ K µ , and d(x µ , K µ )u 2 µ (x µ ) → ∞ as µ goes to infinity. Then, we can repeat the same argument as previously, with x µ instead of xµ . In this case 0, local maximum point of v µ , is a critical point of v. As the only critical point of v is x 0 , we obtain x 0 = 0, whence S = 0 or S = ∞, i.e.

for any R > 0, with Ω µ = u 2 µ (x µ )(Ωx µ ). Actually, (A.1) holds in C 2 (B(0, 2R) ∩ Ω µ ). Indeed, if S = 0, up to a subsequence x µ goes to some x ∈ ∂Ω as µ goes to infinity. Up to a space rotation, we may assume that the tangent space to ∂Ω at x is parallel to the plane x 3 = 0. There exist U neigbourhood of x in R 3 , r > 0 and Ψ :

Then, for any r ′ , 0 < r ′ < r

vµ , continued by reflection in B(0, r ′ /2), satisfies in B(0, r ′ /2) the same equation. The same arguments as previously show that the continuation of vµ converges in

Finally, we show that if S = 0, x µ ∈ ∂Ω for µ large enough. Indeed, assume that x µ ∈ ∂Ω. Since x µ is a local maximum of v µ , y µ = Ψ(x µ ) is a local maximum of vµ , whence ∇v µ (y µ ) = 0. Let y ′ µ be the orthogonal projection of y µ on the plane x 3 = 0. We have

according to the shape of v, whence a contradiction.

To complete the proof of the lemma, it only remains to prove that

for large µ, in contradiction with the assumption that u µ is nonconstant. Let a µ be a point in the closure of

Up to a subsequence and a space rotation, ṽµ converges in C 2 loc (R 3 T ′ ), 0 ≤ T ′ ≤ ∞, to a limit ṽ which satisfies

As a consequence, ṽ ≡ 0 or ṽ ≡ 1. Therefore, |v µ (a µ )θ

0,Rµ/2)∩ Ωµ) goes to a limit l, l = 1 if ṽ ≡ 0, and l = 0 if ṽ ≡ 1. Let us assume that l = 1. In this case, v µ (a µ ) = ṽµ (0) goes to zero as µ goes to infinity. We also know that v µ (0) = 1, whence the existence, for

, and repeating the same argument as above, we see that up to a subsequence and a space rotation, ṽ′ µ converges in C 2 loc (R 3 T ′′ ), 0 ≤ T ′′ ≤ ∞, to a limit ṽ′ . As previously, ṽ′ has to be identically 0 or 1, in contradiction with ṽ′ (0) = 1/2. Therefore, l = 0.

Finally, let us assume that v µ ≡ θ

Therefore, up to a subsequence, ṽµ converges in C 2 loc (R 3 T ′ ) to a limit ṽ which satisfies

and ṽµ L ∞ (R 3 T " ) ≤ 1. It follows that ṽ ≡ 0, in contradiction with | ṽ(0)| = 1. Therefore, v µ is constant in B(0, R µ /2) ∩ Ωµ and, actually, in whole Ωµ . This means that u µ is constant in Ω, a contradiction.

B Proof of Proposition 2.5

Let u µ be a sequence, bounded in H 1 (Ω), of solutions to (P µ ). On one hand, we have

On the other hand, the continuous embedding of H 1 (Ω) into L 6 (Ω) yields

Therefore, u µ H 1 (Ω) ≥ C -3/2 , and |u µ | 6 6 ≥ C -3 . Passing to a subsequence, we may assume that |u µ | 6 6 → l > 0 as µ → ∞. We set

Q µ is continuous and increasing. Let R be such that Ω ⊂ x 0 + B(0, R), for some

for large µ there exist ε µ , 0 < ε µ < R, and a µ ∈ Ω such that

Up to some subsequence, one of the three cases occur :

We are going to prove that, actually, (iii) is the only possible case.

Let us assume, first, that µε 2 µ goes to infinity. (B.1) and the boundedness of

for some b ∈ R 3 . We have

where V is defined by (2.13-14). We notice that

and

(B.12) Indeed, a ∈ ∂R 3 means that a is limit of points of ∂Ω µ , so that we may write

We claim that

where

is the Sobolev constant, and -∆u (1) µ + µu (1) µ = (u (1)+ µ

as µ goes to infinity. If u

µ does not go to zero in H 1 (Ω), we can apply to u

µ the same arguments as those that we used concerning u µ . It is easily checked that the presence of f µ does not affect the situation; in the same way, (B.12) ensures that the normal derivative of u (1) µ on ∂Ω is sufficiently small for our purposes. In particular

Then, we obtain some u

µ which either goes to zero in H 1 (Ω), or may be treated as previously to define some u

µ , and so on. The process must stop after a finite number of steps since, according to (B.13), |∇u| 2 2 loses each time some fixed amount. In the end, we obtain that u µ writes as

Moreover, in view of (B.13), we have

As, on the other hand, we deduce from computations in [START_REF] Bahri | Critical points at infinity in some variational problems[END_REF] that

where C is a strictly positive constant, (B. [START_REF] Cerami | Bifurcation and multiplicity results for nonlinear elliptic problems involving critical Sobolev exponents[END_REF][START_REF] Cerami | Multiplicity of multiple interior peaks solutions for some singularly perturbed Neumann problems[END_REF] imply that the right hand side in (B.17) goes to zero if i = j, and the proof of Proposition 2.5 is complete.

Proof of (B.13). In view of (B.10), we have

with V µ = V µ,λ/εµ,aµ+εµa , for sake of simplicity. On one hand, it follows from (2.16) that

with U µ = U µ,λ/εµ,aµ+εµa , and standard computations yield

On the other hand, still using (2.16) and the boundedness of u µ in H 1 (Ω)

and

whence (B.13).

Proof of (B.14). From (B.10) and (2.15) we know that -∆u (1) µ + µu (1)

≤ Cµε 2 µ |ϕ| 2 as a direct computation shows. Secondly, setting ϕ µ = ϕ µ,λ/εµ,aµ+εµa , we have in view of (2. 16)

using Hölder inequality, the embedding of H 1 (Ω) in L 6 (Ω), the boundedness of (U µ ) in L 6 (Ω) and the fact that ϕ µ goes to zero in L 6 (Ω) as µε 2 µ goes to zero. Consequently, we are left to show that

with C µ independent of ϕ, C µ → 0 as µ → ∞. We proceed as in [START_REF] Cerami | Bifurcation and multiplicity results for nonlinear elliptic problems involving critical Sobolev exponents[END_REF]. Let I µ denote the left hand side integral in (B.18). Setting Ūµ = 3 1/4 U µ , we have

We know that ũµ H 1 (Ω) is bounded, and that ũµ ⇀ ũ in H 1 (K), ũµ → ũ in L q (Ω), 1 ≤ q < 6, for any compact set K ∈ R 3 . The conclusion follows easily.

Let us expand ( j α j V j + v)


Angela Pistoia

Olivier Rey

Multiplicity of solutions to the supercritical Bahri-Coron's problem in pierced domains *

We consider the supercritical Dirichlet problem

where N ≥ 3, ε > 0 and Ω ⊂ IR N is a smooth bounded domain with a small hole of radius d. When Ω has some symmetries, we show that (P ε ) has an arbitrary number of solutions for ε and d small enough.

When Ω has no symmetries, we prove the existence, for d small enough, of solutions blowing up at two or three points close to the hole as ε goes to zero.

Introduction

We consider the problem -∆u = u q in Ω, u > 0 in Ω, u = 0 on ∂Ω,

where N ≥ 3, q > 1 and Ω is a smooth and bounded domain in IR N . In the subcritical case, i.e. q < N +2 N -2 , problem (1.1) has a solution for any domain Ω. In the critical case, i.e. q = N +2 N -2 , Pohozaev's identity (see [START_REF] Pohozaev | On the eigenfunctions of the equation ∆u+λf (u) = 0[END_REF]) shows that (1.1) has no solution when Ω is starshaped, whereas Kazdan and ∆P U λ,ξ = ∆U λ,ξ in Ω, P U λ,ξ = 0 on ∂Ω.

These functions are, as λ goes to zero, approximate solutions to (1.1) in the critical case. Denoting (P ε ) problem (1.1) with q = N +2 N -2 + ε, we set:

Definition 1.1. Let k ∈ IN * and u ε be a solution to (P ε ). We say that (u ε ) ε blows up at k points ξ 1 , . . . , ξ k of Ω, ξ i = ξ j for i = j, as ε goes to zero, if and only if there exist positive parameters λ 1ε , . . . , λ kε and points ξ 1ε , . . . , ξ kε in Ω such that

u ε - k i=1 P U λ iε ,ξ iε → 0 as ε → 0 in H 1 0 (Ω) and C 1 Ω \ k ∪ i=1
B(ξ i , a) , for any a > 0.

In the slightly subcritical case, i.e. q = N +2 N -2 -ε, it was proved that such solutions do exist [START_REF] Bahri | On a variational problem with lack of compactness: the topological effect of the critical points at infinity[END_REF]. In particular it is always true, whatever Ω may be, for k = 1. (Some examples of domains with solutions blowing up at several points are given in [START_REF] Musso | Multispike solutions for a nonlinear elliptic problem involving critical Sobolev exponent[END_REF].) When q = N +2 N -2 + ε, the situation turns out to be different. In particular, solutions blowing up at a single point never exist [START_REF] Ben Ayed | A nonexistence result of single peaked solutions to a supercritical nonlinear problem[END_REF]. However, Del Pino, Felmer and Musso [START_REF] Del Pino | Two-bubble solutions in the super-critical Bahri-Coron's problem[END_REF] proved the existence of solutions blowing up at two points, provided that Ω satisfies some topological assumption -for example, Ω has a small hole. (See also [START_REF] Khenissy | A criterion for existence of solutions to the supercritical Bahri-Coron's problem[END_REF] for a simplified assumption when N = 3.) Existence of a large number of solutions to (P ε ) was found when Ω is a symmetric annulus [START_REF] Del Pino | Multi-bubble solutions for slightly super-critical elliptic problems in domains with symmetries[END_REF] or a symmetric annulus with a "channel" [START_REF] Molle | Positive solutions for slightly super-critical elliptic equations in contractible domains[END_REF] (making the domain contractible): for such domains Ω, there exists an integer k(Ω) such that for any k ≥ k(Ω), problem (P ε ) has, for ε small enough, a solution which blows up at k points as ε goes to zero. (We note that k may need to be chosen very large to obtain a k-peaked solution.) The aim of this paper is to resume the study of the problem in order to obtain existence and multiplicity results which improve the previous ones, both in the symmetric and the nonsymmetric cases.

In Section 2, we consider a symmetric domain with a small hole. Namely, writing x ∈ IR N as x = (x , x ), with x ∈ IR 2 and x ∈ IR N -2 , we assume that (i) Ω is rotationally invariant with respect to x , i.e. if (x , x ) ∈ Ω, then (Ax , x ) ∈ Ω for any central rotation A of the x -plane.

(ii) Ω is symmetric with respect to x j , 3 ≤ j ≤ N, i.e., for any j = 3, . . . , N , (x , x 3 , . . . , x j , . . . , x N ) ∈ Ω implies that (x , x 3 , . . . , -x j , . . . , x N ) ∈ Ω.

(iii) Ω contains a ball centered in zero, i.e. there exists a > 0 such that B(0, a) ⊂ Ω.

For δ ∈ (0, a), we set

Ω δ = x ∈ Ω : |x| > δ
and we consider (P ε ) with Ω = Ω δ . We prove:

Theorem 1.1. Let Ω = Ω δ . For any k ≥ 2, there exists δ k > 0 such that for any δ ∈ (0, δ k ), (P ε ) has, for ε small enough, a solution which blows up at k points in Ω δ as ε goes to zero.

Corollary 1.1. Let Ω = Ω δ . For any h ≥ 1, there exists δ h > 0 such that for any δ ∈ (0, δ h ), (P ε ) has, for ε small enough, h (rotationally) distinct solutions which blow up at two, three, . . . , h + 1 points respectively as ε goes to zero.

In the critical case, a small hole generates the existence of a solution to (1.1) -see [START_REF] Coron | Topologie et cas limite des injections de Sobolev[END_REF], [START_REF] Rey | Sur un probéme variationnel non compact: l'effet de petits trous dans le domain[END_REF]. As such a solution is topologically nontrivial, i.e. induces some difference of topology between the level sets of a functional associated to the problem, this solution continues for ε small enough. This provides us with an additional solution to (P ε ) on Ω δ , which goes to a solution to (P 0 ) as ε goes to zero.

In Section 3, we consider a domain with both a small hole and a small "channel", making it contractible. Namely, for 0 < δ < R and σ > 0 we set

Ω δ,σ = x ∈ IR N : δ < |x| < R, N -1 i=1 x 2 i 1/2 > σx N
and similarly to Theorem 1.1 we prove: Theorem 1.2. Let Ω = Ω δ,σ . For any k ≥ 2, there exist δ k > 0, σ k > 0, such that for any δ ∈ (0, δ k ) and σ ∈ (0, σ k ), (P ε ) has, for ε small enough, a solution which blows up at k points in Ω δ,σ as ε goes to zero.

Corollary 1.2. Let Ω = Ω δ,σ . For any h ≥ 1, there exist δ h > 0, σ h > 0, such that for any δ ∈ (0, δ h ) and σ ∈ (0, σ h ), (P ε ) has, for ε small enough, h (rotationally) distinct solutions which blow up at two, three, . . . , h + 1 points respectively as ε goes to zero.

In Section 4, we drop any symmetry assumption and just consider a domain with a small hole. Namely, Ω being a smooth bounded domain in IR N , we may assume, up to a translation, that Ω contains some ball B(0, r), r > 0. Then, for d ∈ (0, r), we set

Ω d = {x ∈ Ω : |x| > d}
and we first look for a solution blowing up at two points as ε goes to zero. As already stated, existence of such a solution has been proved in [START_REF] Del Pino | Two-bubble solutions in the super-critical Bahri-Coron's problem[END_REF]. We obtain additional informations: Theorem 1.3. Let Ω = Ω d . There exists d 0 > 0 such that for any d ∈ (0, d 0 ), (P ε ) has, for ε small enough, a solution which blows up at two points ξ 1 and ξ 2 in Ω d as ε goes to zero. Moreover |ξ i | ∼ r 0 d and ξ 2 = -ξ 1 + o(d) as ε goes to zero, where r 0 > 1 is the unique solution in (1, ∞) of

1 2 N -1 r N 0 = 1 (r 2 0 -1) N -1 + 1 (r 2 0 + 1) N -1 .
We also look for solutions blowing up at three points, and we obtain a result of the same kind -see Theorem 4.4. Such a result is of qualitative importance, as it shows that the two peaks solutions are not the only blowing up solutions existing. Whereas single peak solutions do not exist, solutions with three peaks or more may be exhibited, without any symmetry assumptions on the domain (the method that we use extends to the study of solutions blowing up at k points, for any k ≥ 2). Such kind of results was expected, but had never been proved because of technical obstacles.

Actually, using a rescaling, we obtain that the nonsymmetric case is asymptotically equivalent, as the hole radius goes to zero, to the symmetric one. Then, we are left with proving that peaked solutions in the symmetric case are stable through C 1 -perturbations. We study this general stability, and check that it is satisfied in the cases we are interested in.

Before turning to the proofs of the results, we introduce some notation. A domain Ω being given, we denote by G Ω the Green's function of the negative Laplacian in Ω with Dirichlet boundary conditions, i.e.

-∆G Ω (•, ξ) = a N δ ξ in Ω, G Ω (•, ξ) = 0 in ∂Ω with a N = (N -2)meas(S N -1 )
-1 , S N -1 being the (N -1)-dimensional unit sphere, and by H Ω its regular part, i.e.

H Ω (x, ξ) = 1 |x -ξ| N -2 -G Ω (x, ξ), (x, ξ) ∈ Ω × Ω.
We also define the Robin function as

R Ω (x) = H(x, x), x ∈ Ω.
We point out that, as a consequence of the maximum principle

P Ω U λ,ξ (x) = U λ,ξ (x) -α N λ N -2 2 H Ω (x, ξ) + O λ N +2 2 (dist (ξ, ∂Ω)) -N .
Such an expansion introduces us to the role that Green's function and its regular part play in that kind of problems (see e.g. [START_REF] Bahri | Critical point at infinity in some variational problems[END_REF], [START_REF] Rey | The role of the Green's function in a nonlinear elliptic equation involving the critical Sobolev exponent[END_REF]), and in the following arguments.

2 Proof of Theorem 1.1

According to assumptions (i), (ii), (iii) in the previous section, let k ≥ 2 be fixed and R > 0 such that (x , 0 ) ∈ Ω δ if and only if δ < |x | < R.

Step 1 -Reduction of the problem to a finite dimensional one. Problem (1.1) has a variational structure. Indeed, defining the functional

J(u) = 1 2 Ω |∇u| 2 - 1 q + 1 Ω (u + ) q+1 , u ∈ H 1 0 (Ω) ∩ L q+1 (Ω)
the strong maximum principle ensures that the critical points u ≡ 0 of J are solutions of the problem. The method initially developed in [START_REF] Bahri | Critical point at infinity in some variational problems[END_REF], [START_REF] Rey | The role of the Green's function in a nonlinear elliptic equation involving the critical Sobolev exponent[END_REF] for q = N +2 N -2 , consists in looking for solutions written as

u ε = α N k i=1 P Ω U λ i ,ξ i + v ε
where v ε is orthogonal for the H 1 0 -scalar product to the derivatives of P Ω U λ i ,ξ i with respect to the parameters λ i , ξ i , and assumed to be small in

H 1 0 -norm. (It is shown that λ = (λ 1 , . . . , λ k ), ξ = (ξ 1 , . . . , ξ k ) and v ε build a good local parametrization of H 1 0 (Ω).) Then, once v ε (λ, ξ) is found such that ∂J ∂v λ, ξ, v ε (λ, ξ) = 0
, the initial problem reduces to a finite dimensional one involving λ and ξ parameters only. Such a method was adapted in [START_REF] Del Pino | Two-bubble solutions in the super-critical Bahri-Coron's problem[END_REF], using weighted Hölder spaces, to the supercritical case q = N +2 N -2 + ε. According to [START_REF] Del Pino | Two-bubble solutions in the super-critical Bahri-Coron's problem[END_REF] the problem reduces, for ε small enough, to finding critical points of a finite dimensional functional written as

J(Λ, ξ) = 1 2 k i=1 R Ω (ξ i )Λ 2 i - k i,j=1 i =j G Ω (ξ i , ξ j )Λ i Λ j + k i=1 ln Λ i + ϕ ε (Λ, ξ) where λ = c N ε 1 N -2 Λ (c N
is a positive constant depending on N only), and

ϕ ε goes to zero in C 1 loc (IR * + ) k × (Ω k δ \ D) as ε goes to zero, with D = ξ ∈ (IR N ) k : ξ i = ξ j for some i = j . As a consequence, defining Ψ(Λ, ξ) = 1 2 k i=1 R Ω (ξ i )Λ 2 i - k i,j=1 i =j G Ω (ξ i , ξ j )Λ i Λ j + k i=1 ln Λ i
stable critical points of Ψ (i.e. critical points which are stable under small C 1 -perturbations) such that ξ i = ξ j for i = j, provide us with solutions to (P ε ) . Now, taking also into account the symmetries of the domain Ω δ , we can look, as in [START_REF] Del Pino | Multi-bubble solutions for slightly super-critical elliptic problems in domains with symmetries[END_REF], [START_REF] Molle | Positive solutions for slightly super-critical elliptic equations in contractible domains[END_REF], for critical points (Λ, ξ) of Ψ such that

Λ i = Λ, ξ i (ρ) = ρ cos 2π(i -1) k , ρ sin 2π(i -1) k , 0 , 1 ≤ i ≤ k (2.2)
with ρ ∈ (δ, R). Then, we are left with the two variables function ψ δ :

IR * + ×(δ, R) → IR defined as ψ δ (Λ, ρ) = k Λ 2 2 γ δ (ρ) + ln Λ with γ δ (ρ) = R Ω δ (ξ 1 (ρ)) - k i=2 G Ω δ (ξ 1 (ρ), ξ i (ρ)).
Step 2 -For δ small enough, ψ δ has a stable critical point. This will conclude the proof of Theorem 1.1. We perform the change of variables

r = ρ δ , µ = Λ δ N -2 2
and we define the function

φ δ : IR * + ×(1, R/δ) → IR as φ δ (µ, r) = ψ δ (Λ, ρ) -k N -2 2 ln δ = k µ 2 2 Γ δ (r) + ln µ with Γ δ (r) = δ N -2 γ δ (δr) = R Ω δ /δ (ξ 1 (r)) - k i=2 G Ω δ /δ (ξ 1 (r), ξ i (r)).
Obviously, (µ, r) is a stable critical point of φ δ if and only if (δ

N -2 2 µ, δr) is a stable critical point of ψ δ . Setting E = x ∈ IR N : |x| > 1
we remark that Ω δ /δ ⊂ E and Ω δ /δ goes to E as δ goes to zero, i.e. for any compact set K ⊂ E, there exists δ K > 0 such that for any δ ∈ (0, δ K ), K ⊂ Ω δ /δ. Moreover, it is easily checked, using the maximum principle, that

R Ω δ /δ (x) = H Ω δ /δ (x, x) goes to R E (x) = H E (x, x) in C 1 loc (E) and G Ω δ /δ (x, y) goes to G E (x, y) in C 1 loc (E 2 \ ∆) as δ goes to zero, where ∆ is the diagonal of E 2 , i.e. ∆ = {(x, y) ∈ E 2 : x = y}. Therefore, φ δ goes to φ E in C 1 loc (IR * + ×(1, ∞)), where φ E : IR * + ×(1, ∞) → IR is defined as φ E (µ, r) = k µ 2 2 Γ E (r) + ln µ with Γ E (r) = R E (ξ 1 (r)) - k i=2 G E (ξ 1 (r), ξ i (r)).
Then, the only thing that remains to be proved is that φ E has a stable critical point. We have

R E (ξ) = 1 (|ξ| 2 -1) N -2 , G E (ξ, ζ) = 1 |ξ -ζ| N -2 - 1 |ζ|ξ -ζ |ζ| N -2 . (2.3) Consequently Γ E (r) = 1 (r 2 -1) N -2 - k-1 i=1 1 2 N -2 2 r N -2 1 -cos 2πi k N -2 2 + k-1 i=1 1 r 4 + 1 -2r 2 cos 2πi k N -2 2
.

(2.4)

Clearly, lim

r→1 + Γ E (r) = +∞ and lim r→+∞ r N -2 Γ E (r) = l < 0. Moreover, the cancellation of Γ E (r) implies that Γ E (r) < 0. Indeed Γ E (r) = -(N -2)   2r (r 2 -1) N -1 - k-1 i=1 1 2 N -2 2 r N -1 1 -cos 2πi k N -2 2 + k-1 i=1 2r r 2 -cos 2π k i r 4 + 1 -2r 2 cos 2πi k N 2   (2.5) whence, in view of (2.4), if Γ E (r) = 0 Γ E (r) = -(N -2)   r 2 + 1 r(r 2 -1) N -1 + k-1 i=1 r 4 -1 r r 4 + 1 -2r 2 cos 2πi k N 2   < 0.
Then, there exists a unique r * ∈ (1, ∞) such that Γ E (r * ) = 0. Γ E (r) < 0 for r > r * , and there is some r 0 > r * which is a global minimum of φ E in (1, ∞) and a isolated critical point because of analyticity. On the other hand, for any r > r * , there exists a unique µ = µ(r) = (-Γ E (r)) -1/2 such that ∂φ E ∂µ (µ(r), r) = 0. Moreover, ∂ 2 φ E ∂µ∂r (µ(r), r) = 0 and ∂ 2 φ E ∂µ 2 (µ(r), r) = 2kΓ E (r) < 0. Therefore, a standard linking argument shows that (µ(r 0 ), r 0 ) is a critical point of φ E which is stable under C 1 -perturbations. This concludes the proof of Theorem 1.1.

3 Proof of Theorem 1.2 Let k ≥ 2 be fixed and R > 0 such that (x , 0 ) ∈ Ω δ,σ if and only if δ < |x | < R.
Step 1 -Reduction of the problem to a finite dimensional one.

We argue exactly as in the previous section. The only change is that the symmetry with respect to x N being lost, we have to set, instead of (2.2)

ξ(ρ, τ ) = ρ cos 2π(i -1) k , ρ sin 2π(i -1) k , 0, . . . , 0, τ , 1 ≤ i ≤ k (3.6) with (ρ, τ ) ∈ S δ,σ = (ρ, τ ) ∈ IR * + × IR : δ 2 < ρ 2 + τ 2 < R, ρ > στ .
Then we have to consider, instead of ψ δ , a three variables function ψ δ,σ : IR * + ×S δ,σ → IR defined as

ψ δ,σ (Λ, ρ, τ ) = k Λ 2 2 γ δ,σ (ρ, τ ) + ln Λ and γ δ,σ (ρ, τ ) = R Ω δ,σ (ξ 1 (ρ, τ )) - k i=2 G Ω δ,σ (ξ 1 (ρ, τ ), ξ i (ρ, τ )).
Step 2 -For δ and σ small enough, ψ δ,σ has a stable critical point. We perform, as previously, a change of variables

r = ρ δ , t = τ δ , µ = Λ δ N -2 2
and we define the function φ δ,σ : Σ δ,σ → IR as

φ δ,σ (µ, r, t) = ψ δ,σ (Λ, ρ, τ ) -k N -2 2 ln δ = k µ 2 2 Γ δ,σ (r, t) + ln µ with Γ δ,σ (r, t) = δ N -2 γ δ,σ (δr, δt), i.e. Γ δ,σ (r, t) = R Ω δ,σ /δ (ξ 1 (r, t)) - k i=2 G Ω δ,σ /δ (ξ 1 (r, t), ξ i (r, t))
and

Σ δ,σ = (r, t) ∈ IR * + × IR : 1 < r 2 + t 2 < R δ 2 , r > σt .
Of course (µ, r, t) is a stable critical point of φ δ,σ if and only if δ

N -2 2 µ, δr, δt is a stable critical point of ψ δ,σ . Setting E * = E \ {(0, . . . , 0, x N ) : x N > 1} , E = x ∈ IR N : |x| > 1
we remark that Ω δ,σ /δ ⊂ E * and Ω δ,σ /δ goes to E * as δ and σ go to zero, i.e. for any compact set K ⊂ E * , there exist δ K > 0, σ K > 0 such that for any δ ∈ (0, δ K ) and σ ∈ (0, σ K ), K ⊂ Ω δ,σ /δ. Moreover, it is easily checked, using the maximum principle, that H Ω δ,σ /δ (x, x) goes to H E (x, x) in C 1 loc (E * ) and G Ω δ,σ /δ (x, y) goes to G E (x, y) in C 1 loc ((E * ) 2 \ ∆) as δ and σ go to zero, where ∆ is the diagonal of (E * ) 2 . Consequently, φ δ,σ goes to

φ E * in C 1 loc (IR * + ×Σ 0 ), where φ E * : IR * + ×Σ 0 → IR is defined as φ E * (µ, r, t) = k µ 2 2 Γ E * (r, t) + ln µ with Γ E * (r, t) = R E (ξ 1 (r, t)) - k i=2 G E (ξ 1 (r, t), ξ i (r, t)) (3.7) and Σ 0 = (r, t) ∈ IR * + × IR : 1 < r 2 + t 2 \ {(0, t) : t > 1}.
In view of the previous arguments, Theorem 1.2 will follow from the existence of a stable critical point of φ E * . From (2.3), (3.6) and (3.7), we deduce that

Γ E * (r, t) = 1 (r 2 + t 2 -1) N -2 - k-1 i=1 1 2 N -2 2 r N -2 1 -cos 2πi k N -2 2 + k-1 i=1 1 (r 2 + t 2 ) 2 + 1 -2r 2 cos 2πi k -2t 2 N -2 2
.

We remark that for any r > 1, there exists a unique t(r) such that ∂Γ E * ∂t (r, t(r)) = 0, i.e. t(r) = 0. Moreover, Γ E * (r, 0) = max t∈IR Γ E * (r, t), and straightforward computations yield ∂ 2 Γ E * ∂t 2 (r, 0) < 0, ∂ 2 Γ E * ∂t∂r (r, 0) = 0. We note that Γ E * (r, 0) coincides with Γ E (r) in the previous section. Then, let r 0 be a global minimum of Γ E * (r, 0) for r ∈ (1, ∞), and µ(r 0 ) = (-Γ E * (r 0 )) -1/2 . (µ(r 0 ), r 0 , 0) is a critical point of φ E * , and

φ E * (µ(r 0 ), r 0 , 0) = k    2Γ E (r 0 ) 0 0 0 (µ(r 0 )) 2 2 Γ E (r 0 ) 0 0 0 (µ(r 0 )) 2 2 ∂ 2 Γ E * ∂t 2 (r 0 , 0)    .
We know that Γ E (r 0 ) < 0, ∂ 2 Γ E * ∂t 2 (r 0 , 0) < 0, and Γ E (r 0 ) ≥ 0. Even if Γ E (r 0 ) = 0, the fact that r 0 is a minimum and an isolated critical point of Γ E (r) (and of r → φ E * (µ(r 0 ), r, 0) as well) would still ensure, through a standard linking argument, that (µ(r 0 ), r 0 , 0), as a critical point of φ E * , is stable under C 1 -perturbations. This concludes the proof of Theorem 1.2.

The nonsymmetric case

The general argument.

In this last section, we drop the symmetry assumptions on the domain. Ω being any smooth bounded domain in IR N , that we may assume, up to a translation, contains a ball B(0, R) for some R > 0, we set, for 0 < d < R

Ω d = {x ∈ Ω : |x| > d} i.e. Ω d = Ω \ B(0, d).
We consider (P ε ) in Ω d and, as previously, we look for a solution blowing up at k points as ε goes to zero, k ≥ 2. According to Section 1, we know that the problem may be reduced to finding, for ε small enough, a critical point of

J(Λ, ξ) = 1 2 k i=1 R Ω d (ξ i 2 i - k i,j=1 i =j G Ω d (ξ i , ξ j )Λ i Λ j + k i=1 ln Λ i + ϕ Ω d ,ε (Λ, ξ) in Σ = (IR * + ) k × (Ω k d \ D), with D = {ξ ∈ (IR N ) k : ξ i = ξ j for some i = j}, ϕ Ω d ,ε (Λ, ξ) going to zero in C 1 loc (Σ).
We consider, as previously, the main part in J(Λ, ξ), that is, we set

ψ(Λ, ξ) = 1 2 k i=1 R Ω d (ξ i )Λ 2 i - k i,j=1 i =j G Ω d (ξ i , ξ j )Λ i Λ j + k i=1 ln Λ i
for (Λ, ξ) ∈ Σ. Through the rescaling

x i = ξ i d , µ i = Λ i d N -2 2 we have ψ(Λ, ξ) = φ(µ, x) + k N -2 2 ln d with φ(µ, x) = 1 2 k i=1 R Ω d /d (x i )µ 2 i - k i,j=1 i =j G Ω d /d (x i , x j )µ i µ j + k i=1 ln µ i as (µ, x) ∈ (IR * + ) k × ((Ω d /d) k \ D). We know that Ω d /d goes to E = {x ∈ IR N : |x| > 1}, H Ω d /d goes to H E in C 1 loc (E) and G Ω d /d goes to G E in C 1
loc (E 2 \ ∆) as d goes to zero (we recall that ∆ is the diagonal of E 2 ). Let us consider the limit function

φ E (µ, x) = 1 2 k i=1 R E (x i )µ 2 i - k i,j=1 i =j G E (x i , x j )µ i µ j + k i=1 ln µ i . (4.8)
As we saw, this function has a critical point (µ * , x * ) which may be written as

µ * i = (-Γ E (r 0 )) -1/2 x * i = r 0 cos 2π(i -1) k , r 0 sin 2π(i -1) k , 0, . . . , 0 , 1 ≤ i ≤ k (4.9)
where r 0 is a global minimizer of Γ E (r) in (1, ∞). Of course, this critical point is degenerate, since φ E is invariant under central rotations of E k . Let M be the manifold of critical points generated by these rotations, i.e.

M = {(µ * , Rx * ) : R is a central rotation of E k }.
Let us assume that φ E (µ * , x * ) is nondegenerate in the orthogonal subspace to T (µ * ,x * ) M (the same is true at any (µ, x) ∈ M). Then, a standard linking argument shows that a C 1 -perturbation of φ E has a critical point in a neighborhood of M. Whence, coming back to J, the existence of d 0 > 0 such that for any 0 < d < d 0 , there exists some ε d > 0 such that, for any 0

< ε < ε d , J has a critical point (Λ, ξ) = d N -2
2 µ, dx , with the distance of (µ, x) to M going to zero as d and ε go to zero.

We remark that dim M = N -1 as k = 2 (x * = (x * 1 , -x * 1 ) and x * 1 runs on a (N -1)-sphere). As k ≥ 3, the position of Rx * is correctly defined by the orientation of the plane in which the x * i 's are, and by a central rotation in that plane. Therefore, for k ≥ 3, dim M = 2(N -2) + 1 = 2N -3 (2(N -2) is the dimension of the 2-Grassmanian in IR N ). Finally, we can state the following general result:

Proposition 4.1. Let k ∈ IN, k ≥ 2, φ E and (µ * , x * ) defined by (4.8) and (4.9). If dim Ker φ E (µ * , x * ) = N -1 as k = 2 dim Ker φ E (µ * , x * ) = 2N -3 as k ≥ 3,
there exists d 0 > 0 such that for any d ∈ (0, d 0 ), (P ε ) has, for ε small enough, a solution which blows up at k points ξ 1 , . . . , ξ k in Ω d as ε goes to zero. Moreover, up to a central rotation, ξ/d goes to x * as d and ε go to zero.

The case k = 2. In view of Proposition 4.1, we just have to check, for proving Theorem 1.3, that dim Ker φ E (µ * , x * ) = N -1. For the sake of simplicity, we write

x = x * 1 = -x * 2 , r = r 0 where r 0 > 1 solves Γ E (r 0 ) = 0, i.e. 1 2 N -1 r N 0 = 1 (r 2 0 -1) N -1 + 1 (r 2 0 + 1) N -1 . (4.10)
Such an r 0 is unique. Indeed, derivating (2.5) while Γ E (r) = 0, we obtain

Γ E (r) = 2(N -2) (N -2)r 2 + N (r 2 -1) N + (N -2)r 2 + N (r 2 + 1) N > 0
whence the unicity of the critical point in (1, ∞). We also write µ

= * = E (r 0 )) -1/2 , i.e. µ = 1 2 N r N -2 0 - 1 (r 2 0 -1) N -2 - 1 (r 2 0 + 1) N -2 -1/2 . (4.11)
Let (e 1 , e 2 , f 1 , . . . , f N , g 1 , . . . , g N ) be the dual basis of the 2N + 2 variables µ 1 , µ 2 , (x 1 ) 1 , . . . , (x 1 ) N , (x 2 ) 1 , . . . , (x 2 ) N . According to the definition of φ E we compute

φ E (µ * , x * ) =                   A B C • • • • • C • • • • • B A -C • • • • • -C • • • • • C -C D • • • • • E • • • • • • • • F • • • • • F • • • • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • • • • • • • F • • • • • F C -C E • • • • • D • • • • • • • • F • • • • • F • • • • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • • • • • • • F • • • • • F                   (4.12)
where horizontal or vertical dots mean zeros, diagonal dots mean F , and, taking account of (4.10) and (4.11)

A = R(x) - 1 µ 2 = - 1 2 N -2 r N -2 + 2 (r 2 -1) N -2 + 1 (r 2 + 1) N -2 B = -G(x, -x) = - 1 2 N -2 r N -2 + 1 (r 2 + 1) N -2 C = µ R f 1 (x) -G f 1 (x, -x) = -µG g 1 (x, -x) = µG f 1 (x, -x) = -µ R g 1 (-x) -G g 1 (x, -x) = -(N -2)µ r (r 2 -1) N -1 D = µ 2 2 R f 1 f 1 (x) -2G f 1 f 1 (x, -x) = µ 2 2 R g 1 g 1 (-x) -2G g 1 g 1 (x, -x) = N -2 2 µ 2 (3N -5)r 2 + (N + 1) (r 2 -1) N + (N -1)r 2 -(N -1) (r 2 + 1) N E = -µ 2 G f 1 g 1 (x, -x) = -µ 2 G f 1 g 1 (x, -x) = N -2 2 µ 2 N -1 (r 2 -1) N -1 + (-N + 3)r 2 + (N + 1) (r 2 + 1) N F = µ 2 2 R f i f i (x) -2G f i f i (x, -x) = -µ 2 G f i g i (x, -x) = -µ 2 G f i g i (x, -x) = µ 2 2 R g i g i (-x) -2G g i g i (x, -x) = - N -2 2 µ 2 1 (r 2 -1) N -1 + r 2 -1 (r 2 + 1) N , 2 ≤ i ≤ N.
In view of (2.3), we used

∂R E ∂ξ i (ξ) = -2(N -2) ξ i (|ξ| 2 -1) N -1 ∂G E ∂ξ i (ξ, ζ) = -(N -2)    ξ i -ζ i |ξ -ζ| N -1 - |ζ| 2 ξ i -ζ i |ζ|ξ -ζ |ζ| N    (4.13) ∂R 2 E ∂ξ i ∂ξ j = -2(N -2) δ ij (|ξ| 2 -1) N -1 -2(N -1) ξ i ξ j (|ξ| 2 -1) N ∂ 2 G E ∂ξ i ∂ξ j (ξ, ζ) = -(N -2) δ ij |ξ -ζ| N -N (ξ i -ζ i )(ξ j -ζ j ) |ξ -ζ| N +2 - |ζ| 2 δ ij |ζ|ξ -ζ |ζ| N + N (|ζ| 2 ξ i -ζ i )(|ζ| 2 ξ j -ζ j ) |ζ|ξ -ζ |ζ| N +2    (4.14)
and

∂ 2 G E ∂ξ i ∂ζ j (ξ, ζ) =(N -2) δ ij |ξ -ζ| N -N (ξ i -ζ i )(ξ j -ζ j ) |ξ -ζ| N +2 + 2ξ i ζ j -δ ij |ζ|ξ -ζ |ζ| N -N (|ζ| 2 ξ i -ζ i )(|ζ| 2 ξ j -ζ j ) |ζ|ξ -ζ |ζ| N +2    (4.15)
Considering (4.12), we find the 2N following eigenvectors for φ E (µ * , x * ): The case k ≥ 3.

1 √ 2 (f i -g i ), 2 ≤ i ≤ N, with eigenvalue 0 1 √ 2 (f i + g i ), 2 ≤ i ≤ N,
For the sake of simplicity, we limit to the case k = 3. The other cases may be treated in the same way, with additional computations. In view of Proposition 4.1, we have to compute the dimension of the kernel of φ E (µ * , x * ), where (µ * , x * ) is given by (4.9), where r 0 > 1 solves Γ E (r 0 ) = 0, i.e. 1

3

N -2 2 r N 0 = 1 (r 2 0 -1) N -1 + 2r 2 0 + 1 (r 4 0 + r 2 0 + 1) N 2 . (4.16)
As previously, such an r 0 is unique. Indeed, derivating (2.5) we find, as

Γ E (r) = 0 Γ E (r) = 2(N -2) (N -2)r 2 + N (r 2 -1) N + N (2r 2 + 1)(r 4 -1) (r 4 + r 2 + 1) N +2 2 > 0
whence again the unicity of the critical point in (1, ∞). We note also that, according to (2.5)

Γ E (2) < -(N -2) 1 3 N -1 - 1 3 N -2 2 2 N < 0 implying that r 0 > 2. Furthermore µ * = (-Γ E (r 0 )) -1/2 , i.e. µ * = 2 3 N -2 2 r N -2 0 - 1 (r 2 0 -1) N -2 - 2 (r 4 0 + r 2 0 + 1) N -2 2 -1/2 . (4.17)
For the sake of simplicity, we write x * = (x 1 , x 2 , x 3 ), r = r 0 and r = (r 4 0 +r 2 0 + 1) 1 4 . Let (e 1 , e 2 , e 3 , f 1 , . . . , f N , g 1 , . . . , g N , h 1 , . . . , h N ) be the dual basis of the 3N +3 variables µ 1 , µ 2 , µ 3 , (x 1 ) 1 , . . . , (x 1 ) N , (x 2 ) 1 , . . . , (x 2 ) N , (x 3 ) 1 , . . . , (x 3 ) N .

The subspaces E i = Span(f i , g i , h i ), 3 ≤ i ≤ N, are stable for φ E (µ * , x * ), and with multiplicity 1, and 0 with multiplicity 2 -whence 2(N -2) vectors in the kernel of φ E (µ * , x * ). Because of the symmetries of φ E , we know three additional eigenvectors:

φ E (µ * , x * ) E i = α   1 1 1 1 1 1 1 1 1   with α = µ * 2 2 R f i f i (x 1 ) -2G f i f i (x 1 , x 2 ) -2G f i f i (x 1 , x 3 ) = -µ * 2 G f i g i (x 1 , x 2 ) = -µ * 2 G f i h i (x 1 , x 3 ) = (N -2)µ * 2 2 3 N 2 r N - 1 (r 2 -1) N -1 -
u = 1 √ 3 (e 1 + e 2 + e 3 ), with eigenvalue Γ E (r) < 0; v = 1 √ 3 f 1 + - 1 2 g 1 + √ 3 2 g 2 + - 1 2 h 1 - √ 3 2 h 2 , with eigenvalue µ 2 2 Γ E (r) > 0; w = 1 √ 3 f 2 + - √ 3 2 g 1 - 1 2 g 2 + √ 3 2 h 1 - 1 2 h 2 ,
with eigenvalue 0.

Therefore, Proposition 4.1 will provide us with a solution to (P ε ) blowing up at three points as ε goes to zero, if the restriction of φ E (µ * , x * ) to the stable 6-dimensional orthogonal subspace E to the E i 's, 3 ≤ i ≤ N, and to u, v, w, is nondegenerate. As a basis of E, we can take

k 1 = √ 2 √ 3 e 1 - 1 2 e 2 - 1 2 e 3 k 2 = 1 √ 2 (e 2 -e 3 ) k 3 = 1 √ 3 (f 1 + g 1 + h 1 ) k 4 = 1 √ 3 (f 2 + g 2 + h 2 ) k 5 = 1 √ 3 f 1 + - 1 2 g 1 - √ 3 2 g 2 + - 1 2 h 1 + √ 3 2 h 2 k 6 = 1 √ 3 f 2 + √ 3 2 g 1 - 1 2 g 2 + - √ 3 2 h 1 - 1 2 h 2 .
In this basis we have

φ E (µ * , x * ) E =         A 0 B 0 C 0 0 A 0 B 0 -C B 0 D 0 E 0 0 B 0 D 0 -E C 0 E 0 F 0 0 -C 0 -E 0 F         (4.18) with A = 2 (r 2 -N -2 - 1 3 N -2 2 r N -2 + 1 r 2(N -2) B = - N -2 √ 2 µ * r 2 (r 2 -1) N -1 + r 2 -1 r 2N C = N -2 √ 2 µ * r 1 (r 2 -1) N -1 - r 2 -1 r 2N D = (N -2)µ * 2 (N -2)r 2 + 1 (r 2 -1) N - r 2 -2 r 2N + N r 2 (r 2 -1) 2 2r 2(N +2) E = (N -2)µ * 2 (N -1)r 2 (r 2 -1) N + r 2 r 2N + N r 2 (r 4 + r 2 -2) 2r 2(N +2) F = N -2 2 µ * 2 (N -2)r 2 + N (r 2 -1) N - 2(N -1)r 2 + N 2r 2N + N r 2 (r 2 + 2) 2 2r 2(N +2) . ( 4

.19)

For example

A = k t 1 .φ E (µ * , x * ).k 1 = 2 3 ∂ 2 φ E ∂µ 2 1 - 1 2 ∂ 2 φ E ∂µ 1 ∂µ 2 - 1 2
∂ 2 φ E ∂µ 1 ∂µ 3 - 1 2 ∂ 2 φ E ∂µ 1 ∂µ 2 - 1 2
∂ 2 φ E ∂µ 2 2 - 1 2
∂ 2 φ E ∂µ 2 ∂µ 3 - 1 2 ∂ 2 φ E ∂µ 1 ∂µ 3 - 1 2 ∂ 2 φ E ∂µ 2 ∂µ 3 - 1 2 ∂ 2 φ E ∂µ 2 3 .
We have

∂ 2 φ E ∂µ 2 1 = ∂ 2 φ E ∂µ 2 2 = ∂ 2 φ E ∂µ 2 3 = R(x 1 ) - 1 µ * 2 ∂ 2 φ E ∂µ 1 ∂µ 2 = ∂ 2 φ E ∂µ 1 ∂µ 3 = ∂ 2 φ E ∂µ 2 ∂µ 3 = -G(x 1 , x 2 )
whence the announced result, observing (2.3) and (4.9). The other quantities are obtained in the same way, using also (4.13), (4.14), (4.15) and (4.16).

All these quantities are strictly positive, except B which is strictly negative. Actually, B < 0 and E > 0 are trivial. C > 0, D > 0 and F > 0 follow directly from the fact that, according to the definition of r, r 2 -1 < r 2 . Lastly, we deduce from (4.16)

A = r 2 -2 (r 2 -1) N -1 - r 4 -1 r 2N .
Therefore, A > 0 is equivalent to (r 2 -2N > (r 2 + 1)(r 2 -1) N or r 4 + r 2 + 1 r 4 -2r 2 + 1

N 2 > r 2 + 1 r 2 -2 = 1 + 3 r 2 -2 . ( 4

.20)

As (1 + X) α > 1 + αX for X > 0 and α > 1

r 4 + r 2 + 1 r 4 -2r 2 + 1 N 2 > 1 + N 2 3r 2 r 4 -2r 2 + 1
and, in view of (4.20), A > 0 follows from the fact that (as r > 2)

N 2 1 r 2 -2 + 1 r 2 > 1 r 2 -2 or (N -2)(r 2 -2) > 2 r 2 .
We note now that through suitable exchanges between lines and columns, we deduce from (4. As C, D, E, F are strictly positive and B is strictly negative, the right hand side is strictly positive. As we have also A > 0, the inequality DF -E 2 < 0 is sufficient to conclude. Considering that D, F are strictly positive, the result will follow from the two inequalities D < E and F < E. In view of (4.19), D < E is obvious and F < E is equivalent to

N (r 2 -1) N -1 + 2N r 2 + N 2r 2N + N r 2 (r 4 -2r 2 -8) 2r 2(N +2) > 0
As r 2 -1 < r 2 , this inequality will be satisfied if (4r 2 -1)r 4 + r 2 (r 4 -2r 2 -8) = 5r 6 + r 4 -5r 2 -1 > 0 which is true since r > 1.

Collecting the previous informations, we know that the dimension of the kernel of φ E (µ * , x * ) is exactly 2N -3. Then, in view of Proposition 4.1 we can state:

1 - 1 √ 2 (e 1 -e 2 ), 1 √ 2 (f 1 + 1 and

 112121211 g 1 ), with eigenvalue D -E and in the remaining orthonormal basis g 1 ), φ E (µ * , x * ) writes as A -B 2C 2C D + E . It is easily checked that F < 0, A + B < 0 (because of (4.10)), D -E > 0 and (A -B)(D + E) -4C 2 > 0. Consequently dim Ker φ E (µ * , x * ) = N -Theorem 1.2 follows from Proposition 4.1.

2r 2 r

 2 2N < 0 using (4.14), (4.15) and (4.16). The eigenvalues φ E (µ * , x * ) E i are 3α

  [START_REF] Passaseo | Multiplicity of positive solutions of nonlinear elliptic equations with critical Sobolev exponent in some contractible domains[END_REF] detφ E (µ * , x * ) (µ * , x * ) E = A(DF -E 2 ) -(F B 2 + DC 2 -2EBC) 2where A, B, C, D, E, F are defined by (4.16) and(4.19). Then, the nondegeneracy of φ E (µ * , x * ) restricted to E is equivalent toA(DF -E 2 ) = F B 2 + DC 2 -2EBC.

Theorem 4 . 4 .

 44 There exists d 0 > 0 such that for any d ∈ (0, d 0 ), (P ε ) has, for ε small enough, a solution which blows up at three points ξ 1 , ξ 2 , ξ 3 of Ω d as ε goes to zero. Moreover,|ξ i | ∼ r 0 d and there is a rotation R in (IR N ) 3 , R 3 = id, such that ξ 2 = Rξ 1 + o(d) and ξ 3 = R 2 ξ 1 + o(d).

Acknoledgements. The authors are very grateful to the referee for pointing out several inaccuracies in the first version of the paper, the correction of which allowed to improve the formulation of the results.

M.U.