Olivier Rey
email: rey@math.polytechnique.fr

Juncheng Wei
email: wei@math.cuhk.edu.hk.

Blowing up Solutions for an Elliptic Neumann Problem with Sub-or Supercritical Nonlinearity Part I: N = 3

Blowing up solutions for an elliptic Neumann problem with sub-and supercritical nonlinearity. Part I: N=3

Introduction

In this paper we consider the nonlinear Neumann elliptic problem (P q,µ )

-∆u + µu = u q u > 0 in Ω ∂u ∂n = 0 on ∂Ω where 1 < q < +∞, µ > 0 and Ω is a smooth and bounded domain in R 3 .

Equation (P q,µ ) arises in many branches of the applied sciences. For example, it can be viewed as a steady-state equation for the shadow system of the Gierer-Meinhardt system in biological pattern formation ( [START_REF] Gierer | A theory of biological pattern formation[END_REF], [START_REF] Ni | Diffusion, cross-diffusion, and their spike-layer steady states[END_REF]) or of parabolic equations in chemotaxis, e.g. Keller-Segel model ( [START_REF] Lin | Large amplitude stationary solutions to a chemotaxis system[END_REF]). When q is subcritical, i.e. q < 5, Lin, Ni and Takagi proved that the only solution, for small µ, is the constant one, whereas nonconstant solutions appear for large µ [START_REF] Lin | Large amplitude stationary solutions to a chemotaxis system[END_REF] which blow up, as µ goes to infinity, at one or several points. The least energy solution blows up at a boundary point which maximizes the mean curvature of the frontier [START_REF] Ni | On the shape of least-energy solutions to a semi-linear problem Neumann problem[END_REF] [START_REF] Ni | Locating the peaks of least-energy solutions to a semi-linear Neumann problem[END_REF]. Higher energy solutions exist which blow up at one or several points, located on the boundary [START_REF] Dancer | Multipeak solutions for a singularly perturbed Neumann problem[END_REF] [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][33] [START_REF] Gui | Multiple boundary peak solutions for some singularly perturbed Neumann problems[END_REF], in the interior of the domain [START_REF] Bates | Equilibria with many nuclei for the Cahn-Hilliard equation[END_REF][7] [START_REF] Grossi | On the effect of critical points of distance function in superlinear elliptic problems[END_REF] [START_REF] Grossi | Existence of multipeak solutions for a semilinear elliptic problem via nonsmooth critical point theory[END_REF] [START_REF] Gui | Multiple interior peak solutions for some singularly perturbed Neumann problems[END_REF][21] [START_REF] Wei | On the interior spike layer solutions of singularly perturbed semilinear Neumann problems[END_REF] [START_REF] Yan | On the number of interior multipeak solutions for singularly perturbed Neumann problems[END_REF], or some of them on the boundary and others in the interior [START_REF] Gui | On multiple mixed interior and boundary peak solutions for some singularly perturbed Neumann problems[END_REF]. (A good review can be found in [START_REF] Ni | Diffusion, cross-diffusion, and their spike-layer steady states[END_REF].) In the critical case, i.e. q = 5, Zhu [START_REF] Zhu | Uniqueness results through a priori estimates, I. A three dimensional Neumann problem[END_REF] proved that, for convex domains, the only solution is the constant one for small µ (see also [START_REF] Wei | Uniqueness and a priori estimates for some nonlinear elliptic Neumann equations in R 3[END_REF]). For large µ, nonconstant solutions exist [START_REF] Adimurthi | The Neumann problem for elliptic equations with critical nonlinearity, "A tribute in honour of G. Prodi[END_REF] [START_REF] Wang | Neumann problem of semilinear elliptic equations involving critical Sobolev exponents[END_REF]. As in the subcritical case the least energy solution blows up, as µ goes to infinity, at a unique point which maximizes the mean curvature of the boundary [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] Ni | Singular behavior of least-energy solutions of a semilinear Neumann problem involving critical Sobolev exponents[END_REF]. Higher energy solutions have also been exhibited, blowing up at one [START_REF] Adimurthi | Geometry and topology of the boundary in the critical Neumann problem[END_REF] [START_REF] Wang | The effect of domain geometry on the number of positive solutions of Neumann problems with critical exponents[END_REF] [START_REF] Rey | An elliptic Neumann problem with critical nonlinearity in three dimensional domains[END_REF] [START_REF] Gui | Estimates for boundary-bubbling solutions to an elliptic Neumann problem[END_REF] or several boundary points [START_REF] Maier-Paape | On Neumann problems for semilinear elliptic equations with critical nonlinearity : existence and symmetry of multi-peaked solutions[END_REF][36] [START_REF] Wang | Construction of multi-peaked solution for a nonlinear Neumann problem with critical exponent[END_REF] [START_REF] Gui | On the existence of arbitrary number of bubbles for some semilinear elliptic equations with critical Sobolev exponent[END_REF]. The question of interior blow-up is still open. However, in contrast with the subcritical situation, at least one blow-up point has to lie on the boundary [START_REF] Rey | The question of interior blow-up points for an elliptic Neumann problem : the critical case[END_REF]. Very few is known about the supercritical case, save the uniqueness of the radial solution on a ball for small µ [START_REF] Lin | On the Diffusion Coefficient of a Semilinear Neumann Problem[END_REF].

Our aim, in this paper, is to study the problem for fixed µ, when the exponent q close to the critical one, i.e. q = 5 + ε and ε is a small nonzero number. Whereas the previous results, concerned with peaked solutions, always assume that µ goes to infinity, we are going to prove that a single peak solution may exist for finite µ, provided that q is close enough to the critical exponent. Such a solution blows up, as q goes to 5, at one point which may be characterized.

In order to state a precise result, some notations have to be introduced. Let G µ (x, y) denote the Green's function of the operator -∆ + µ in Ω with Neumann boundary conditions. Namely, for any y ∈ Ω, x → G µ (x, y) is the unique solution of

-∆G µ + µG µ = 4πδ y x ∈ Ω ; ∂G µ ∂n = 0 x ∈ ∂Ω. (1.1)
G µ writes as

G µ (x, y) = e -µ 1/2 |x-y| |x -y| -H µ (x, y)
where H µ (x, y), regular part of the Green's function, satisfies

-∆H µ + µH µ = O x ∈ Ω ; ∂H µ ∂n = 1 ∂n e -µ 1/2 |x-y| |x -y| x ∈ ∂Ω. (1.2)
We set

ϕ µ (x) = µ 1 2 + H µ (x, x).
It is to be noticed that

H µ (x, x) → -∞ as d(x, ∂Ω) → 0 (1.3) implying that M µ = sup x∈Ω ϕ µ (x)
is achieved in Ω. (See (5.10) in Proposition 5.2 for the proof of (1.3).) Denoting

f α = {x ∈ Ω, f (x) ≤ α}
the level sets of a function f defined in Ω, we have :

Theorem 1.1 Assume that there exist b and c, b < c < 0, such that c is not a critical value of ϕ µ and the relative homology H * (ϕ c µ , ϕ b µ ) = 0. (P 5+ε,µ ) has a nontrivial solution, for ε > 0 close enough to zero, which blows up as ε goes to zero at a point a ∈ Ω, such that b < ϕ µ (a) < c. The same result holds, for ε < 0, assuming that 0 < b < c.

We notice that, M µ < 0 (resp. > 0) when µ is small (resp. large) enough (see (5.12) and (5.13) of Proposition 5.2). Consequently, we deduce from the previous result : Theorem 1.2 There exist µ 0 and µ 1 , 0 < µ 0 ≤ µ 1 , such that : 1) If 0 < µ < µ 0 , (P 5+ε,µ ) has a nontrivial solution, for ε > 0 close enough to zero, which blows up as ε goes to zero at a maximum point a of H µ (a, a).

2) If µ > µ 1 , (P 5+ε,µ ) has a nontrivial solution, for ε < 0 close enough to zero, which blows up as ε goes to zero at a maximum point a of H µ (a, a).

Remarks. 1) In the critical case, i.e. ε = 0, further computations suggest that a nontrivial solution should exist for µ > µ 0 close enough to µ 0 , such that M µ > 0 and M µ0 = 0. This solution would blow up, as previously, at a maximum point of H µ0 (a, a) as µ goes to µ 0 . (This contrasts to previous results for P 5,0 on the nonexistence of solutions for µ small ( [START_REF] Wei | Uniqueness and a priori estimates for some nonlinear elliptic Neumann equations in R 3[END_REF], [START_REF] Zhu | Uniqueness results through a priori estimates, I. A three dimensional Neumann problem[END_REF]) and nonexistence of interior bubble solutions for µ large ( [START_REF] Ghoussoub | On a singularly perturbed Neumann problem with the critical exponent[END_REF], [START_REF] Rey | An elliptic Neumann problem with critical nonlinearity in three dimensional domains[END_REF]).)

2) In a forthcoming paper, we shall treat the case N ≥ 4, which appears to be qualitatively different.

The scheme of the proof is the following. In the next section, we define a two-parameters set of approximate solutions to the problem, and we look for a true solution in a neighborhood of this set. Considering in Section 3 the linearized problem at an approximate solution, and inverting it in suitable functional spaces, the problem reduces to a finite dimensional one, which is solved in Section 4. Some useful facts and computations are collected in Appendix.

Approximate solutions and rescaling

For sake of simplicity, we consider in the following the supercritical case, i.e. we assume that ε > 0. The subcritical case may be treated exactly in the same way.

For normalization reasons, we consider throughout the paper the equation

-∆u + µu = 3u 5+ε , u > 0 (2.1)
instead of the original one. The solutions are identical, up to the multiplicative constant 3 -1 4+ε . We recall that, according to [START_REF] Caffarelli | Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth[END_REF], the functions

U λ,a (x) = λ 1 2 (1 + λ 2 |x -a| 2 ) 1 2 λ > 0 , a ∈ R 3 (2.2)
are the only solutions to the problem

-∆u = 3u 5 , u > 0 in R 3 .
As a ∈ Ω and λ goes to infinity, these functions provide us with approximate solutions to the problem that we are interested in. However, in view of the additional linear term µu which occurs in (P 5+ε,µ ), the approximation needs to be improved. Actually, we define in Ω the following functions :

Ũλ,a,µ (x) = U λ,a (x) - 1 λ 1 2
1 -e -µ We are going to seek for solutions in a neighborhood of such functions, with the a priori assumption that a remains far from the boundary of the domain, that is there exists some number δ > 0 such that d(a, ∂Ω) > δ.

(2.4)

Moreover, integral estimates (see Appendix) suggest to make the additional a priori assumption that λ behaves as 1/ε as ε goes to zero. Namely, we set

λ = 1 Λε 1 δ < Λ < δ (2.5)
with δ some strictly positive number.

In fact, in order to avoid the singularity which appears in the right hand side of (2.3), and to cancel the normal derivative on the boundary, we modify slightly the definition of Ũλ,a,µ , setting

V Λ,a,µ,ε (x) = Ũ 1 Λε ,a,µ (x) - µ 2 (Λε) 1 2 |x -a|(1 -e -ε 2 |x-a| 2 ) + θ Λ,a,µ,ε (x) (2.6)
θ Λ,a,µ,ε = θ being the unique solution to the problem

   -∆θ + µθ = 0 in Ω ∂θ ∂n = ∂ ∂n -U 1 Λε ,a (x) + (Λε) 1 2
|x-a| + µ 2 (Λε)

1 2 |x -a|(1 -e -ε 2
|x-a| 2 ) on ∂Ω.

From the above assumption (2.4) we know that

H µ (a, x) = O(1) θ λ,a,µ,ε = O(ε 5 2 ) (2.7) in C 2 (Ω). We note that V Λ,a,µ,ε = V satisfies        -∆V + µV = 3U 5 1 Λε ,a + µ U 1 Λε ,a -(Λε) 1 2 |x-a| e -ε 2 |x-a| 2 -µΛ 1 2 ε 5 2 |x-a| 3 1 + 2ε 2 |x-a| 2 e -ε 2 |x-a| 2 -µ 2 ε 2 2 (Λε) 1 2 |x -a|(1 -e -ε 2 |x-a| 2 ) in Ω ∂V ∂n = 0 on ∂Ω.
(2.8) The V Λ,a,µ,ε 's are the suitable approximate solutions in the neigbourhood of which we shall find a true solution to the problem. In order to make further computations easier, we proceed to a rescaling. We set

Ω ε = Ω ε
and we define in Ω ε the functions

W Λ,ξ,µ,ε (x) = ε 1 2 V Λ,a,µ,ε (εx) ξ = a ε (2.9)
which write as

W Λ,ξ,µ,ε (x) = U 1 Λ ,ξ (x) -Λ 1 2 1 -e -µ 1 2 ε|x-ξ| |x -ξ| + H µ,ε (ξ, x) - µε 2 2 Λ 1 2 |x -ξ|(1 -e -1 |x-ξ| 2 ) + θΛ,ξ,µ,ε (x) (2.10)
H µ,ε denoting the regular part of the Green's function of the operator -∆+µε 2 with Neumann boundary conditions in Ω ε , and θΛ,ξ,µ,ε (x) = ε 1 2 θ Λ,a,µ,ε (εx). We notice that, taking account of (2.7)

H µ,ε (ξ, x) = O(ε) θΛ,ξ,µ,ε (x) = O(ε 3 ) (2.11) in C 2 (Ω ε ).
We notice also that assumption (2.4) is equivalent to

d(ξ, ∂Ω ε ) > δ ε (2.12) and that W Λ,ξ,µ,ε = W satisfies the uniform estimate |W Λ,ξ,µ,ε | ≤ CU 1 Λ ,ξ in Ω ε . Moreover, we have        -∆W + µε 2 W = 3U 5 1 Λ ,ξ + µε 2 U 1 Λ ,a -Λ 1 2 |x-ξ| e -1 |x-ξ| 2 -µΛ 1 2 ε 2 |x-ξ| 3 1 + 2 |x-ξ| 2 e -1 |x-ξ| 2 -µ 2 ε 4 2 (Λε) 1 2 |x -ξ|(1 -e -1 |x-ξ| 2 ) in Ω ε ∂W ∂n = 0 on ∂Ω ε .
(2.13) Finding a solution to (P 5+ε,µ ) in a neighbourhood of the functions V Λ,a,µ,ε is equivalent, through the rescaling, to solving the problem

(P 5+ε,µ ) -∆u + µε 2 u = 3u 5+ε u > 0 in Ω ε ∂u ∂n = 0 on ∂Ω ε (2.14)
in a neigbourhood of the functions W Λ,ξ,µ,ε . For that purpose, we have to use some local inversion procedure. Namely, we are going to look for a solution to (P ε,µ ) writing as

w = W Λ,ξ,µ,ε + ω
with ω small and orthogonal at W Λ,ξ,µ,ε , in a suitable sense, to the manifold

M = W Λ,ξ,µ,ε , (Λ, ξ) satisfying (2.5) (2.12) . (2.15)
The general strategy consists in finding first, using an inversion procedure, a smooth map (Λ, ξ) → ω(Λ, ξ) such that W Λ,ξ,µ,ε + ω(Λ, ξ, µ, ε) solves the problem in an orthogonal space to M . Then, we are left with a finite dimensional problem, for which a solution may be found using the topological assumption of the theorem. In the subcritical or critical case, the first step may be performed in H 1 (see e.g. [START_REF] Bahri | Critical points at infinity in some variational problems[END_REF][30] [START_REF] Rey | An elliptic Neumann problem with critical nonlinearity in three dimensional domains[END_REF]). However, this approach is not valid any more in the supercritical case, for H 1 does not inject into L q as q > 6. Following [START_REF] Del Pino | Two-bubble solutions in the super-critical Bahri-Coron's problem[END_REF], we use instead weighted Hölder spaces to reduce the problem to a finite dimensional one.

3 The finite dimensional reduction

Inversion of the linearized problem

We first consider the linearized problem at a function W Λ,ξ,µ,ε , and we invert it in an orthogonal space to M . From now on, we omit for sake of simplicity the indices in the writing of

W Λ,ξ,µ,ε . Equipping H 1 (Ω ε ) with the scalar product (u, v) ε = Ωε (∇u • ∇v + µε 2 uv)
orthogonality to the functions

Y 0 = ∂W ∂Λ Y i = ∂W ∂ξ i 1 ≤ i ≤ 3 (3.1)
in that space is equivalent, setting

Z 0 = -∆ ∂W ∂Λ + µε 2 ∂W ∂Λ Z i = -∆ ∂W ∂ξ i + µε 2 ∂W ∂ξ i 1 ≤ i ≤ 3 (3.2)
to the orthogonality in L 2 (Ω ε ), equipped with the usual scalar product •, • , to the functions Z i , 0 ≤ i ≤ 3. Then, we consider the following problem : h ∈ L ∞ (Ω ε ) being given, find a function φ which satisfies

   -∆φ + µε 2 φ -3(5 + ε)W 4+ε + φ = h + i c i Z i in Ω ε ∂φ ∂n = 0 on ∂Ω ε < Z i , φ > = 0 0 ≤ i ≤ 3 (3.3)
for some numbers c i .

Existence and uniqueness of φ will follow from an inversion procedure in suitable functional spaces. Namely, for f a function in Ω ε , we define the following weighted L ∞ -norms

f * = sup x∈Ωε (1 + |x -ξ| 2 ) 1 2 f (x) and f * * = sup x∈Ωε (1 + |x -ξ| 2 ) 2 f (x) .
Writing U instead of U 1 Λ ,ξ , the first norm is equivalent to U -1 f ∞ and the second one to U -4 f ∞ , uniformly with respect to ξ and Λ.

We have the following result :

Proposition 3.1 There exists ε 0 > 0 and a constant C > 0, independent of ε and ξ, Λ satisfying (2.12) (2.5), such that for all 0 < ε < ε 0 and all

h ∈ L ∞ (Ω ε ), problem (3.3) has a unique solution φ ≡ L ε (h). Besides, L ε (h) * ≤ C h * * |c i | ≤ C h * * . (3.4)
Moreover, the map L ε (h) is C 2 with respect to Λ, ξ and the L ∞ * -norm, and

D (Λ,ξ) L ε (h) * ≤ C h * * D 2 (Λ,ξ) L ε (h) * ≤ C h * * . (3.5)
Proof. The argument follows closely the ideas in [START_REF] Del Pino | Two-bubble solutions in the super-critical Bahri-Coron's problem[END_REF]. We repeat it for convenience of the reader. The proof relies on the following result :

Lemma 3.1 Assume that φ ε solves (3.3) for h = h ε . If h ε * * goes to zero as ε goes to zero, so does φ ε * .

Proof of the lemma. For 0 < ρ < 1, we define

f ρ = sup x∈Ωε |(1 + |x -ξ| 2 ) 1 2 (1-ρ) f (x)|
and we first prove that φ ε ρ goes to zero. Arguing by contradiction, we may assume that φ ε ρ = 1. Multiplying the first equation in (3.3) by Y j and integrating in Ω ε we find

i c i Z i , Y j = -∆Y j + µε 2 Y j -3(5 + ε)W 4+ε + Y j , φ ε -h ε , Y j .
On one hand we check, in view of the definition of

Z i , Y j Z 0 , Y 0 = Y 0 2 ε = γ 0 + o(1) Z i , Y i = Y i 2 ε = γ 1 + o(1) 1 ≤ i ≤ 3 (3.6)
where γ 0 , γ 1 are strictly positive constants, and

Z i , Y j = o(1) i = j. (3.7)
On the other hand, in view of the definition of Y j and W , straightforward computations yield

-∆Y j + µε 2 Y j -3(5 + ε)W 4+ε + Y j , φ ε = o( φ ε ρ ) and h ε , Y j = O( h ε * * ).
Consequently, inverting the quasi diagonal linear system solved by the c i 's, we find

c i = O( h ε * * ) + o( φ ε ρ ). (3.8)
In particular, c i = o(1) as ε goes to zero. The first equation in (3.3) may be written as

φ ε (x) = 3(5 + ε) Ωε G ε (x, y) W 4+ε + φ ε + h ε + i c i Z i dy (3.9)
for all x ∈ Ω ε , G ε denoting the Green's function of the operator (-∆ + µε 2 ) in Ω ε with Neumann boundary conditions. We notice that by scaling and (5.11) of Proposition 5.2,

G ε (x, y) = εG µ ( x ε , y ε ) ≤ C |x -y| (3.10)
and hence we obtain

Ωε G ε (x, y)W 4+ε + φ ε dy ≤ C φ ε ρ Ωε 1 |x -y| 1 (1 + |x -ξ| 2 ) 1 2 (3+ε+ρ) dy ≤ C φ ε ρ (1 + |x -ξ| 2 ) -1 2 Ωε G ε (x, y)h ε dy ≤ C h ε * * Ωε 1 |x -y| 1 (1 + |x -ξ| 2 ) 2 dy ≤ C h ε * * (1 + |x -ξ| 2 ) -1 2 Ωε G ε (x, y)Z i dy ≤ C Ωε 1 |x -y| 1 (1 + |x -ξ| 2 ) 5 2 dy ≤ C(1 + |x -ξ| 2 ) -1 2 (3.11)
from which we deduce

(1 + |x -ξ| 2 ) 1 2 (1-ρ) |φ ε (x)| ≤ C(1 + |x -ξ| 2 ) -ρ 2 . φ ε ρ = 1 implies the existence of R > 0, γ > 0 independent of ε such that φ ε L ∞ (B R (ξ)) > γ.
Then, elliptic theory shows that along some subsequence, φε (x) = φ ε (x -ξ) converges uniformly in any compact subset of R 3 to a nontrivial solution of

-∆ φ = 15U 4 Λ,0 φ for some Λ > 0. Moreover, | φ(x)| ≤ C/|x|. As a consequence, φ writes as φ = α 0 ∂U Λ,0 ∂ Λ + 3 i=1 α i ∂U Λ,0 ∂a i
(see e.g. [START_REF] Rey | The role of the Green's function in a nonlinear elliptic problem involving the critical Sobolev exponent[END_REF]). On the other hand, equalities < Z i , φ ε >= 0 provide us with the equalities

R 3 -∆ ∂U Λ,0 ∂ Λ φ = R 3 U 4 Λ,0 ∂U Λ,0 ∂ Λ φ = 0 R 3 -∆ ∂U Λ,0 ∂a i φ = R 3 U 4 Λ,0 ∂U Λ,0 ∂a i φ = 0 1 ≤ i ≤ 3.
As we have also

R 3 |∇ ∂U Λ,0 ∂ Λ | 2 = γ 0 > 0 R 3 |∇ ∂U Λ,0 ∂a i | 2 = γ 1 > 0 1 ≤ i ≤ 3 and R 3 ∇ ∂U Λ,0 ∂ Λ .∇ ∂U Λ,0 ∂a i = R 3 ∇ ∂U Λ,0 ∂a j .∇ ∂U Λ,0 ∂a i = 0 i = j
the α j 's solve a homogeneous quasi diagonal linear system, yielding α j = 0, 0 ≤ α j ≤ 3, and φ = 0, hence a contradiction. This proves that φ ε ρ = o(1) as ε goes to zero. Furthermore, (3.9), (3.11) and (3.8) show that

φ ε * ≤ C( h ε * * + φ ε ρ )
whence also φ ε * = o(1) as ε goes to zero.

Proof of Proposition 3.1 completed. We set

H = φ ∈ H 1 (Ω ε ), < Z i , φ >= 0 0 ≤ i ≤ 3 equipped with the scalar product (•, •) ε . Problem (3.3) is equivalent to finding φ ∈ H such that (φ, θ) ε = 3(5 + ε)W 4+ε + φ + h , θ ∀θ ∈ H that is φ = T ε (φ) + h (3.12)
h depending linearly on h, and T ε being a compact operator in H. Fredholm's alternative ensures the existence of a unique solution, provided that the kernel of Id -T ε is reduced to 0. We notice that φ ε ∈ Ker(Id -T ε ) solves (3.3) with h = 0. Thus, we deduce from Lemma 3.1 that φ ε * = o(1) as ε goes to zero. As Ker(Id -T ε ) is a vector space, Ker(Id -T ε ) = {0}. The inequalities (3.4) follow from Lemma 3.1 and (3.8). This completes the proof of the first part of Proposition 3.1.

The smoothness of L ε with respect to Λ and ξ is a consequence of the smoothness of T ε and h, which occur in the implicit definition (3.12) of φ ≡ L ε (h), with respect to these variables. Inequalities (3.5) are obtained differentiating (3.3), writing the derivatives of φ with respect Λ and ξ as a linear combination of the Z i ' and an orthogonal part, and estimating each term using the first part of the proposition -see [START_REF] Del Pino | Two-bubble solutions in the super-critical Bahri-Coron's problem[END_REF] [20] for detailed computations.

The reduction

In view of (2.13), a first correction between the approximate solution W and a true solution to (P ε,µ ) writes as

ψ ε = L ε (R ε ) (3.13) with R ε =3W 5+ε + -(-∆W + µε 2 W ) =3W 5+ε + -3U 5 1 Λ ,ξ -µε 2 U 1 Λ ,a - Λ 1 2 |x -ξ| e -1 |x-ξ| 2 + µΛ 1 2 ε 2 |x -ξ| 3 1 + 1 |x -ξ| 2 e -1 |x-ξ| 2 + µ 2 ε 4 2 (Λε) 1 2 |x -ξ|(1 -e -1
|x-ξ| 2 ).

(3.14) We have : Lemma 3.2 There exists C, independent of ξ, Λ satisfying (2.12) (2.5), such that

R ε * * ≤ Cε D (Λ,ξ) R ε * * ≤ Cε D 2 (Λ,ξ) R ε * * ≤ Cε. Proof. According to (2.10), W = U + O(ε) uniformly in Ω ε . Consequently, noticing that U ≥ Cε in Ω ε , C independent of ε U 5 -W 5+ε + = O(εU 5 | ln U | + εU 4 )
uniformly in Ω ε , whence

U 5 -W 5+ε + * * ≤ C U -4 (U 5 -W 5+ε + ) ∞ = O(ε).
On the other hand

(1 + |x -ξ| 2 ) 2 µε 2 U 1 Λ ,a - Λ 1 2 |x -ξ| e -1 |x-ξ| 2 - µΛ 1 2 ε 2 |x -ξ| 3 1 + 1 |x -ξ| 2 e -1 |x-ξ| 2 - µ 2 ε 4 2 (Λε) 1 2 |x -ξ|(1 -e -1 |x-ξ| 2 ) = O(ε) uniformly for x ∈ Ω ε , since U 1 Λ ,a - Λ 1 2
|x -ξ| e Lemma 3.3 There exists C, independent of ξ, Λ satisfying (2.12) (2.5), such that

ψ ε * ≤ Cε D (Λ,ξ) ψ ε * ≤ Cε D 2 (Λ,ξ) ψ ε * ≤ Cε.
We consider now the following nonlinear problem : finding φ such that, for some numbers

c i    -∆(W + ψ + φ) + µε 2 (W + ψ + φ) -3(W + ψ + φ) 5+ε + = i c i Z i in Ω ε ∂φ ∂n = 0 on ∂Ω ε < Z i , φ > = 0 0 ≤ i ≤ 3. (3.15) Setting N ε (η) = (W + η) 5+ε + -W 5+ε + -(5 + ε)W 4+ε + η (3.16)
the first equation in (3.15) writes as

-∆φ + µε 2 φ -3(5 + ε)W 4+ε + φ = 3N ε (ψ + φ) + i c i Z i (3.17)
for some numbers c i . Assuming that η * is bounded, say η * ≤ M for some constant M , we have

N ε (η) * * ≤ C η 2 *
whence, assuming that φ * ≤ 1 and using Lemma 3.3

N ε (ψ + φ) * * ≤ C( φ 2 * + ε 2 ). (3.18)
We state the following result : Proposition 3.2 There exists C, independent of ε and ξ, Λ satisfying (2.12) (2.5), such that for small ε problem (3.15) has a unique solution φ = φ(Λ, ξ, µ, ε) with

φ * ≤ Cε 2 . (3.19)
Moreover, (Λ, ξ) → φ(Λ, ξ, µ, ε) is C 2 with respect to the L ∞ * -norm, and

D (Λ,ξ) φ * ≤ Cε 2 D 2 (Λ,ξ) φ * ≤ Cε 2 . (3.20)
Proof. Following [START_REF] Del Pino | Two-bubble solutions in the super-critical Bahri-Coron's problem[END_REF], we consider the map

A ε from F = {φ ∈ H 1 ∩ L ∞ (Ω ε ) : φ * ≤ ε} to H 1 ∩ L ∞ (Ω ε ) defined as A ε (φ) = L ε (3N ε (φ + ψ))
and we remark that finding a solution φ to problem (3.15) is equivalent to finding a fixed point of A ε . One the one hand we have, for φ ∈ F

A ε (φ) * ≤ L ε (N ε (φ + ψ)) * ≤ N ε (φ + ψ) * * ≤ Cε 2 ≤ ε
for ε small enough, implying that A ε sends F into itself. On the other hand A ε is a contraction. Indeed, for φ 1 and φ 2 in F, we write

A ε (φ 1 ) -A ε (φ 2 ) * ≤ N ε (ψ + φ 1 ) -N ε (ψ + φ 2 ) * * ≤ U -4 N ε (ψ + φ 1 ) -N ε (ψ + φ 2 ) ∞ .
In view of (3.16) we have

∂ η N ε (η) = (5 + ε) (W + η) 4+ε + -W 4+ε + ) (3.21) whence |N ε (ψ + φ 1 ) -N ε (ψ + φ 2 )| ≤ CU 3 |ψ + tφ 1 + (1 -t)φ 2 ||φ 1 -φ 2 |
for some t ∈ (0, 1). Then

U -4 N ε (ψ + φ 1 ) -N ε (ψ + φ 2 ) ∞ ≤ C U -1 (ψ + tφ 1 + (1 -t)φ 2 )(φ 1 -φ 2 ) ∞ ≤ C( ψ * + φ 1 * + φ 2 * ) φ 1 -φ 2 * ≤ ε φ 1 -φ 2 * .
This implies that A ε has a unique fixed point in F, that is problem (3.15) has a unique solution φ such that φ * ≤ ε. Furthermore, the definition of φ as a fixed point of A ε yields

φ * = L ε (N ε (φ + ψ)) * ≤ C N ε (φ + ψ) * * ≤ Cε 2
using (3.18), whence (3.19).

In order to prove that (Λ, ξ) → φ(Λ, ξ) is C 2 , we remark that setting for η ∈ F

B(Λ, ξ, η) ≡ η -L ε (3N ε (η + ψ)) φ is defined as B(Λ, ξ, φ) = 0. (3.22)
We have

∂ η B(Λ, ξ, η)[θ] = θ -3L ε θ (∂ η N ε )(η + ψ)
and, using (3.21)

L ε θ (∂ η N ε )(η + ψ) * ≤ C θ (∂ η N ε )(η + ψ) * * ≤ C U -3 (∂ η N ε )(η + ψ) ∞ θ * ≤ C η + ψ * θ * ≤ Cε θ * .
Consequently, ∂ η B(Λ, ξ, φ) is invertible in L ∞ * with uniformly bounded inverse. Then, the fact that (Λ, ξ) → φ(Λ, ξ) is C 2 follows from the fact that (Λ, ξ, η) → L ε (N ε (η + ψ)) is C 2 and the implicit functions theorem.

Finally, let us show how estimates (3.20) may be obtained. Derivating (3.22) with respect to Λ, we have

∂ Λ φ = 3(∂ η B(Λ, ξ, φ)) -1 (∂ Λ L ε )(N ε (φ + ψ)) + L ε ((∂ Λ N ε )(φ + ψ)) + L ε ((∂ η N ε )(φ + ψ)∂ Λ ψ)
whence, according to Proposition 3.1

∂ Λ φ * ≤ C N ε (φ + ψ) * * + (∂ Λ N ε )(φ + ψ) * * + (∂ η N ε )(φ + ψ)∂ Λ ψ * * .
From (3.18) and (3.19) we know that

N ε (φ + ψ) * * ≤ Cε 2 .
Concerning the next term, we notice that according to the definition (3.16)

of N ε |(∂ Λ N ε )(φ + ψ)| = (5 + ε) (W + φ + ψ) 4+ε + -W 4+ε + -(4 + ε)W 3+ε + (φ + ψ) |∂ Λ W | ≤ CU 5 φ + ψ 2 * ≤ CU 5 ε 2
using again (3.18) and (3.19), whence

(∂ Λ N ε )(φ + ψ) * * ≤ Cε 2 .
Lastly, from (3.21) we deduce

|(∂ η N ε )(φ + ψ)∂ Λ ψ| ≤ U 5 φ + ψ * ∂ Λ ψ * yielding (∂ η N ε )(φ + ψ)∂ Λ ψ * * ≤ Cε 2 .
Finally we obtain

∂ Λ φ * ≤ Cε 2 .
The other first and second derivatives of φ with respect to Λ and ξ may be estimated in the same way (see [START_REF] Khenissy | A criterion for existence of solutions to the supercritical Bahri-Coron's problem[END_REF] for detailed computations concerning the second derivatives). This concludes the proof of Proposition 3.2.

Coming back to the original problem

We introduce the following functional defined in H 1 (Ω) ∩ L 6+ε (Ω)

J ε (u) = 1 2 Ω (|∇u| 2 + µu 2 ) - 3 6 + ε Ω u 6+ε + (3.23)
whose nontrivial critical points are solutions to (P 5+ε,µ ) (up to the multiplicative constant 3

1 4+ε
). We consider also the rescaled functions defined in Ω

Ŵ (Λ, a)(x) = ε -ζ W Λ,ξ (ε -1 x) = ε 1 2 -ζ V Λ,a (x) (3.24)
with ζ = 1 2 + 1 2 ε a = εξ.
We define also

ψ(Λ, a)(x) = ε -ζ ψ(Λ, ξ)(ε -1 x) φ(Λ, a)(x) = ε -ζ φ(Λ, ξ)(ε -1 x) (3.25)
and we set

I ε (Λ, a) ≡ J ε ( Ŵ + ψ + φ)(Λ, a) . (3.26)
We have :

Proposition 3.3 The function u = 3 1 4+ε ( Ŵ + ψ + φ) is a solution to problem (P 5+ε,µ ) if and only if (Λ, a) is a critical point of I ε . Proof. For v in H 1 (Ω ε ) ∩ L 6+ε (Ω ε ), we set K ε (v) = 1 2 Ωε (|∇v| 2 + µε 2 v 2 ) - 3 6 + ε Ωε v 6+ε + (3.27)
whose nontrivial critical points are solutions to (P 5+ε,µ ). According to the definition I ε we have

I ε (Λ, a) = ε 1-2ζ K ε (W + ψ + φ)(Λ, ξ) . (3.28)
We notice that u = 3 1 4+ε ( Û + ψ+ φ) being a solution to (P 5+ε,µ ) is equivalent to W +ψ+φ being a solution to (P 5+ε,µ ), that is a critical point of K ε . It is also equivalent to the cancellation of the c i 's in (3.15) or, in view of (3.6) (3.7)

K ε (W + ψ + φ)[Y i ] = 0 0 ≤ i ≤ 3. (3.29)
On the other hand, we deduce from (3.28) that I ε (Λ, a) = 0 is equivalent to the cancellation of K ε (W + ψ + p) applied to the derivatives of W + ψ + p with respect to Λ and ξ. According to the definition (3.1) of the Y i 's, Lemma 3.3 and Proposition 3.2 we have

∂(W + ψ + φ) ∂Λ = Y 0 + y 0 ∂(W + ψ + φ) ∂ξ j = Y j + y j 1 ≤ j ≤ 3 with y i L ∞ * = o(1), 0 ≤ i ≤ 3.
Writing

y i = y i + 3 j=0 a ij Y j y i , Z j = (y i , Y j ) ε = 0 0 ≤ i, j ≤ 3 and K ε (W + ψ + p)[Y i ] = α i it turns out that I ε (Λ, a) = 0 is equivalent, since K ε (W + ψ + p)[θ] = 0 for θ, Z j = (θ, Y j ) ε = 0, 0 ≤ j ≤ 3, to (Id + [a ij ])[α i ] = 0.
As a ij = O( y i * ) = o(1), we see that I ε (Λ, a) = 0 means exactly that (3.29) is satisfied.

4 Proof of Theorem 1.1

In view of Proposition 3.3 we have, for proving the theorem, to find critical points of I ε . We establish first a C 2 -expansion of I ε .

Expansion of I ε

Proposition 4.1 There exist A, B, C, strictly positive constants such that

I ε (Λ, a) = A + A 4 ε ln(εΛ) + 1 2 (C + A 6 )ε + 3BΛ 2 µ 1/2 + H µ (a, a) ε + εσ ε (Λ, a)
with σ ε , D (Λ,a) σ ε and D 2 (Λ,a) σ ε going to zero as ε goes to zero, uniformly with respect to a, Λ satisfying (2.4) and (2.5).

Proof. In view of the definition (3.26) of I ε , we first estimate J ε ( Ŵ ). We have

ε 2ζ-1 J ε ( Ŵ ) = ε 2ζ-1 J ε (ε 1 2 -ζ V ) = J ε (V ) + 3 1 -ε ε 2 6 + ε Ω V 6+ε + = J ε (V ) + 1 2 - ε 2 ln ε + o(ε) Ω V 6+ε
+ from which we deduce, using the integral estimates (5.8), (5.9) and Proposition 5.1 in Appendix,that

J ε ( Ŵ ) = A + A 4 ε ln(εΛ) + 1 2 (C + A 6 )ε + 3BΛ 2 (µ 1/2 + H µ (a, a))ε + o(ε). (4.1)
Then, we prove that

I ε (Λ, a) -J ε ( Ŵ + ψ) = o(ε). (4.2)
Indeed, from a Taylor expansion and the fact that J ε ( Ŵ + ψ + φ)[φ] = 0, we have

I(Λ, a) -J ε ( Ŵ + ψ) = J ε ( Ŵ + ψ + φ) -J ε ( Ŵ + ψ) = 1 0 J ε ( Ŵ + ψ + t φ)[ φ, φ]tdt = ε 1-2ζ 1 0 K ε (W + ψ + φ)[φ, φ]tdt = ε 1-2ζ 1 0 Ωε |φ| 2 + µε 2 φ 2 -3(5 + ε)(W + ψ + φ) 4+ε + φ 2 tdt = ε 1-2ζ 1 0 Ωε N ε (φ + ψ)φ + 3(5 + ε) W 4+ε + -(W + ψ + tφ) 4+ε + φ 2 tdt.
The desired result follows from (3.18), Lemma 3.3 and (3.19). Similar computations show that estimate (4.2) is also valid for the first and second derivatives of I ε (Λ, a)-J ε ( Ŵ + ψ) with respect to Λ and a. Then, the proposition will follow from an estimate of J ε ( Ŵ + ψ) -J ε ( Ŵ ). We have

J ε ( Ŵ + ψ) -J ε ( Ŵ ) =ε 1-2ζ K ε (W + ψ) -K ε (W ) =ε 1-2ζ K ε (W )[ψ] + 1 0 (1 -t)K ε (W + tψ)[ψ, ψ] .
By definition of ψ and R ε

K ε (W )[ψ] = - Ωε R ε ψ
and we have

K ε (W + tψ)[ψ, ψ] = Ωε (|∇ψ| 2 + µε 2 ψ 2 ) -3(5 + ε) Ωε (W + tψ) 4+ε + ψ 2 .
Then, integration by parts and ψ = L ε (R ε ) yield

K ε (W + tψ)[ψ, ψ] = Ωε R ε ψ -3(5 + ε) Ωε (W + tψ) 4+ε + -W 4+ε + ψ 2 . Consequently J ε ( Ŵ + ψ) -J ε ( Ŵ ) = ε 1-2ζ - 1 2 Ωε R ε ψ -3(5 + ε) 1 0 (1 -t)( Ωε (W + tψ) 4+ε + -W 4+ε + ψ 2 )dt
and Lemmas 3.2 and 3.3 yield

J ε ( Ŵ + ψ) -J ε ( Ŵ ) = o(ε).
The same estimate holds for the first and second derivatives with respect to Λ and a, obtained similarly with more delicate computations -see Proposition 3.4 in [START_REF] Khenissy | A criterion for existence of solutions to the supercritical Bahri-Coron's problem[END_REF]. This concludes the proof of Proposition 4.1.

Proof of Theorem 1.1 completed

According to the statement of Theorem 1.1, we assume the existence of b and c, b < c < 0, such that c is not a critical value of ϕ µ (x) = µ

∂I ε ∂Λ (Λ, a) = Aε 4Λ + 3B 2 ϕ µ (a)ε + o(ε) and ∂ 2 I ε ∂Λ 2 (Λ, a) = - Aε 4Λ 2 + o(ε)
The previous arguments prove the existence, for ε small enough, of a nontrivial solution u ε to the problem

-∆u + µu = u 5+ε + in Ω ; ∂u ∂n = 0 on ∂Ω.
Then, the strong maximum principle shows that u ε > 0 in Ω. The fact that u ε blows up, as ε goes to zero, at a point a such that b < ϕ µ (a) < c, ∇ϕ µ (a) = 0, follows from the construction of u ε . In particular, ∇ϕ µ (a) = 0 is a straightforward consequence of (4.3) as ε goes to zero. This concludes the proof of the theorem.

Appendix

Integral estimates

In this subsection, we collect the integral estimates which are needed in the previous section. We recall that according to the definitions of Section 2, we have

V Λ,a,µ,ε (x) = U 1 Λε ,a (x) -(Λε) 1 2 1 -e -µ 1 2 |x-a| |x -a| + H µ (a, x) + ρ Λ,a,µ,ε (x) (5.1) with ρ Λ,a,µ,ε = O(|ε| 3
2 ) (5.2)
uniformly in Ω and with respect to a and Λ satisfying (2.4) (2.5), and the same estimate holds for the derivatives of ρ Λ,a,µ,ε with respect to a and Λ. We recall also that V Λ,a,µ,ε satisfies

   -∆V Λ,a,µ,ε + µV Λ,a,µ,ε = 3U 5 1 Λε ,a + µ U 1 Λε ,a -(Λε) 1 2 |x-a| + ρ Λ,a,µ,ε in Ω ∂VΛ,a,µ,ε ∂n = 0 on ∂Ω (5.3) with ρ Λ,a,µ,ε = µ (Λε) 1 2 |x -a| (1 -e -ε 2 |x-a| 2 ) -µ(Λε) 1 2 ε 2 |x -a| 3 + 2ε 4 |x -a| 5 e -ε 2 |x-a| 2 + O(|ε| 7
2 ) (5.4) and such an expansion holds for the derivatives of ρ Λ,a,µ,ε with respect to a and Λ.

Omitting, for sake of simplicity, the indices Λ, a, µ, ε, we state : Proposition 5.1 Assuming that a and Λ satisfy (2.4) (2.5), we have the uniform expansions as ε goes to zero with

J ε (V ) = A + A 4 ε ln(|ε|Λ) + 1 2 (C + A 6 )ε + 3BΛ 2 µ 1/2 +
A = R 3 U 6 1,0 = π 2 4 B = R 3 U 5 1,0 = 4π 3 C = - 1 2 R 3 U 6
1,0 ln U 1,0 > 0.

Proof. For sake of simplicity, we assume that ε > 0 (the computations are equivalent as ε < 0), and we set r = |x -a|. In view of (5.3), we write (Ω-a)/(Λε)

Ω (|∇V | 2 + µV 2 ) = Ω (-∆V + µV )V = Ω 3U 5 + µ(U - (Λε)
U 5 1 -e -µ Concerning the second term in the right hand side of (5.5), denoting by R the diameter of Ω and using (2.6), we have = O(ε 2 ).

Lastly, noticing that V = O(U ) uniformly in Ω and with respect to the parameters a, Λ satisfying (2.4) and (2.5), we have, using (5.4)

Ω ρ V = O Ω ε 1 2
r

(1 -e -ε 2 r 2 ) + ε and the conclusion follows from the previous computations, noticing that

Ω U 4 = O(ε).
Then, we write

Ω V 6+ε + = Ω V 6 + + Ω V 6 + (V ε + -1). Noticing that 0 ≤ V + ≤ 2ε -1 2 V ε + -1 = ε ln V + + O(ε 2 (ln ε) 2 )

- 1 |x-ξ| 2 =

 12 O(|x -ξ| -3 ) as |x -ξ| goes to infinity, and |x -ξ| = O(1/ε) in Ω ε . The first estimate of the lemma follows. The others are obtained in the same way, differentiating (3.14) and estimating each term as previously. Lemma 3.2 and Proposition 3.1 yield :

1 2 +

 2 H µ (x, x) and the relative homology H * (ϕ c µ , ϕ b µ ) = 0. In view of Proposition 3.3, we have to prove the existence of a critical point of I ε (Λ, a). According to Proposition 4.1, we have

U 6 = 3 U 6 4 .ΩU 5 1 -

 63641 A + O(ε 3 ) with A = R On the other hand, since d(a, ∂Ω) ≥ δ > 0

r 5 e -ε 2 r 2 U + ε 4 = O ε 2 | 2 + 6 +

 524226 µV 2 ) = 3A -3BΛ µ 1/2 + H µ (a, a) ε + O(ε 2 ). (5.7)In the same way we haveΩ V = A -6BΛ µ 1/2 + H µ (a, a) ε + O(ε 2 ).(5.8)Namely, from (5.1)(5.2) and V = O(U ) we derive Ω

  H µ (a, a) |ε| + O(ε 2 (ln |ε|) 2 ) + H µ (a, a) |ε| + O(ε 2 (ln |ε|) 2 ) (a, a) |ε| + O(ε 2 (ln |ε|) 2 ) ∂ 2 J ε ∂Λ 2 = -Aε 4Λ 2 + O(ε 2 (ln |ε|) 2 )

	∂J ε ∂Λ µ 1/2 ∂J ε = Aε 4Λ + 3B 2 ∂a = 3BΛ 2 ∂ ∂a H µ ∂ 2 J ε ∂Λ∂a = 3B 2 ∂ ∂a H

µ (a, a) |ε| + O(ε 2 (ln |ε|) 2 ).

Research supported in part by an Earmarked Grant from RGC of HK.

uniformly with respect to a and Λ satisfying (2.4) (2.5). For δ > 0, η > 0, we define Ω δ,γ = a ∈ Ω s.t. d(a, ∂Ω) > δ, ϕ µ (a) < -γ .

The implicit functions theorem provides us, for ε small enough, with a C 1 -map a ∈ Ω δ,γ → Λ(a) such that ∂I ε ∂Λ (Λ(a), a) = 0 Λ(a) = -A 6B (ϕ µ (a)) -1 + o(1).

Then, finding a critical point of (Λ, a) → I ε (Λ, a) reduces to finding a critical point of a → Ĩε (a), with Ĩε (a) = I ε (Λ(a), a).

We deduce from Proposition 4.1 yields the C 1 -expansion

Therefore, up to an additive and to a multiplicative constant, we have to look for critical points in Ω δ,γ of

as ε goes to zero, uniformly with respect to a ∈ Ω δ,γ .

Arguing by contradiction, we assume

We are going to use the gradient of I ε to build a continuous deformation of ϕ c µ onto ϕ b µ , a contradiction with the assumption H * (ϕ c µ , ϕ b µ ) = 0. We first remark that ϕ µ has isolated critical values, since ϕ µ is analytic in Ω and ϕ µ = -∞ on the boundary of Ω. Therefore, the assumption that c is not a critical value of ϕ µ implies the existence of η > 0 such that ϕ µ has no critical value in (b, b+η]∪(c-η, c]. Moreover, ϕ c µ retracts by deformation onto ϕ c-η µ , ϕ b+η µ retracts by deformation onto ϕ b µ , and H * (ϕ c-η µ , ϕ b+η µ ) = 0. Secondly, we choose δ > 0 such that ϕ µ (x) < b for d(x, ∂Ω) ≤ δ. We choose also γ > 0 such that -γ > c. Then, a point x in the complementary of Ω δ,γ in Ω is either in

a(t) is defined as long as the boundary of Ω δ,γ is not achieved. I ε (a(t)) being decreasing, (4.3) shows that for ε small enough, a(t) remains in ϕ c µ . Then, the boundary of Ω δ,γ may only be achieved by a(t) in ϕ b µ . This means that a(t) is well defined as long as b < ϕ µ (a(t)) < c, and according to assumption (H), I ε (a(t)) is strictly decreasing in that region. Therefore (4.3) proves, for ε small enough, the existence of t > 0 such that ϕ µ (a(t)) = b + η. Composing the flow with a retraction of ϕ c µ onto ϕ c-η µ , we obtain a continuous deformation of ϕ c-η µ onto ϕ b+η µ , a contradiction with H * (ϕ c-η µ , ϕ b+η µ ) = 0. and using the fact that

2 ) we have

R in Ω) whence

We find easily

and noticing that

(5.9)

The first expansion of Proposition 5.1 follows from (5.7), (5.8), (5.9) and the definition (3.23) of J ε .

The expansions for the derivatives of J ε are obtained exactly in the same way.

Green's function

We study the properties of Green's function G µ (x, y) and its regular part H µ (x, y). We summarize their properties in the following proposition Proposition 5.2 Let G µ (x, y) and H µ (x, y) be defined in (1.1) and (1.2), respectively. Then we have H µ (x, x) → -∞ as d(x, ∂Ω) → 0 (5.10)

(5.13)

Proof. (5.10) follows from standard argument. Let x ∈ Ω be such that d = d(x, ∂Ω) is small. So there exists a unique point x ∈ ∂Ω such that d = |x -x|. Without loss of generality, we may assume x = 0 and the outer normal at x is pointing toward x N -direction. Let x * be the reflection point x * = (0, ..., 0, -d) and consider the following auxiliary function We now prove (5.12). For µ small, we can decompose H µ as follows:

where

and H 0 satisfies


Estelle Frey
email: estelle.frey@utbm.fr

Egon Ostrosi
email: egon.ostrosi@utbm.fr

Samuel Gomes
email: samuel.gomes@utbm.fr

Lionel Roucoules
email: lionel.roucoules@ensam.eu

TOOL BASED SUPPORTING COLLABORATIVE DESIGN FROM SPECIFICATION DEVELOPMENT TO CAD MODELLING

Keywords: Product development, collaborative design tools, functional analysis, product data management, communication tools

Management of dynamic information interdependencies and as well as management of multiple perspectives from specification development to conceptual design, as well as from conceptual design to detailed design are critical elements of collaborative and distributed design. Supporting collaborative design with computer technology is a strategy to enhance the ability of actors to interact with each other and with computational resources during different phases of product design and development. However, many observations from the industry and research support the evidence that both specification development phase and conceptual design are poorly supported by collaborative tools. Furthermore, though CAD tools support detail design, it is imperative to develop new collaborative design tools or renovate traditional standalone CAD systems by making it collaborative-native. Consequently, the first goal of this paper is to propose the development of collaborative tools related to the conceptual design. The second goal is the development of a mechanism for communication between the developed collaborative conceptual design tools and actually CAD tools. The mechanism and architecture of the developed tool for collaborative conceptual design is presented. A design problem illustrates the concepts discussed in the paper.

INTRODUCTION

Design is a problem-solving process that transforms an ill-defined problem into a final product [START_REF] Ullman | The mechanical design process[END_REF]. Indeed, engineering design problems are fuzzy and complex. Requirements or the specifications are incomplete, ambiguous and instable. During the design problem evolution, new specifications and constraints yield from the initial requirements and this process is a recursive one. Instability comes from incompleteness, variability and sensitivity of design specifications and data produced during design process [START_REF] Antonsson | Imprecision in Engineering Design[END_REF][START_REF] Deciu | Configurable product design using multiple fuzzy models[END_REF]. Specification development and conceptual design are crucial phases in design. Identifying the customer, determining their requirements and specification, translating customer requirements into measurable engineering requirements is a highly interdisciplinary task. It is concerns more with understanding the problem and making general rather specific, decisions about the solution [START_REF] French | Conceptual Design for Engineers, The Design Council[END_REF]. Conceptual design succeeds the high-level description of requirements and proceeds to a high level description of a solution [START_REF] Mcneil | Understanding conceptual electronic design using protocol analysis[END_REF]. The objective of the conceptual design is to find an optimal and feasible concept meeting functional specifications. It starts with clarified engineering specifications followed by the definition of functions and the establishment of function structure of the product (Pahl and Beitz, 1996). Developing concepts for each function and their combining permit to generate a set of potential concepts. Management of combinatorial explosion (Al-Salka et al., 1998) and functional design verification [START_REF] Deng | Constraint-based functional design verification for conceptual design[END_REF] are challenges of the designers and engineers during conceptual design. The generated concepts give no details about the product's final form, but they affect the basic physical embodiment in detailed design phase [START_REF] Ullman | The mechanical design process[END_REF].

Solving fuzzy and complex problems requires collaboration of actors from multiple distributed disciplines. Collaborative and distributed design is a complex, decision-making engineering process. This complexity results from the conjugation of an important number of heterogeneous data (domain, actors, perspectives, organization, and methods) interacting between them (Movahed-Khah et al., 2007). Self-organization, dynamics and selfsimilarity are some properties of the collaborative and distributed design organisation (Movahed-Khah et. al., 2009). Collaborative and distributed design then could face the most difficult situation in which the information and task interdependencies, related to design phases of product design and development, are not fully defined à priori [START_REF] Lu | A Methodology for Collaborative Design Process and Conflict Analysis, CIRP Annals[END_REF][START_REF] Lu | A scientific foundation of collaborative engineering[END_REF]).

However, management of information interdependencies [START_REF] Frey | A model to develop multi-domain product taking into account the initial requirements until the modelling and simulation tools[END_REF]) and as well as management of multiple perspectives are critical elements of collaborative and distributed design [START_REF] Détienne | Collaborative design: Managing task interdependencies and multiple perspectives[END_REF]. Indeed, each actor has its design model and limited understanding of models of other actors [START_REF] Kusiak | Negotiation in Constraint-Based Design[END_REF][START_REF] Frey | A model to develop multi-domain product taking into account the initial requirements until the modelling and simulation tools[END_REF]). Different actors, pursuing goals that are specific to them, will often have different notions concerning what kind of design is best [START_REF] Klein | Supporting conflict resolution in cooperative design systems[END_REF]. Furthermore, due to the multi-actors interaction, disagreement emerges between designers about incompatible and interdependent propositions [START_REF] Ouertani | Tracking product specification dependencies in collaborative design for conflict management[END_REF][START_REF] Ouertani | Supporting conflict management in collaborative design: An approach to assess engineering change impacts[END_REF]. Unexpected interdependencies can be found when solutions generated by different designers are recombined [START_REF] Case | Discourse model for collaborative design[END_REF] because latent requirements or specification emerge. All these lead to complex and dynamic iteration from requirements to the physical embodiment of the design product.

Supporting collaborative design with computer technology is a strategy to enhance the ability of actors to interact with each other and with computational resources. Research in design technology systems aims at the integration of various design tools into a single holistic system, local or distributed, that is able to support all actions of designers [START_REF] Horváth | A treatise on order in engineering design research[END_REF]. Many observations from the industry and research support the evidence that conceptual design is poorly supported by collaborative tools [START_REF] Wang | Collaborative conceptual designstate of the art and future trends[END_REF].

Fuzziness, incompleteness, and instability of information during the conceptual design are principal causes of difficulties of representation of the designed product during this phase. Though CAD tools support detailed design, it is imperative to develop new collaborative design tools or renovate traditional standalone CAD systems by making it collaborative-native [START_REF] Roucoules | A cooperative design environment -a new generation of CAD systems[END_REF][START_REF] Li | Collaborative computer-aided designresearch and development status[END_REF][START_REF] Fuh | Advances in collaborative CAD: the-state-of-the art[END_REF]. Because the conceptual phase is poorly supported by collaborative tools, and the detailed phase is supported by traditional CAD systems, the communication between these tools is quasi-inexistent. Highly distributed nature of the design, diversity of the engineering tools, complexity and dynamic environment are some real difficulties to develop tools for collaborative conceptual design as well as well collaborative CAD.

Consequently, the first goal of this paper is to propose the development of collaborative tools related to the conceptual design. The second goal is the development of a mechanism for communication between the developed collaborative conceptual design tools and actually CAD tools. The framework of the proposed research is based on a real model extracted from real industrial experiences. The second section of the paper analyse the extracted model. A new model is proposed to overcome the shortcomings of the real model. The developed tools, called TDC system are presented in the third section. The fourth section describes an application. Finally, the fifth section concludes the paper.

2. COMMUNICATION MODEL IN COLLABORATIVE PRODUCT DESIGN

Current communication model

According to our experience in many enterprises, we should note that the product design is often realised by two teams working together. On the one hand, it is the team that develop specifications and defines requirements that product should meet. On the other hand, it is the architect who defines the detailed structure of the product and gives the modelling instructions to the designer. The list of specifications usually regroups the set of requirements and constraints the final product should respect. Modelling will permit the elaboration of detailed drawings, assembly drawings and parts lists. This allows using of drawings for product production. Both documents specification list as well as drawings, must be therefore coherent in order to assure the quality of the final product. However, the information communication from specification development to product modelling is only oral (Figure 1). Indeed, today, from thirteen consulted enterprises, 77% confirm having only an oral communication between actors corresponding to these two tasks. On the same survey, 77% of interrogated enterprises affirm to develop the specifications and requirements in group of different experts representing different product views, but this development is done without a formal approach. They use the basic soft office tools to put in form and stock the information on the specifications and requirements. A printed list of the specifications and requirements is given then to the architect who defines the product structure meeting these specifications and requirements.

Once the architectural structure has been determined, the architect stocks and put it in the disposal of the CAD designer. Still, on the same survey, 85% of interrogated enterprises affirm that they stock the architectural structure product in a PDM. In the most cases, this storage is done in the form of an e-BOM (engineering Bill Of Material).

Then, modelling instructions are given to the CAD designer. CAD designer models the product with the help of CAD tools and stock the organization of the CAD modelling files in a PDM in the form of a cad-BOM (computing aided design Bill Of Material).

In this situation, it can be mentioned that only architect's rigor can insure that the final product will meet the list of specifications and requirements. Effectively, the communication between specifications and requirements on the one hand, and the product modelling on the other, is only realised via the oral exchanges between actors.

Discussion

The first problem in current model comes out from an overall oral communication between actors of product specification and requirements development on the one hand and the architects on the other. Indeed, during data exchange via oral communications, there are a lot of problems related to the human issues such as different vocabulary utilization by different actors, uncommon ontology, disagreement and conflicts, difficulties The second problem resides in the representation form of specifications and requirements. Today, most enterprises are in difficulties on this early phase of the product design. They often have interrogations as how to define and develop adequately the list of specifications and requirements. They frequently develop the list of specifications and requirement with the help of soft office tools but they have not any methodological approach to formalise it properly.

The third problem is related to the parallel definition of the product architecture as well as the development and the evolution of the list of specifications and requirements. This necessarily has a negative impact on the searching and the choices of solutions which therefore can not fully meet the specifications and requirements because of an absent dynamic communication. Consequently, there is not a certainty that e-BOM ensues of logical manner the specification and requirements.

Therefore, three problems to be improved can be identified (Figure 2):

1. The consistency of data related to the specifications and product modelling must be improved because the only oral communication is not sufficient; 2. The development of the specifications and requirements must be improved because it lacks methodological support and is not sufficiently formalized;

3. The adequacy between the structure of the product on the one hand and specification on the other must be improved because the product structure does not reflect always the later.

Proposed Model

For solving these problems, we propose a communication model (Figure 3). The proposed model allows carrying out the product design through communication according to the following phases: The first phase, definition and development of list of specifications and requirements referred to Figure 3 as Functional Performance Specifications (FPS), is proposed to be carried out with the help of External Functional Analysis (EFA) and Internal Functional Analysis (IFA) methodologies. The External Functional Analysis allows identifying the global service functions that designed product The fourth phase enables us to have essentially the product modelling based on predefined the cad-BOM. Modelling files have been created previously in the third phase. Thus, modelling designer can open these files and follow the part or sub-assembly modelling according to the files' modelling instructions. The communication between different BOMs permit to have the necessary information on the part or assembly modelling such as the global service functions realised by the part or sub-assembly, criteria and levels of flexibility related to the functions that the part or sub-assembly model realise, and so on. All these information allow the CAD designer to carry out the product modelling in consistency with the developed specifications and requirements.

TOOLS AND THEIR ARCHITECTURE FOR COLLABORATIVE PRODUCT DESIGN

In order to implement the proposed model, it is necessary to develop software tools allowing supporting each phase of the model. For that raison, the following tools are proposed for each of described phases (Figure 4):

Phase 1: The software tool called, Suite TDC, is proposed to carry out the first phase. TDC Suite is methodological tool permitting to lead the actors to carry out the Functional Analysis. The Suite TDC is composed of two modules: TDC Need that permits to implement the Exterior Functional Analysis and TDC Structure that allows applying the Interior Functional Analysis.

Phase 2:

The platform TDC System supports the second phase. TDC System restructures the functional and technical specifications into a linked structure of BOMs: requirement BOM, (r-BOM); functional BOM (f-BOM) and conceptual BOM (c-BOM).

Phase 3: Product Data Management (Smarteam) tool supports the third phase. It allows the architect to be able to create the e-BOM and the cad-BOM from the c-BOM transferred from TDC System.

Phase 4: CAD modeller (CATIA V5.R16) is used in the forth phase. It gives the possibility to the designer to carry out product modelling using the data and the instructions from cad-BOM.

Between these four tools, three interfacing mechanisms can be distinguished (Figure 4):

Interfacing mechanism A: It allows the integration of data in TDC System from specification and requirement defined in the Suite TDC. Here, the term integration is used because the Suite TDC and TDC System tools have a common core. The application is a hinge of car door. It is a mechanical system decomposed in two main subsets: the hinge side door and the hinge side case. These subsets are composed of several components as shown in Figure 5. BOMs from the data represented in Figure 6, Figure 7 and Figure 8. Nevertheless, further points remain to be considered on the issue of collaborative tool development for conceptual as well as collaborative CAD. As we continue our research, our aim is to search for the granularity of the requirements, their propagation and feedback mechanisms in different stages of collaborative design. View management and augmented annotation in CAD system [START_REF] Shen | Product information visualization and augmentation in collaborative design[END_REF]) related to the developed x-BOMs is another direction of our work.

Interface mechanism B:

CONCLUSION AND FUTURE PROSPECTS

Figure 1

 1 Figure 1 Current communication model

Phase 1 :Phase 2 :Phase 3 :Phase 4 :

 1234 Specifications and requirements definition and its formalisation with the help of methodological tools; Structuring of data related to specifications and requirements into requirement, functional and conceptual BOMs (respectively r-BOM, f-BOM and c-BOM) ; Definition of the product structure (e-BOM) from c-BOM, generation of cad-BOM from e-BOM as well as the definition of CAD modelling instructions; CAD product modelling.

Figure 2 Figure 3

 23 Figure 2 Shortcomings of the current

Figure 4

 4 Figure 4 Architecture of inter-communicated tools

Phase 1 :

 1 Specification and Requirements Definition. For specification and requirement development, the Exterior Functional Analysis and the Interior Functional Analysis are carried out with the Suite TDC. First, Exterior Functional Analysis is realised for hinge of car door through the following steps: • Service functions searching and definition using interactions diagram (Figure 6); • Hierarchical diagram based service functions ordering; • Service functions characterization through criteria, values and flexibility (Figure 7); Second, Interior Functional Analysis is applied for hinge of car door. The following models are generated: • List of components and hierarchical diagram based component ordering ; • Components characterization through criteria, values and flexibility; • Initial product structure representation through Functional Block Diagram and the definition which component realises which function (Figure 8).

Phase 2 :

 2 Structuring of data into BOMs. It consists in translating functional and technical specifications and requirements into BOM representations. Figure 9 shows the generation of

Figure 5 Figure 7

 57 Figure 5 Hinge of car door

Figure 6 diagram 8 Estelle

 68 Figure 6 Interaction diagram

Phase 3 :

 3 Definition of the product structure (e-BOM). Thus, the engineering-BOM (e-BOM) for hinge of car door is deduced from conceptual BOM (c-BOM). Here, for instance, based on the c-BOM, the architect duplicates door screw in e-BOM of hinge of car door. The e-BOM as well as cad-BOM are shown in the Figure10.

Phase 4 :

 4 Hinge of car door CAD modelling. CAD modeller (CATIA V5.R16) is used in the forth phase. It gives the possibility to the designer to carry out Hinge of car door modelling using the data and the instructions from cad-BOM. The files for the parts and assemblies modelling are generated automatically. The set of information in CAD files allows the designer to carry out the hinge of car door modelling in consistency with the developed specifications and requirements.

  well as management of multiple perspectives from specification development to conceptual design, as well as from conceptual design to detailed design are critical elements of collaborative and distributed design. Supporting collaborative design with computer technology is a strategy to enhance the ability of actors to interact with each other and with computational resources during different phases of product design and development. The observations from the industry and research support the evidence specification development phase as well as conceptual design are poorly supported by collaborative tools. The proposed research proposes the development of collaborative tools related to the conceptual design. The development of a mechanism for communication between the developed collaborative conceptual design tools and actual CAD tools is proposed. In the proposed model, specification development can be assisted by collaborative tools. The mapping from specification to the CAD model is realised through different BOMs. Requirements and specifications are then structured in requirement BOM (r-BOM), functional BOM (f-BOM) and conceptual BOM (c-BOM). The structure of the product, represented by engineering BOM (e-BOM) is inferred based on the conceptual BOM (c-BOM). The proposed mechanism translates the conceptual BOM (c-BOM) in computer-aided-design (cad-BOM).Among the advantages of the proposed model, the following ones are noteworthy. The proposed model permits to assist the collaborative product design according to specifications handled by collaborative tools. Furthermore, the tools allows the capitalization and sharing of the know-how of the various actors.

Figure 8 Figure 9 Figure 10

 8910 Figure 8 Functional Block Diagram