Nassif Ghoussoub
email: nassif@math.ubc.ca

F R Éd

Éric Robert

Shaya Shakerian

Mingefeng Zhao
email: mingefeng@math.ubc.ca

Frédéric Robert
email: frederic.robert@univ-lorraine.fr

MASS AND ASYMPTOTICS ASSOCIATED TO FRACTIONAL HARDY-SCHR ÖDINGER OPERATORS IN CRITICAL REGIMES

We consider linear and non-linear boundary value problems associated to the fractional Hardy-Schrödinger operator Lγ,α := (-∆) α 2 -γ |x| α on domains of R n containing the singularity 0, where 0 < α < 2 and 0 ≤ γ < γ H (α), the latter being the best constant in the fractional Hardy inequality on R n . We tackle the existence of least-energy solutions for the borderline boundary

on Ω, where 0 ≤ s < α < n and 2 α (s) = 2(n-s) n-α is the critical fractional Sobolev exponent. We show that if γ is below a certain threshold γ crit , then such solutions exist for all 0 < λ < λ 1 (Lγ,α), the latter being the first eigenvalue of Lγ,α. On the other hand, for γ crit < γ < γ H (α), we prove existence of such solutions only for those λ in (0, λ 1 (Lγ,α)) for which the domain Ω has a positive fractional Hardy-Schrödinger mass m γ,λ (Ω). This latter notion is introduced by way of an invariant of the linear equation (Lγ,α -λI)u = 0 on Ω.

.

Introduction

We study various linear and non-linear equations involving the fractional Hardy-Schrödinger operator L γ,α := (-∆) α 2 -γ |x| α , where (-∆) α 2 is the so-called fractional Laplacian, defined below. Throughout this paper, we shall assume that [START_REF] Abdellaoui | Caffarelli-Kohn-Nirenberg type inequalities of fractional order and applications[END_REF] 0 < α < n and 0 ≤ γ < γ H (α) = 2 α Γ 2 ( n+α 4 ) Γ 2 ( n-α 4 ) , the latter being the best constant in the fractional Hardy constant on R n (see below). Our main focus will be on the case when α < 2, that is when (-∆) α 2 is not a local operator. We shall study problems on bounded domains, but will start by recalling the properties of (-∆) α 2 on the whole of R n , where it can be defined on the Schwartz class S (the space of rapidly decaying C ∞ functions on R n ) via the Fourier transform, (-∆)

α 2 u = F -1 (|2πξ| α F(u)).
Here F(u) is the Fourier transform of u, F(u)(ξ) = R n e -2πix.ξ u(x)dx. See Servadei-Valdinoci [START_REF]The Brezis-Nirenberg result for the fractional Laplacian[END_REF] and references therein for the basics on the fractional Laplacian. For α ∈ (0, 2), the fractional Sobolev space H

α 2 0 (R n ) is defined as the completion of C ∞ c (R n ) under the norm u 2 H α 2 0 (R n ) = R n |2πξ| α |Fu(ξ)| 2 dξ = R n |(-∆) α 4 u| 2 dx.
By Proposition 3.6 in Di Nezza-Palatucci-Valdinoci [START_REF] Di Nezza | Hitchhiker's guide to the fractional Sobolev spaces[END_REF] (see also Frank-Lieb-Seiringer [START_REF] Frank | Hardy-Lieb-Thirring inequalities for fractional Schrödinger operators[END_REF]), the following relation holds: For u ∈ H where λ 0 , λ ∞ > 0 and β -(γ) (resp., β + (γ)) is the unique solution in 0, n-α 2 (resp., in n-α 2 , n -α ) of the equation Ψ n,α (t) = γ. In particular, there exists C 1 , C 2 > 0 such that β+(γ) for all x ∈ R n \ {0}.

α 2 0 (R n ), R n |2πξ| α |Fu(ξ)| 2 dξ = C n,α 2 R n R n
C 1 |x| β-(γ) + |x| β+(γ) ≤ u(x) ≤ C 2 |x| β-(γ) + |x|
Remark 1.3. Note that a direct consequence of Theorem 1.2 is (4) and the corresponding control by Dipierro-Montoro-Peral-Sciunzi [START_REF] Dipierro | Qualitative properties of positive solutions to nonlocal critical problems involving the Hardy-Leray potential[END_REF].

Also note that if α = 2, that is when the fractional Laplacian is the classical Laplacian, the best constant in the Hardy inequality is then γ H (2) = (n-2) 2

4

. The best constant associated with the Hardy-Sobolev inequality is

µ γ,s,2 (R n ) := inf u∈D 1,2 (R n )\{0} R n |∇u| 2 dx -γ R n |u| 2 |x| 2 dx ( R n |u| 2 (s) |x| s dx) 2 2 * (s)
,

where s ∈ [0, 2), 2 (s) := 2(n-s) n-2 , 0 ≤ γ < γ H (2) = (n-2) 2 4
and D 1,2 (R n ) is the completion of C ∞ c (Ω) with respect to the norm u 2 = R n |∇u| 2 dx. The extremals for µ γ,s,2 (R n ) are then explicit and are given by multiples of the functions u (x) = -n-2 2 U ( x ) for > 0, where

U (x) = 1 |x| (2-s)σ -(γ) n-2 + |x| (2-s)σ + (γ) n-2 n-2 2-s for R n \ {0}, and
σ ± (γ) = n -2 2 ± (n -2) 2 4 -γ.
Note that the radial function u(x) = |x| -β is a solution of L γ,2 (u) = 0 on R n \ {0} if and only if β ∈ {σ -(γ), σ + (γ)}.

Back to the case 0 < α < 2, we now turn to when Ω is a smooth bounded domain in R n with 0 in its interior. The best constant in the corresponding fractional Hardy-Sobolev inequality is then,

µ γ,s,α (Ω) := inf u∈H α 2 0 (Ω)\{0} Cn,α 2 R n R n |u(x)-u(y)| 2 |x-y| n+α dxdy -γ Ω |u| 2 |x| α dx ( Ω |u| 2 α (s) |x| s dx) 2 2 α (s)
, where H α 2 0 (Ω) is the closure of C ∞ c (Ω) with respect to the norm

u 2 H α 2 0 (Ω) = C n,α 2
R n R n |u(x) -u(y)| 2 |x -y| n+α dxdy = R n |(-∆) α 4 u| 2 dx.
In Proposition 6.1, we note that -just like the case when α = 2-we have µ γ,s,α (Ω) = µ γ,s,α (R n ), and therefore (3) restricted to Ω, with Dirichlet boundary condition has no extremal, unless Ω is essentially R n . We therefore resort to a setting popularized by Brezis-Nirenberg [START_REF] Brézis | Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents[END_REF] by considering the following boundary value problem:

(7)        (-∆) α 2 u -γ u |x| α = u 2 α (s)-1 |x| s + λu in Ω u ≥ 0 in Ω, u = 0 in R n \ Ω
where 0 < λ < λ 1 (L γ,α ) and λ 1 (L γ,α ) is the first eigenvalue of the operator L γ,α = (-∆)

α 2 -γ |x| α
with Dirichlet boundary condition, that is,

λ 1 := λ 1 (L γ,α ) = inf u∈H α 2 0 (Ω)\{0} Cn,α 2 R n R n |u(x)-u(y)| 2 |x-y| n+α dxdy -γ Ω u 2 |x| α Ω u 2 dx .
One then considers the quantity

µ γ,s,α,λ (Ω) = inf u∈H α 2 0 (Ω)\{0} Cn,α 2 R n R n |u(x)-u(y)| 2 |x-y| n+α dxdy -γ Ω u 2 |x| α dx -λ Ω u 2 dx Ω u 2 α (s) |x| s dx 2 2 α (s)
, and uses the fact that compactness is restored as long as µ γ,s,α,λ (Ω) < µ γ,s,α (R n ); see Proposition 4.1 and also [START_REF] Brézis | Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents[END_REF] for more details. This type of condition is now classical in borderline variational problems; see Aubin [START_REF] Aubin | Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire[END_REF] and Brezis-Nirenberg [START_REF] Brézis | Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents[END_REF]. When α = 2, i.e., in the case of the standard Laplacian, the minimization problem µ γ,s,α,λ (Ω) has been extensively studied, see for example Lieb [START_REF] Elliott | Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities[END_REF], Chern-Lin [START_REF] Chern | Minimizers of Caffarelli-Kohn-Nirenberg inequalities with the singularity on the boundary[END_REF], Ghoussoub-Moradifam [START_REF] Ghoussoub | Functional inequalities: new perspectives and new applications[END_REF] and Ghoussoub-Robert [START_REF] Ghoussoub | On the Hardy-Schrödinger operator with a boundary singularity[END_REF]. The non-local case has also been the subject of several studies, but in the absence of the Hardy term, i.e., when γ = 0. In [START_REF] Servadei | A critical fractional Laplace equation in the resonant case[END_REF], Servadei proved the existence of extremals for µ 0,0,α,λ (R n ), and completed the study of problem [START_REF] Chern | Minimizers of Caffarelli-Kohn-Nirenberg inequalities with the singularity on the boundary[END_REF] which has been initiated by [START_REF]The Brezis-Nirenberg result for the fractional Laplacian[END_REF]. Recently, it has been shown by Yang-Yu [START_REF] Yang | Fractional Hardy-Sobolev elliptic problems[END_REF] that there exists a positive extremal for µ 0,s,α,λ (R n ) when s ∈ [0, 2). In this paper, we consider the remaining cases.

In the spirit of Jannelli [START_REF] Jannelli | The role played by space dimension in elliptic critical problems[END_REF], who dealt with the Laplacian case, we observe that problem [START_REF] Chern | Minimizers of Caffarelli-Kohn-Nirenberg inequalities with the singularity on the boundary[END_REF] is deeply influenced by the value of the parameter γ. Roughly speaking, if γ is sufficiently small then µ γ,s,α,λ (Ω) is attained for any 0 < λ < λ 1 . This is essentially what was obtained by Servadei-Valdinoci [START_REF]The Brezis-Nirenberg result for the fractional Laplacian[END_REF] when s = γ = 0 and n ≥ 2α via local arguments. This is, however not the case, when γ is closer to γ H (α), which amounts to dealing with low dimensions: see for instance Servadei-Valdinoci [START_REF] Servadei | A Brezis-Nirenberg result for non-local critical equations in low dimension[END_REF]. In this context of low dimension, the local arguments generally fail, and it is necessary to use global arguments via the introduction of a notion of mass in the spirit of Schoen [START_REF] Schoen | Conformal deformation of a Riemannian metric to constant scalar curvature[END_REF]. In the present case, and as in the work of Ghoussoub-Robert [START_REF]The Hardy-Schrödinger operator with interior singularity: The remaining cases[END_REF], we define a notion of mass for the operator L γ,α -λI, which again turns out to be critical for this non-local case. The mass is defined via the following key result.

Theorem 1.4. Let Ω be a bounded smooth domain in R n (n > α) and consider, for 0 < α < 2, the boundary value problem

(8)      (-∆) α 2 H -γ |x| α + a(x) H = 0 in Ω \ {0} H ≥ 0 in Ω \ {0} H = 0 in R n \ Ω,
where a(x) ∈ C 0,τ (Ω) for some τ ∈ (0, 1). Assuming the operator (-∆)

α 2 -( γ |x| α + a(x)) coercive, there exists then a threshold -∞ < γ crit (α) < γ H (α) such that for any γ with γ crit (α) < γ < γ H (α),
there exists a unique solution to (8) (in the sense of Definition 2.2) H : Ω → R, H ≡ 0, and a constant c ∈ R such that

H(x) = 1 |x| β+(γ) + c |x| β-(γ) + o 1 |x| β-(γ)
as x → 0.

We define the fractional Hardy-singular internal mass of Ω associated to the operator L γ,α to be m α γ,a (Ω) := c ∈ R. We then prove the following existence result, which complements those in [START_REF] Servadei | A critical fractional Laplace equation in the resonant case[END_REF] and [START_REF] Yang | Fractional Hardy-Sobolev elliptic problems[END_REF] to the case when γ > 0.

Theorem 1.5. Let Ω be a smooth bounded domain in R n (n > α) such that 0 ∈ Ω, and let 0 ≤ s < α, 0 ≤ γ < γ H (α).Then, there exist extremals for µ γ,s,α,λ (Ω) under one of the following two conditions:

(1) 0 ≤ γ ≤ γ crit (α) and 0

< λ < λ 1 (L γ,α ), ( 2
) γ crit (α) < γ < γ H (α), 0 < λ < λ 1 (L γ,α
) and m α γ,λ (Ω) > 0. The idea of studying how critical behavior occurs while varying a parameter γ on which an operator L γ,α continuously depends goes back to [START_REF] Jannelli | The role played by space dimension in elliptic critical problems[END_REF], who considered the classical Hardy-Schrödinger operator L γ,2 := -∆ -γ |x| 2 , and showed the existence of extremals for any λ > 0 provided 0

≤ γ ≤ (n-2) 2 4 -1. In this case, γ crit (2) = (n-2) 2 4 -1.
The definition of the mass and the counterpart of Theorem 1.5 for the operator L γ,2 was established by Ghoussoub-Robert [START_REF]The Hardy-Schrödinger operator with interior singularity: The remaining cases[END_REF]. The complete picture can be described as follows.

Hardy term Dimension

Singularity Analytic. cond. Extremals 0 ≤ γ ≤ γ crit (α) n ≥ 2α s ≥ 0 λ > 0 Yes γ crit (α) < γ < γ H (α) n ≥ 2α s ≥ 0 m α γ,λ (Ω) > 0 Yes 0 ≤ γ < γ H (α) α < n < 2α s ≥ 0 m α γ,λ (Ω) > 0 Yes
Even though the constructions and the methods are heavily inspired by the work of Ghoussoub-Robert [START_REF]The Hardy-Schrödinger operator with interior singularity: The remaining cases[END_REF] on the Laplacian case, the fact that the operator is nonlocal here induces several fundamental difficulties that had to be overcome. First, the construction of the mass in the local case uses a precise classification of singularities for solutions of corresponding elliptic equations, that follows from the comparison principle stating that behavior in a domain is governed by the behavior on its boundary. In the nonlocal case, this fails since one needs to consider the whole complement of the domain, and not only its boundary. We were able to bypass this difficulty by using sharp regularity results available for the fractional Laplacian. Another difficulty we had to face came from the test-functions estimates in the presence of the mass. In the classical local case, one estimates the associated functional on a singular test-function, counting on the mass to appear after suitable integrations by parts. In the nonlocal context, this strategy fails. We overcome this difficulty by looking at the integral on the boundary of a domain as a limit of integrals on the domain after multiplying by a cut-off functions whose support converge to the boundary. This process is well-defined in the nonlocal context and proves to be efficient in tackling the estimates involving the mass.

2. The fractional Hardy-Schrödinger operator L γ,α on R n

In this section, we study the local behavior of solutions of the fractional Hardy-Schrödinger operator L γ,α := (-∆) α 2 -γ |x| α on R n . The most basic solutions for L γ,α u = 0 on R n are of the form u(x) = |x| -β , and a straightforward computation yields (see [START_REF] Frank | Hardy-Lieb-Thirring inequalities for fractional Schrödinger operators[END_REF])

(-∆) α 2 |x| -β = Ψ n,α (β)|x| -β-α in the sense of S (R n ) when 0 < β < n -α, where (9) Ψ n,α (β) := 2 α Γ( n-β 2 )Γ( α+β 2 ) Γ( n-β-α 2 )Γ( β 2 )
.

Recall that the best constant in the fractional Hardy inequality

γ H (α) := µ 0,α,α (R n ) = inf    R n |(-∆) α 4 u| 2 dx R n |u| 2 |x| α dx ; u ∈ H α 2 0 (R n ) \ {0}    is never achieved (see Fall [12]), is equal to Ψ n,α ( n-α 2 ) = 2 α Γ 2 ( n+α whenever α → 2.
We summarize some properties of the function β → Ψ n,α (β) which will be used freely in this section. They are essentially consequences from known properties of Gamma function Γ.

Proposition 2.1 (Frank-Lieb-Seiringer [START_REF] Frank | Hardy-Lieb-Thirring inequalities for fractional Schrödinger operators[END_REF]). The following properties hold:

(1) Ψ n,α (β) > 0 for all β ∈ (0, n -α).

(2) The graph of Ψ n,α in (0, n -α) is symmetric with respect to n-α 2 , that is, Ψ n,α (β) = Ψ n,α (n -α -β) for all β ∈ (0, n -α).

(3) Ψ n,α is strictly increasing in (0, n-α

2 ), and strictly decreasing in

( n-α 2 , n -α). (4) Ψ n,α n -α 2 = γ H (α).
(5) lim

β 0 Ψ n,α (β) = lim β n-α Ψ n,α (β) = 0. ( 6
)
For any γ ∈ (0, γ H (α)), there exists a unique

β -(γ) ∈ (0, n-α 2 ) such that Ψ n,α (β -(γ)) = γ. (7) For any 0 < β ≤ n -α, we have that (10) (-∆) α 2 |x| -β = Ψ n,α (β)|x| -α-β + c n,α 1 {β=n-α} δ 0 in S (R n ),
where we define Ψ n,α (n -α) = 0 and c n,α > 0 is a constant.

In particular, for 0 < β < n -α,

(-∆) α 2 - γ |x| α |x| -β = 0 in S (R n ) if and only if β ∈ {β + (γ), β -(γ)}, where 0 < β -(γ) < n-α 2 is as in Proposition 2.1 and β + (γ) := n -α -β -(γ) ∈ n-α 2 , n -α . In particular, it follows from Proposition 2.1 that β -(γ), β + (γ) are the only solutions to Ψ n,α (β) = γ in (0, n -α). Since 0 < β -(γ) < n-α 2 < β + (γ) < n -α, we get that x → |x| -β-(γ) is locally in H α 2 0 (R n ).
It is the"small" or variational solution, while x → |x| -β+(γ) is the"large" or singular solution. We extend β -(γ), β + (γ) to the whole interval [0, γ H (α)] by defining [START_REF] Dipierro | A density property for fractional weighted Sobolev spaces[END_REF] β -(0) := 0, β + (0) := n -α, and

β -(γ H (α)) = β + (γ H (α)) = n -α 2 ,
which is consistant with Proposition 2.1.

We now proceed to define a critical threshold γ crit (α) as follows. Assuming first that n > 2α, then n-α 2 < n 2 < n -α and therefore, by Proposition 2.1, there exists γ(α)

∈ (0, γ H (α)) such that            n 2 < β + (γ) < n -α if γ ∈ (0, γ(α)) β + (γ) = n 2 if γ = γ(α) n -α 2 < β + (γ) < n 2 if γ ∈ (γ(α), γ H (α)).
We then set

(12) γ crit (α) :=      γ(α) if n > 2α 0 if n = 2α -1 if n < 2α.
One can easily check that for γ ∈ [0, γ H (α)), we have that

γ ∈ (γ crit (α), γ H (α)) ⇔ β + (γ) < n 2 ⇔ x → |x| -β+(γ) ∈ L 2 loc (R n ).
We now introduce the following terminology in defining a notion of solution on a punctured domain.

Definition 2.2. Let Ω be a smooth domain (not necessarily bounded) of R n , n > 1. Let f be a function in L 1 loc (Ω \ {0}). We say that u : Ω → R is a solution to (-∆)

α 2 u = f in Ω \ {0} u = 0 in ∂Ω, provided (1) For any η ∈ C ∞ c (R n \ {0}), we have that ηu ∈ H α 2 0 (Ω); (2) Ω |u(x)| 1+|x| n+α dx < ∞; (3) For any ϕ ∈ C ∞ c (Ω \ {0}), we have that C n,α 2 R n R n (u(x) -u(y))(ϕ(x) -ϕ(y)) |x -y| n+α dxdy = R n f (x)ϕ(x) dx.
Note that the third condition is consistent thanks to the two preceding it. If Ω is bounded, the second hypothesis rewrites as u ∈ L 1 (Ω).

Profile of solutions

Throughout this paper, we shall frequently use the following fact:

Proposition 3.1. A measurable function u : R n → R belongs to H α 2 0 (R n ) if and only if R n |u| 2 α (0) dx < +∞ and (R n ) 2 |u(x)-u(y)| 2 |x-y| n+α dxdy < +∞.
The proof consists of approximating u by a compactly supported function satisfying the same properties. Then, by convoluting with a smooth mollifier, this approximation is achieved by a smooth compactly supported function. The rest is classical and the details are left to the reader.

To prove Theorem 1.2, we shall use a similar argument as in Dipierro-Montoro-Peral-Sciunzi [START_REF] Dipierro | Qualitative properties of positive solutions to nonlocal critical problems involving the Hardy-Leray potential[END_REF]. The main idea is to transform problem (3) into a different nonlocal problem in a weighted fractional space by using a representation introduced in Frank-Lieb-Seiringer [START_REF] Frank | Hardy-Lieb-Thirring inequalities for fractional Schrödinger operators[END_REF]. Lemma 3.2 (Ground State Representation [START_REF] Frank | Hardy-Lieb-Thirring inequalities for fractional Schrödinger operators[END_REF]; Formula (4.3)). Assume

0 < α < 2, n > α, 0 < β < n-α 2 . For u ∈ C ∞ c (R n \{0}), we let v β (x) = |x| β u(x) in R n \{0}. Then, C n,α 2 R n R n |u(x) -u(y)| 2 |x -y| n+α dxdy = Ψ n,α (β) R n u 2 (x) |x| α dx + C n,α 2 R n R n |v β (x) -v β (y)| 2 |x -y| n+α dx |x| β dy |y| β . Let now u ∈ H α 2
0 (R n ) be a positive weak solution to (3). Then by (4) and Remark 4.4 in [START_REF] Dipierro | Qualitative properties of positive solutions to nonlocal critical problems involving the Hardy-Leray potential[END_REF], we have

C n,α 2 R n R n |u(x) -u(y)| 2 |x -y| n+α dxdy = γ R n u 2 (x) |x| α dx + R n u 2 * α (s) (x) |x| s dx. Set v(x) = |x| β-(γ) u(x) on R n \{0}. It follows from Lemma 3.2 and the definition of β -(γ) that C n,α 2 R n R n |v(x) -v(y)| 2 |x -y| n+α dx |x| β-(γ) dy |y| β-(γ) = C n,α 2 R n R n |u(x) -u(y)| 2 |x -y| n+α dxdy -Ψ n,α (β -(γ)) R n u 2 (x) |x| α dx = γ R n u 2 (x) |x| α dx + R n u 2 * α (s) (x) |x| s dx -Ψ n,α (β -(γ)) R n u 2 (x) |x| α dx = R n u 2 * α (s) (x) |x| s dx = R n v 2 * α (s) (x) |x| s+β-(γ)2 * α (s) dx. For 0 < β < n-α 2 , define the space H α 2 ,β 0 (R n ) as the completion of C ∞ c (R n \ {0}) with respect to the norm φ H α 2 ,β 0 (R n ) := R n R n |φ(x) -φ(y)| 2 |x -y| n+α dx |x| β dy |y| β 1 2
.

Many of the properties of the space H α 2 ,β 0 (R n ) were established in [START_REF] Dipierro | A density property for fractional weighted Sobolev spaces[END_REF]. By Lemma 3.2, Remark 4.4 in [START_REF] Dipierro | Qualitative properties of positive solutions to nonlocal critical problems involving the Hardy-Leray potential[END_REF] and [START_REF] Abdellaoui | Caffarelli-Kohn-Nirenberg type inequalities of fractional order and applications[END_REF], we have that v ∈ H α 2 ,β 0 (R n ). Now, we introduce the operator (-∆ β ) α 2 , whose action on a function w is given via the following duality:

For φ ∈ H α 2 ,β 0 (R n ), (-∆ β ) α 2 w, φ = C n,α 2 R n R n (w(x) -w(y))(φ(x) -φ(y)) |x -y| n+α dx |x| β dy |y| β .
This means that v is a weak solution to

(-∆ β-(γ) ) α 2 v = v 2 * α (s)-1 |x| s+β-(γ)2 * α (s) in R n , (13)
in the sense that for any

φ ∈ H α 2 ,β-(γ) (R n ), we have that C n,α 2 R n R n (v(x) -v(y))(φ(x) -φ(y)) |x -y| n+α dx |x| β-(γ) dy |y| β-(γ) = R n v 2 * α (s)-1 |x| s+β-(γ)2 * α (s) φ dx.
The following proposition gives a regularity result and a Harnack inequality for weak solutions of [START_REF] Moustapha | Unique continuation property and local asymptotics of solutions to fractional elliptic equations[END_REF].

Proposition 3.3. Assume 0 < s < α < 2, n > α and 0 < β < n-α 2 , and let v ∈ H α 2 ,β 0 (R n ) be a non-negative, non-zero weak solution to the problem (-∆ β ) α 2 v = v 2 * α (s)-1 |x| s+β2 * α (s) in R n . Then, v ∈ L ∞ (R n ) and there exist constants R > 0 and C > 0 such that C ≤ v(x) in B R (0).
Proof. The statement that v(x) ≥ C in B R (0) is essentially the Harnack inequality for superharmonic functions associated to the nonlocal operator (-∆ β ) α 2 , which is just Theorem 3.4 in Abdellaoui-Medina-Peral-Primo [START_REF] Abdellaoui | The effect of the Hardy potential in some Calderón-Zygmund properties for the fractional Laplacian[END_REF]. See also the proof of Lemma 3.10 in [START_REF] Abdellaoui | The effect of the Hardy potential in some Calderón-Zygmund properties for the fractional Laplacian[END_REF] and also [START_REF] Dipierro | Qualitative properties of positive solutions to nonlocal critical problems involving the Hardy-Leray potential[END_REF]. We now show that v ∈ L ∞ (R n ) by using a similar argument as in [START_REF] Dipierro | Qualitative properties of positive solutions to nonlocal critical problems involving the Hardy-Leray potential[END_REF]. For any p ≥ 1 and T > 0, define the function

φ p,T (t) = t p if 0 ≤ t ≤ T pT p-1 (t -T ) + T p if t > T .
It is easy to check that the function φ p,T (t) has the following properties:

• φ p,T (t) is convex and Lipschitz in [0, ∞).

• φ p,T (t) ≤ t p for all t ≥ 0.

• tφ p,T (t) ≤ 2pφ p,T (t) for all t ≥ 0, since tφ p,T (t) = pφ p,T (t) if 0 < t < T pT p-1 t if t > T . • If T 2 > T 1 > 0, then φ p,T1 (t) ≤ φ p,T2 (
t) for all t ≥ 0. Since φ p,T (t) is convex and Lipschitz, then as noted in [START_REF] Leonori | Basic estimates for solutions of a class of nonlocal elliptic and parabolic equations[END_REF],

(-∆ β ) α 2 φ p,T (v) ≤ φ p,T (v)(-∆ β ) α 2 v in R n . (14) Since φ p,T (t) is Lipschitz and φ p,T (0) = 0, then φ p,T (v) ∈ H α 2 ,β 0 (R n ).
By the weighted fractional Hardy-Sobolev inequality, the ground state representation formula, Lemma 3.2, and (2), we get that there exists some constant C 0 > 0 which only depends on n, α, s and β such that

R n |φ p,T (v)| 2 * α (s) |x| s+β2 * α (s) dx 2 2 * α (s) ≤ C 0 2 R n R n |φ p,T (v(x)) -φ p,T (v(y))| 2 |x -y| n+α dx |x| β dy |y| β . ( 15
)
Since φ p,T (t) ≥ 0 for all t ≥ 0, we get from ( 14) that

R n R n |φ p,T (v(x)) -φ p,T (v(y))| 2 |x -y| n+α dx |x| β dy |y| β = R n φ p,T (v)(-∆ β ) α 2 φ p,T (v) dx ≤ R n φ p,T (v)φ p,T (v)(-∆ β ) α 2 v dx = R n φ p,T (v)φ p,T (v) v 2 * α (s)-1 |x| s+β2 * α (s) dx ≤ 2p R n |φ p,T (v)| 2 v 2 * α (s)-2 |x| s+β2 * α (s) dx.
Note that the last inequality holds, since tφ p,T (t) ≤ 2pφ(t) for all t ≥ 0. By [START_REF] Frank | Hardy-Lieb-Thirring inequalities for fractional Schrödinger operators[END_REF], we have

R n |φ p,T (v)| 2 * α (s) |x| s+β2 * α (s) dx 2 2 * α (s) ≤ pC 0 R n |φ p,T (v)| 2 v 2 * α (s)-2 |x| s+β2 * α (s) dx. (16)
Letting

p 1 = 2 * α (s) 2 , then R n |φ p1,T (v)| 2 * α (s) |x| s+β2 * α (s) dx 2 2 * α (s) ≤ p 1 C 0 R n |φ p1,T (v)| 2 v 2 * α (s)-2 |x| s+β2 * α (s) dx. (17)
For m > 0, a simple computation and Hölder's inequality yield that

p 1 C 0 R n |φ p1,T (v)| 2 v 2 * α (s)-2 |x| s+β2 * α (s) dx = p 1 C 0 v(x)≤m |φ p1,T (v)| 2 v 2 * α (s)-2 |x| s+β2 * α (s) dx +p 1 C 0 v(x)>m |φ p1,T (v)| 2 v 2 * α (s)-2 |x| s+β2 * α (s) dx ≤ p 1 C 0 m 2 * α (s)-2 v(x)≤m |φ p1,T (v)| 2 |x| s+β2 * α (s) dx +p 1 C 0 v(x)>m |φ p1,T (v)| 2 |x| 2(s+β2 * α (s)) 2 * α (s) v 2 * α (s)-2 |x| s+β2 * α (s)- 2(s+β2 * α (s)) 2 * α (s) dx ≤ p 1 C 0 m 2 * α (s)-2 R n |φ p1,T (v)| 2 |x| s+β2 * α (s) dx +p 1 C 0 v(x)>m |φ p1,T (v)| 2 * α (s) |x| s+β2 * α (s) dx 2 2 * α (s) v(x)>m v 2 * α (s) |x| s+β2 * α (s) dx α-s n-s ≤ p 1 C 0 m 2 * α (s)-2 R n |φ p1,T (v)| 2 |x| s+β2 * α (s) dx +p 1 C 0 R n |φ p1,T (v)| 2 * α (s) |x| s+β2 * α (s) dx 2 2 * α (s) v(x)>m v 2 * α (s) |x| s+β2 * α (s) dx α-s n-s . Recall that v ∈ H α 2 ,β 0 (R n ), hence R n v 2 * α (s)
|x| s+β2 * α (s) dx < ∞. Thus, we can take a large M 0 1 and fix it in such a way that

p 1 C 0 v(x)>M0 v 2 * α (s) |x| s+β2 * α (s) dx α-s n-s ≤ 1 2 .
Since φ p1,T (t) ≤ t p1 for all t ≥ 0, then by [START_REF] Ghoussoub | On the Hardy-Schrödinger operator with a boundary singularity[END_REF] and the fact that

p 1 = 2 * α (s) 2 , we get R n |φ p1,T (v)| 2 * α (s) |x| s+β2 * α (s) dx 2 2 * α (s) ≤ 2p 1 C 0 M 2 * α (s)-2 0 R n |φ p1,T (v)| 2 |x| s+β2 * α (s) dx ≤ 2p 1 C 0 M 2 * α (s)-2 0 R n |v| 2p1 |x| s+β2 * α (s) dx = 2p 1 C 0 M 2 * α (s)-2 0 R n v 2 * α (s) |x| s+β2 * α (s) dx. (18)
Let

C 1 = 2C 0 M 2 * α (s)-2 0
. By taking T → ∞ in [START_REF]The Hardy-Schrödinger operator with interior singularity: The remaining cases[END_REF] and applying Fatou's lemma, we get that

R n v p12 * α (s) |x| s+β2α * (s) dx 2 2α * (s) ≤ p 1 C 1 R n v 2 * α (s) |x| s+β2 * α (s) dx < ∞.
Define now recursively the sequence {p k } ∞ k=2 as follows:

2p k+1 + 2 * α (s) -2 = p k 2 * α (s) for all k ≥ 1. ( 19
)
Using ( 16) and ( 19), we have

R n |φ p k+1 ,T (v)| 2 * α (s) |x| s+β2 * α (s) dx 2 2 * α (s) ≤ p k+1 C 0 R n |φ p k+1 ,T (v)| 2 v 2 * α (s)-2 |x| s+β2 * α (s) dx ≤ p k+1 C 0 R n v 2p k+1 v 2 * α (s)-2 |x| s+β2 * α (s) dx = C 0 p k+1 R n v p k 2 * α (s) |x| s+β2 * α (s) dx. (20)
We also have used the fact that φ p k+1 ,T (t) ≤ t p k+1 for all t ≥ 0. By taking T → ∞ in [START_REF] Herbst | Spectral theory of the operator (p 2 + m 2 ) 1/2 -Ze 2 /r[END_REF] and applying Fatou's lemma, we get that

R n v p k+1 2 * α (s) |x| s+β2 * α (s) dx 2 2 * α (s) ≤ C 0 p k+1 R n v p k 2 * α (s) |x| s+β2 * α (s) dx for all k ≥ 1.
Hence, by [START_REF] Ghoussoub | Borderline variational problems involving fractional Laplacians and critical singularities[END_REF], we obtain that

R n v p k+1 2 * α (s) |x| s+β2 * α (s) dx 1 2 * α (s)(p k+1 -1) ≤ (C 0 p k+1 ) 1 2(p k+1 -1) R n v p k 2 * α (s) |x| s+β2 * α (s) dx 1 2(p k+1 -1) = (C 0 p k+1 ) 1 2(p k+1 -1) R n v p k 2 * α (s) |x| s+β2 * α (s) dx 1 2 * α (s)(p k -1)
.

For k ≥ 1, set

I k := R n v p k 2 * α (s) |x| s+β2 * α (s) dx 1 2 * α (s)(p k -1)
and

D k = (C 0 p k+1 ) 1 2(p k+1 -1) .
We have I k+1 ≤ D k I k for all k ≥ 1, and

ln I k+1 ≤ ln D k + ln I k ≤ k j=1 ln D j + ln I 1 ≤ k j=1 ln C 0 + ln p j+1 2(p j+1 -1) + ln I 1 .
It follows from ( 19) that p k+1 = p k 1 (p 1 -1) + 1 for all k ≥ 0. This coupled with the fact that p 1 > 1 yield

ln I k+1 ≤ k j=1 ln C 0 2p j 1 (p 1 -1) + k j=1 ln[p j 1 (p 1 -1) + 1] 2p j 1 (p 1 -1) + ln I 1 ≤ k j=1 ln C 0 2p j 1 (p 1 -1) + k j=1 ln p j+1 1 2p j 1 (p 1 -1) + ln I 1 < C 2 < ∞.
For any fix R ≥ 1, we then have

|x|≤R v p k 2 * α (s) |x| s+β2 * α (s) dx 1 2 * α (s)(p k -1) ≤ I k ≤ e C2 =: C 3 for all k ≥ 1. Since s + β2 * α (s) > 0, we then get |x|≤R v p k 2 * α (s) dx 1 2 * α (s)p k ≤ C 3 R s+β2 * α (s) 2 * α (s)p k for all k ≥ 1. Since lim k→∞ p k = ∞, we have v L ∞ (B R (0)) = lim k→∞ |x|≤R v p k 2 * α (s) dx 1 2 * α (s)p k ≤ C 3 , and finally, that v L ∞ (R n ) ≤ C 3 . Proof of Theorem 1.2. Let v(x) = |x| β-(γ) u(x) in R n \{0}
, by the discussion before at the beginning of section 3, we know that v ∈ H α 2 ,β 0 (R n ) is a positive weak solution to [START_REF] Moustapha | Unique continuation property and local asymptotics of solutions to fractional elliptic equations[END_REF]. We deduce from Proposition 3.3 that for all R > 0, there exist some constant

C > 1 such that C -1 ≤ v(x) ≤ C in B R (0). Since v(x) = |x| β-(γ) u(x) in R n \{0}, then (21) C -1 |x| β-(γ) ≤ u(x) ≤ C |x| β-(γ) in B R (0)\{0}.
In order to prove the asymptotic behavior at zero, it is enough to show that lim

x→0 |x| β-(γ) u(x) exists.
To that end, we proceed as follows:

Claim 1: u ∈ C 1 (R n \ {0}).
This is consequence of regularity theory and we only sketch the proof. First we define f 0 (x) := γ|x| -α u + u 2 α (s)-1 |x| -s , so that for any ω ⊂⊂ R n \ {0}, we have that (-∆)

α/2 u = f 0 in ω in the sense that u ∈ H α 2 0 (R n ) and C n,α 2 R n R n |u(x) -u(y)| 2 |x -y| n+α dxdy = ω f 0 ϕ dx for all ϕ ∈ C ∞ c (ω).
It follows from ( 21) that f 0 ∈ L ∞ (ω). Since u ≥ 0 and f 0 ∈ L ∞ (ω), it follows from Remark 2.5 (see also Theorem 2.1) in Jin-Li-Xiong [START_REF] Jin | On a fractional Nirenberg problem, part I: blow up analysis and compactness of solutions[END_REF] that there exists τ > 0 such that u ∈ C 0,τ loc (R n \ {0}). Then, using recursively Theorem 2.1 in Jin-Li-Xiong [START_REF] Jin | On a fractional Nirenberg problem, part I: blow up analysis and compactness of solutions[END_REF], we get that u ∈ C 1 (R n \ {0}). This proves the claim.

Claim 2: There exists

C > 0 such that |x| β-(γ)+1 |∇u(x)| ≤ C for all x ∈ B 1 (0) \ {0}. If not, then there exists a sequence (x i ) i∈N ∈ B 1 (0) \ {0} such that lim i→+∞ |x i | β-(γ)+1 |∇u(x i )| = +∞.
For simplicity, we write β -:= β -(γ). It follows from from Claim 1, that lim i→+∞ x i = 0. We define r i := |x i | and we set

u i (x) := r β- i u(r i x) for all x ∈ R n \ {0}. It is easy to see that u i ∈ H α 2 0 (R n ), u i ≥ 0 for all i ∈ N and (-∆) α/2 u i = f i in ω ⊂⊂ R n \ {0} where f i (x) := γ|x| -α u i + r (2 α (s)-2)( n-α 2 -β-) i u 2 α (s)-1 i |x| -s for all x ∈ R n \ {0}.
Using the apriori bound of Remark 2.5 (see also Theorem 2.1) in Jin-Li-Xiong [START_REF] Jin | On a fractional Nirenberg problem, part I: blow up analysis and compactness of solutions[END_REF], we get that there exists τ > 0 such that for any R > 1, there exists C(R) > 0 such that u i C 0,τ (B R (0)-B R -1 (0)) ≤ C(R) for all i ∈ N. Using recursively Theorem 2.1 of [START_REF] Jin | On a fractional Nirenberg problem, part I: blow up analysis and compactness of solutions[END_REF] as in Step 1, we get that for any ω ⊂⊂ R n \ {0}, there exists C(ω) > 0 such that u i C 1 (ω) ≤ C(ω). Taking ω large enough and estimating |∇u i ( xi |xi| )|, we get a contradiction, which proves Claim 2.

Set now h(x

) := u 2 α (s)-2 |x| s , so that (-∆) α/2 u -γ |x| α u = h(x)u in R n .
It follows from Claims 1 and 2, that h ∈ C 1 (R n \ {0}), and for some C > 0,

|h(x)| + |x| • |∇h(x)| ≤ C|x| θ-α for all x ∈ B 1 (0) \ {0}, where θ := (2 α (s) -2)( n-α 2 -β -) > 0.
It then follows from Lemma 5.4 below that there exists λ 0 > 0 such that lim

x→0 |x| β-u(x) = λ 0 > 0.
In order to deal with the behavior at infinity, let w be the fractional Kelvin transform of u, that is,

w(x) = |x| α-n u(x * ) := |x| α-n u x |x| 2 in R n \{0}.
By Lemma 2.2 and Corollary 2.3 in [START_REF] Moustapha | Nonexistence results for a class of fractional elliptic boundary value problems[END_REF], we have that w ∈ H α 2 (R n ). A simple calculation gives us that w is also a positive weak solution to [START_REF] Aubin | Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire[END_REF]. Indeed, we have

(-∆) α 2 w(x) = 1 |x| n+α (-∆) α 2 u x |x| 2 = γ w(x) |x| α + w 2 * α (s)-1 (x) |x| s .
Arguing as in the first part of the proof, we get that there exists λ ∞ > 0 such that lim

x→0 |x| β-(γ) w(x) = λ ∞ > 0.
Coming back to u, this implies that

lim |x|→∞ |x| β+(γ) u(x) = λ ∞ > 0.
This ends the proof of Theorem 1.2.

Analytic Conditions for The Existence of Extremals

Let a ∈ C 0,τ (Ω) for some τ ∈ (0, 1), and define the functional

J Ω a : H α 2 0 (Ω) -→ R by J Ω a (u) := Cn,α 2 R n R n |u(x)-u(y)| 2 |x-y| n+α dxdy -γ Ω u 2 |x| α dx -Ω au 2 dx Ω |u| 2 α (s) |x| s dx 2 2 α (s)
, in such a way that

µ γ,s,α,a (Ω) := inf J Ω a (u) : u ∈ H α 2
0 (Ω) \ {0} . We now prove the following proposition, which gives analytic conditions for the existence of extremals for µ γ,s,α,a (Ω). Let Ω be a bounded domain in R n (n > α) such that 0 ∈ Ω, and assume that 0 ≤ γ < γ H (α) and 0 ≤ s ≤ α.

(1) If µ γ,s,α,a (Ω) < µ γ,s,α (R n ), then there are extremals for µ γ,s,α,a (Ω) in H

α 2 0 (Ω). (2) If a(x) is a constant λ, with 0 < λ < λ 1 (L γ,α ) and if s < α, then µ γ,s,α,a (Ω) > 0. Proof. Let (u k ) k∈N ⊂ H α 2 0 (Ω) \ {0} be a minimizing sequence for µ γ,s,α,a (Ω), that is, J Ω a (u k ) = µ γ,s,α,a (Ω) + o(1)
as k → ∞. Up to multiplying by a constant, we may assume that ( 22)

Ω |u k | 2 α (s) |x| s dx = 1 (23) C n,α 2 R n R n |u k (x) -u k (y)| 2 |x -y| n+α dxdy - Ω γ |x| α + a u 2 k dx = µ γ,s,α,λ (Ω) + o(1)
as k → +∞. By [START_REF] Jin | On a fractional Nirenberg problem, part I: blow up analysis and compactness of solutions[END_REF], we have 0 (Ω), strongly in L 2 (Ω) as k → ∞, and θ k (x) → 0 for a.e. x ∈ Ω as k → +∞. Hence, by the Brezis-Lieb lemma (see [START_REF] Brézis | A relation between pointwise convergence of functions and convergence of functionals[END_REF] and [START_REF] Yang | Fractional Sobolev-Hardy inequality in R N[END_REF]), we get that

Ω u 2 k dx ≤ C < ∞ for all k. Since 0 ≤ γ < γ H (α),
R n R n |u k (x) -u k (y)| 2 |x -y| n+α dxdy = R n R n |θ k (x) -θ k (y)| 2 |x -y| n+α dxdy + R n R n |u(x) -u(y)| 2 |x -y| n+α dxdy + o(1), 1 = Ω |u k | 2 α (s) |x| s dx = Ω |θ k | 2 α (s) |x| s dx + Ω |u| 2 α (s) |x| s dx + o(1), Ω u 2 k |x| α dx = Ω θ 2 k |x| α dx + Ω u 2 |x| α dx + o(1),
and

Ω u 2 k dx = Ω u 2 dx + o(1),
as k → ∞. Thus, we have

µ γ,s,α,a (Ω) = C n,α 2 R n R n |u(x) -u(y)| 2 |x -y| n+α dxdy - Ω γ |x| α + a u 2 dx + C n,α 2 R n R n |θ k (x) -θ k (y)| 2 |x -y| n+α dxdy -γ Ω θ 2 k |x| α dx + o(1) (24)
as k → +∞. The definition of µ γ,s,α,a (Ω) and

H α 2 0 (Ω) ⊂ H α 2 0 (R n ) yield µ γ,s,α,a (Ω) Ω |u| 2 α (s) |x| s dx 2 2 * α (s) ≤ C n,α 2 R n R n |u(x) -u(y)| 2 |x -y| n+α dxdy - Ω γ |x| α + a u 2 dx, and (25) µ γ,s,α (R n ) Ω |θ k | 2 α (s) |x| s dx 2 2 * α (s) ≤ C n,α 2 R n R n |θ k (x) -θ k (y)| 2 |x -y| n+α dxdy -γ Ω θ 2 k |x| α dx.
Summing these two inequalities and using ( 22) and ( 24), and passing to the limit k → ∞, we obtain

µ γ,s,α (R n ) 1 - Ω |u| 2 α (s) |x| s dx 2 2 * α (s) ≤ µ γ,s,α,a (Ω)   1 - Ω |u| 2 α (s) |x| s dx 2 2 * α (s)   .
Finally, the fact that µ γ,s,α,a (Ω) < µ γ,s,α (R n ) implies that Ω |u| 2 α (s) |x| s dx = 1. It remains to show that u is an extremal for µ γ,s,α,a (Ω). For that, note that since Ω |u| 2 α (s) |x| s dx = 1, the definition of µ γ,s,α,a (Ω) yields that

µ γ,s,α,a (Ω) ≤ C n,α 2 R n R n |u(x) -u(y)| 2 |x -y| n+α dxdy - Ω γ |x| α + a u 2 dx.
The second term in the right-hand-side of ( 24) is nonnegative due to [START_REF] Elliott | Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities[END_REF]. Therefore, we get that

µ γ,s,α,a (Ω) = C n,α 2 R n R n |u(x) -u(y)| 2 |x -y| n+α dxdy - Ω γ |x| α + a u 2 dx.
This proves the first claim of the Proposition. Now assume that λ ∈ (0, λ 1 (L γ,α )) and 0 ≤ γ < γ H (α), then for all u ∈ H α 2 0 (Ω) \ {0},

J Ω λ (u) = Cn,α 2 R n R n |u(x)-u(y)| 2 |x-y| n+α dxdy -Ω γ |x| α + λ u 2 dx Ω |u| 2 α (s) |x| s dx 2 2 * α (s) ≥ 1 - λ λ 1 (L γ,α ) Cn,α 2 R n R n |u(x)-u(y)| 2 |x-y| n+α dxdy -γ Ω u 2 |x| α dx Ω |u| 2 α (s) |x| s dx 2 2 * α (s) ≥ 1 - λ λ 1 (L γ,α ) 1 - γ γ H (α) µ 0,s,α,0 (Ω) = 1 - λ λ 1 (L γ,α ) 1 - γ γ H (α) µ 0,s,α,0 (R n ) > 0.
Therefore, µ γ,s,α,λ (Ω) > 0.

The fractional Hardy singular interior mass of a domain in the critical case

In this section, we define the fractional Hardy singular interior mass of a domain by proving Theorem 1.4. We shall need the following five lemmae. Lemma 5.1. Assume 0 < β ≤ n-α, and let η ∈ C ∞ c (Ω) be a cut-off function such that 0 ≤ η(x) ≤ 1 in Ω, and η(x) ≡ 1 in B δ (0), for some δ > 0 small. Then x → η(x)|x| -β ∈ H α 2 0 (Ω) and there exists

f β ∈ L ∞ loc (R n ) with f β (x) ≥ 0 on B δ (0) and f β ∈ C 1 (B δ (0)) such that (26) (-∆) α 2 (η|x| -β ) = Φ n,α (β)|x| -α η|x| -β + f β in D (Ω \ {0}), in the sense that, if v β (x) := η(x)|x| -β , then for all ϕ ∈ C ∞ c (Ω \ {0}), C n,α 2 R n R n (v β (x) -v β (y))(ϕ(x) -ϕ(y)) |x -y| n+α dxdy = Φ n,α (β) Ω v β ϕ |x| α dx + Ω f β ϕ(x) dx. Moreover, if β < n-α 2 , then v β ∈ H α 2
0 (Ω) and equality (26) holds in the classical sense of H

α 2 0 (Ω). Proof. When β < n-α 2 , it follows from Proposition 3.1 that x → η(x)|x| -θ ∈ H α 2 0 (Ω). In the general case, for ϕ ∈ C ∞ c (Ω \ {0}), straightforward computations yield C n,α 2 R n R n (v β (x) -v β (y))(ϕ(x) -ϕ(y)) |x -y| n+α dxdy = (-∆) α 2 |x| -β , ηϕ + Ω f β ϕ dx,
where

f β (x) := C(n, α) lim →0 |x-y|> η(x) -η(y) |x -y| n+α • 1 |y| β dy for all x ∈ R n . Note that f β ∈ L ∞ loc (R n )
, and for x ∈ B δ (0), we have that

f β (x) := C(n, α) R n 1 -η(y) |x -y| n+α • 1 |y| β dy ≥ 0, yielding that f β ∈ C 1 (B δ (0)
). Since ϕ ≡ 0 around 0, the lemma is a consequence of (10).

Lemma 5.2 (A comparison principle via coercivity).

Suppose Ω be a bounded smooth domain in R n , 0 < α < 2, γ < γ H (α) and a(x) ∈ C 0,τ (Ω) for some τ ∈ (0, 1). Assume that the operator

(-∆) α 2 -( γ |x| α + a(x)) is coercive. Let u be a function in H α 2 0 (Ω) that satisfies    (-∆) α 2 u - γ |x| α + a(x) u ≥ 0 in Ω u ≥ 0 on ∂Ω, in the sense that u ≥ 0 in R n \ Ω and C n,α 2 R n R n (u(x) -u(y))(v(x) -v(y)) |x -y| n+α dxdy -γ Ω u.v |x| α dx - Ω a(x)uvdx ≥ 0 for all v ∈ H α 2
0 (Ω) with v ≥ 0 a.e. in Ω. Then, u ≥ 0 in Ω. Proof. Let u -(x) = -min(u(x), 0) be the negative part of u. It follows from Proposition 3.1 that u -∈ H α 2 0 (Ω). We can therefore use it as a test function to get

Lu, u -:= C n,α 2 R n R n (u(x) -u(y))(u -(x) -u -(y)) |x -y| n+α dxdy -γ Ω uu - |x| α dx - Ω a(x)uu -dx ≥ 0 Let Ω + := {x : u(x) ≥ 0} and Ω -:= {x : u(x) < 0}. Straightforward computations yield 0 ≤ -Lu -, u -- C n,α 2
Ω -Ω + (u(x) -u(y))u -(y) |x -y| n+α dxdy + C n,α 2
Ω + Ω - (u(x) -u(y))u -(x) |x -y| n+α dxdy, which yields via coercivity c u - 2 H α 2 0 (R n ) ≤ Lu -, u -≤ 0.
Thus, u -≡ 0, and therefore, u ≥ 0 on Ω.

Lemma 5.3. Assume that u ∈ H α 2 0 (Ω) is a weak solution of (-∆) α 2 u - γ + O(|x| τ ) |x| α u = 0 in H α 2 0 (Ω),
for some τ > 0. If u ≡ 0 and u ≥ 0, then there exists a constant C > 0 such that

C -1 ≤ |x| β-(γ) u(x) ≤ C for x → 0, x ∈ Ω.
Proof. We use the weak Harnack inequality to prove the lower bound. Indeed, using Theorem 3.4 and Lemma 3.10 in [START_REF] Abdellaoui | The effect of the Hardy potential in some Calderón-Zygmund properties for the fractional Laplacian[END_REF], we get that there exists C 1 > 0 such that for δ 1 > 0 small enough,

u(x) ≥ C 1 |x| -β-(γ) in B δ1 .
The other inequality goes as in the iterative scheme used to prove Proposition 3.3.

Lemma 5.4 (See Fall-Felli [START_REF] Moustapha | Unique continuation property and local asymptotics of solutions to fractional elliptic equations[END_REF]). Consider an open subset ω ⊂ Ω with 0 ∈ ω, and a function h ∈ C 1 (ω) such that for some τ > 0,

|h(x)| + |x| • |∇h(x)| ≤ C|x| τ -α for all x ∈ ω \ {0}. Let u ∈ H α 2
0 (Ω) be a weak solution of (-∆)

α 2 u - γ |x| α u = h(x)u in ω ⊂ Ω, in the sense that for all ϕ ∈ C ∞ c (ω), C n,α 2 R n R n (u(x) -u(y))(ϕ(x) -ϕ(y)) |x -y| n+α dxdy -γ Ω uϕ |x| α dx = Ω h(x)uϕ dx.
Assume further that there exists C > 0 such that

C -1 ≤ |x| β-(γ) u(x) ≤ C for x → 0, x ∈ Ω.
Then there exists l > 0 such that lim

x→0 |x| β-(γ) u(x) = l.
Proof. This result is an extension of Theorem 1.1 proved by Fall-Felli [START_REF] Moustapha | Unique continuation property and local asymptotics of solutions to fractional elliptic equations[END_REF], who showed that under these conditions, one has ( 27) lim

τ →0 |τ x| n-α 2 -( n-α 2 ) 2 +µ u(τ x) = ψ 0, x |x| in C 1 loc (B 1 (0)) \ {0}
where µ ∈ R and ψ : S n+1 + := {θ ∈ S n+1 θ 1 > 0} → R are respectively an eigenvalue and an eigenfunctions for the problem [START_REF] Schoen | Conformal deformation of a Riemannian metric to constant scalar curvature[END_REF] -div(θ 1-α

1 ∇ψ) = µψ in S n+1 + -lim θ1→0 θ 1 1-α ∂ ν ψ(θ 1 , θ ) = γk α/2 for θ ∈ ∂S n+1 + ,
where k α/2 is a positive constant. We refer to [START_REF] Moustapha | Unique continuation property and local asymptotics of solutions to fractional elliptic equations[END_REF] for the explicit definition of this eigenvalue problem, in particular the relevant spaces used via the Caffarelli-Silvestre classical representation [START_REF] Caffarelli | An extension problem related to the fractional Laplacian[END_REF]. It then follows from the pointwise control ( 21) that

β -(γ) := n -α 2 - n -α 2 2 + µ,
and by Proposition 2.3 in Fall-Felli [START_REF] Moustapha | Unique continuation property and local asymptotics of solutions to fractional elliptic equations[END_REF], that µ is the first eigenvalue of the eigenvalue problem [START_REF] Schoen | Conformal deformation of a Riemannian metric to constant scalar curvature[END_REF]. Then, using classical arguments, we get that the corresponding eigenspace is one-dimensional and is spanned by any positive eigenfunction of (28) (no matter the value of µ, it must necessarily be the first eigenvalue). We are left with proving that ψ(0, x/|x|) is independant of x. In view of the remarks above, this amonts to prove the existence of a positive eigenfunction that is constant on the boundary.

We now exhibit such an eigenfunction by following the argument in Proposition 2.3 in [START_REF] Moustapha | Unique continuation property and local asymptotics of solutions to fractional elliptic equations[END_REF]. First, use ( [START_REF] Moustapha | Semilinear elliptic equations for the fractional Laplacian with Hardy potential[END_REF], Lemma 3.1) to obtain

Γ ∈ C 0 ([0, +∞) × R n ) ∩ C 2 ((0, +∞) × R n ) such that (29)    -div(t 1-α ∇Γ) = 0 in (0, +∞) × R n -lim t→0 t 1-α ∂ ν Γ(t, x) = k α/2 γ |x| α for x ∈ R n = ∂((0, +∞) × R n ) Γ(0, x) = |x| -β-(γ) for x ∈ R n = ∂((0, +∞) × R n ).
Moreover, Γ is in the relevant function space, Γ > 0 and satisfies

Γ(z) = |z| -β-(γ) Γ z |z| for all z ∈ (0, +∞) × R n where |z| = t 2 + |x| 2 if z = (t, x).
In particular, we have that Γ(z) = |z| -β-(γ) ψ 0 (θ) for θ := z/|z| and some ψ 0 ∈ C 0 (S n+1 + ) ∩ C 2 (S n+1 + ). Following [START_REF] Moustapha | Unique continuation property and local asymptotics of solutions to fractional elliptic equations[END_REF], we get that ψ 0 is an eigenvalue for the problem [START_REF] Schoen | Conformal deformation of a Riemannian metric to constant scalar curvature[END_REF]. Moreover, ψ 0 > 0. Therefore, ψ 0 corresponds to the first eigenvalue and spans the corresponding eigenspace. Finally, we remark that for θ ∈ ∂S n+1 + , we have that

ψ 0 (0, θ) = Γ(0, θ) = |θ| -β-(γ) = 1.
Since the eigenspace is one-dimensional, there exists l ∈ R such that ψ = l • ψ 0 . Therefore ψ(0, x/|x|) = l for all x ∈ B 1 (0) \ {0} ⊂ R n . It then follows from [START_REF] Pucci | Critical exponents and critical dimensions for polyharmonic operators[END_REF] that

lim x→0 |x| β-(γ) u(x) = l > 0,
which complete the proof of Lemma 5.4.

Proof of Theorem 1.4. We first prove the existence of a solution. For δ > 0 small enough, let η ∈ C ∞ c (Ω) be a cut-off function as in Lemma 5.1 such that η(x) ≡ 1 in B δ (0). Set β := β + (γ) ≤ n-α in ( 26) and define

f (x) := -(-∆) α 2 - γ |x| α + a(x) (η|x| -β+(γ) ) = -f β+(γ) + aη |x| β+(γ) in Ω \ {0}
in the distribution sense. In particular, f ∈ C 1 (B δ (0) \ {0}) and there exists a positive constant C > 0 such that [START_REF]The Brezis-Nirenberg result for the fractional Laplacian[END_REF] |f

(x)| + |x| • |∇f (x)| ≤ C|x| -β+(γ) for x = 0 close to 0.
In the sequel, we write β + := β + (γ) and β -:= β -(γ). Note that the assumption γ > γ crit (α) implies that β + < n 2 < n+α 2 . Thus, using [START_REF]The Brezis-Nirenberg result for the fractional Laplacian[END_REF] and the fact that

β + < n+α 2 , we get that f ∈ L 2n n+α (Ω). Since L 2n n+α (Ω) = L 2n n-α (Ω) ⊂ H α 2 0 (Ω) , there exists g ∈ H α 2 0 (Ω) such that (-∆) α 2 - γ |x| α + a(x) g = f weakly in H α 2 0 (Ω). Set (31) H(x) := η(x) |x| β+ + g(x) for all x ∈ Ω \ {0}.
Thanks to [START_REF] Lu | Symmetry and regularity of extremals of an integral equation related to the Hardy-Sobolev inequality[END_REF],

H : Ω → R is a solution to (32) (-∆) α 2 H -γ |x| α + a(x) H = 0 in Ω \ {0} H = 0 in R n \ Ω,
in the sense of Definition 2.2. The idea is to now write f as the difference of two positive C 1 functions. The decomposition f = |f | -2f -does not work here since the resulting functions are not necessarily C 1 . To smooth out the functions x → |x| and x → x -, we consider ϕ 1 (x) := 1 + x 2 and ϕ 2 (x) := ϕ 1 (x) -x for all x ∈ R.

It is clear that ϕ 1 , ϕ 2 ∈ C 1 (R) and there exists C > 0 such that

(33) 0 ≤ ϕ i (x) ≤ C(1 + |x|) , |ϕ i (x)| ≤ C and x = ϕ 1 (x) -ϕ 2 (x) for all x ∈ R and i = 1, 2.
Define

f i := ϕ i • f for i = 1, 2. In particular, f = f 1 -f 2 . Let g 1 , g 2 ∈ H α 2 0 (Ω) be solutions to (34) (-∆) α 2 g i - γ |x| α + a(x) g i = f i weakly in H α 2 0 (Ω) for i = 1, 2. Since f 1 , f 2 ≥ 0, Lemma 5.2 yields g 1 , g 2 ≥ 0. Also (-∆) α 2 - γ |x| α + a(x) (g -(g 1 -g 2 )) = f -(f 1 -f 2 ) = 0.
It follows from coercivity that g = g 1 -g 2 . Assuming g 1 ≡ 0, it follows from Lemma 3.10 in [START_REF] Abdellaoui | The effect of the Hardy potential in some Calderón-Zygmund properties for the fractional Laplacian[END_REF] that there exists K > 0 such that

g 1 (x) ≥ K |x| -β-in B δ (0) \ {0}. Since g 1 ∈ H α 2
0 (Ω), it follows from (34) and Theorem 2.1 of Jin-Li-Xiong [START_REF] Jin | On a fractional Nirenberg problem, part I: blow up analysis and compactness of solutions[END_REF] that g 1 ∈ C 0,τ loc (Ω \ {0}) for some τ > 0. Arguing as in the proof of Theorem 1.2, we get that

g 1 ∈ C 1 (Ω \ {0}). Setting h(x) := f 1 (x) g 1 (x)
for x close to 0, we have that h ∈ C 1 (B δ (0)). Now use [START_REF]The Brezis-Nirenberg result for the fractional Laplacian[END_REF] and [START_REF] Yang | Fractional Sobolev-Hardy inequality in R N[END_REF] to get that

f 1 (x) ≤ C(1 + |f (x)|) ≤ C|x| -β+ = C|x| -β-|x| α-(β+-β-) |x| -α ≤ K 1 |x| -α+(α-(β+-β-)) g 1 (x).
Using the fact that γ > γ crit (α) if and only if α -(β + -β -) > 0, we get that |h(x)| ≤ C|x| τ -α for x → 0 where τ := α -(β + -β -) > 0. Therefore, we have that

(-∆) α 2 g 1 - γ + O(|x| (α-(β+-β-)) ) |x| α g 1 = 0 weakly in H α 2 0 (Ω),
with g 1 ≥ 0 and g 1 ≡ 0. It then follows from Lemma 5.3 that there exists c > 0 such that c -1 ≤ |x| β-g 1 (x) ≤ c for x ∈ Ω, x = 0 close to 0. Arguing as in Claim 2 in the proof of Theorem 1.2, we get that there exists C > 0 such that

(35) c -1 ≤ |x| β-g 1 (x) ≤ c and |x| β-+1 |∇g 1 (x)| ≤ C for all x ∈ B δ (0).
We now deal with the differential of h. With the controls ( 30), ( 33) and ( 35), we get that

|x| • |∇h(x)| ≤ C|x| τ -α for x ∈ B δ/2 (0) \ {0}. Now, writing (-∆) α 2 g 1 -γ |x| α g 1 = h(x)g 1
in Ω and using Lemma 5.4, we get that |x| -β-g 1 (x) has a finite limit as x → 0. Note that this is also clearly the case if g 1 ≡ 0. The same holds for g 2 . Therefore, there exists a constant c ∈ R such that |x| -β-g(x) → c as x → 0. In other words,

H(x) = 1 |x| β+ + c |x| β-+ o 1 |x| β- as x → 0,
and there exists C > 0 such that |g(x)| ≤ C|x| -β-for all x ∈ Ω.

We now prove that H > 0 in Ω \ {0}. Indeed, from the above asymptotic expansion we have that

H(x) > 0 for x → 0, x = 0. Since χH ∈ H α 2 0 (R n ) for all χ ∈ C ∞ c (R n \ {0}), it follows from Proposition 3.1 that H -∈ H α 2
0 (Ω \ B (0)) for some > 0 small. We then test [START_REF] Yafaev | Sharp constants in the Hardy-Rellich inequalities[END_REF] against H - and, arguing as in the proof of Lemma 5.2 we get that H -≡ 0, and then H ≥ 0. Since H ≡ 0, H ∈ C 1 (Ω \ {0}), it follows from the Harnack inequality (see Lemma 3.10 in [START_REF] Abdellaoui | The effect of the Hardy potential in some Calderón-Zygmund properties for the fractional Laplacian[END_REF]) that H > 0 in Ω \ {0}. This proves the existence of a solution u to Problem [START_REF] Cotsiolis | Best constants for Sobolev inequalities for higher order fractional derivatives[END_REF] with the relevant asymptotic behavior.

We now deal with uniqueness. Assume that there exists another solution u satisfying the hypothesis of Theorem 1.4. We define ū := u -u . Then ū : Ω → R is a solution to

(-∆) α 2 ū -γ |x| α + a(x) ū = 0 in Ω \ {0} ū = 0 in R n \ Ω,
in the sense of Definition 2.2. Since |ū(x)| ≤ C|x| -β-for all x ∈ Ω where C > 0 is some uniform constant, then by using Proposition 3.1 one concludes that ū ∈ H

α 2 0 (Ω) is a weak solution to (-∆) α 2 ū - γ |x| α + a(x) ū = 0 in Ω, that is, for all ϕ ∈ H α 2 0 (Ω), C n,α 2 R n R n (ū(x) -ū(y))(ϕ(x) -ϕ(y)) |x -y| n+α dxdy - R n γ |x| α + a(x) ūϕ dx = 0.
Taking ϕ := ū and using the coercivity, we get that ū ≡ 0, and then u ≡ u , which yields the uniqueness.

Existence of extremals

This section is devoted to prove the main result, which is Theorem 1.5. By choosing a suitable test function, we estimate the functional J Ω a (u), and we show that the condition µ γ,s,α,a (Ω) < µ γ,s,α (R n ) holds under suitable conditions on the dimension or on the mass of the domain. Recall that Proposition 4.1 implies that it is this strict inequality that guarantees the existence of extremals for µ γ,s,α,a (Ω).

We fix a ∈ C 0,τ (Ω), τ ∈ (0, 1) and

η ∈ C ∞ c (Ω) such that (36) η ≡ 1 in B δ (0) and η ≡ 0 in R n \ B 2δ (0) with B 4δ (0) ⊂ Ω. Let U ∈ H α 2
0 (R n ) be an extremal for µ γ,s,α,0 (R n ). It follows from Theorem 1.2 that, up to multipliying by a nonzero constant, U satisfies for some κ > 0, (37) (-∆)

α 2 U - γ |x| α U = κ U 2 α (s)-1 |x| s weakly in H α 2 0 (R n ). Moreover, U ∈ C 1 (R n \ {0}), U > 0 and (38) lim |x|→∞ |x| β+ U (x) = 1. Set J Ω a (u) := Cn,α 2 (R n ) 2 (u(x)-u(y)) 2 |x-y| n+α dxdy -Ω γ |x| α + a u 2 dx Ω |u| 2 α (s) |x| s dx 2 2 α (s) = A(u) B(u) 2 2 α (s) , (39) where
(R n ) 2 (u(x) -u(y))(v(x) -v(y)) |x -y| n+α dxdy (41) - R n γ |x| α uv dx for u, v ∈ H α 2 0 (R n ). Consider u (x) := -n-α 2 U ( -1 x) for x ∈ R n \ {0}. 2
It follows from Proposition 3.1, that ηu ∈ H α 2 0 (Ω) 6.1. General estimates for ηu . We define the following bilinear form B η on H

α 2 0 (R n ) as follows: For any ϕ, ψ ∈ H α 2 0 (R n ), B η (ϕ, ψ) := ηϕ, ψ -ϕ, ηψ = C n,α 2 (R n ) 2 η(x) -η(y) |x -y| n+α (ϕ(y)ψ(x) -ϕ(x)ψ(y)) dxdy. ( 42
)
This expression makes sense since η ≡ 1 around 0 and η ≡ 0 around ∞. Note that

ηu , ηu = u , η 2 u + β+-β-B η u β + -β - 2 , ηu β + -β - 2 . ( 43
)
It follows from (37) and the definition of u that

u , ϕ = κ R n u 2 α (s)-1 |x| s ϕ dx for all ϕ ∈ H α 2 0 (R n ).
By a change of variable, we get as → 0,

u , η 2 u = κ R n η 2 u 2 α (s) |x| s dx = κ R n u 2 α (s) |x| s dx + O R n \B δ (0) u 2 α (s) |x| s dx = κ R n U 2 α (s) |x| s dx + O R n \B -1 δ (0) U 2 α (s) |x| s dx .
With (38), we get that

(44) u , η 2 u = κ R n U 2 α (s) |x| s dx + O 2 α (s) 2 (β+-β-) = κ R n U 2 α (s) |x| s dx + o β+-β-.
We now deal with the second term of (43). First note that

B η u β + -β - 2 , ηu β + -β - 2 = C n,α 2 (R n ) 2 (η(x) -η(y)) 2 |x -y| n+α u (x) β + -β - 2 • u (y) β + -β - 2
dxdy.

It follows from (38) and the pointwise control of Theorem 1.2 that there exists C > 0 such that for any x ∈ R n \ {0}, we have that

(45) lim →0 u (x) β + -β - 2 = S(x) := 1 |x| β+ and u (x) β + -β - 2 ≤ C |x| β+ .
Since η(x) = 1 for all x ∈ B δ (0) and β + (γ) < n, Lebesgue's convergence theorem yields (46) lim

→0 B η u β + -β - 2 , ηu β + -β - 2 = C n,α 2 (R n ) 2 (η(x) -η(y)) 2
|x -y| n+α S(x)S(y) dxdy = B η (S, ηS).

By plugging together (43), ( 44) and (46), we get as → 0, (47) ηu , ηu = κ

R n U 2 α (s) |x| s dx + B η (S, ηS) β+-β-+ o β+-β-.
Arguing as in the proof of (44), we obtain as → 0, (48)

R n (ηu ) 2 α (s) |x| s dx = R n U 2 α (s) |x| s dx + o( β+-β-).
As an immediate consequence, we get Proposition 6.1. Suppose that 0 ≤ s < α < n, 0 < α < 2 and 0 ≤ γ < γ H (α). Then,

µ γ,s,α,0 (Ω) = µ γ,s,α (R n ).
Proof. It follows from the definition of µ γ,s,α (Ω) that µ γ,s,α,0 (Ω) ≥ µ γ,s,α (R n ). We now show the reverse inequity. Using the estimates (47) and (48) above, we have as → 0,

J Ω 0 (U ) = κ R n U 2 α (s) |x| s dx R n U 2 α (s) |x| s dx 2 2 α (s) + O( β+-β-) = J R n 0 (U ) + O( β+-β-) = µ γ,s,α (R n ) + O( β+-β-).
Letting → 0 yields µ γ,s,α,0 (Ω) ≤ µ γ,s,α (R n ) from which follows that µ γ,s,α,0 (Ω) = µ γ,s,α (R n ).

6.2.

Test functions for the non-critical case 0 ≤ γ ≤ γ crit (α). We now estimate J(ηu ) when 0 ≤ γ ≤ γ crit (α), that is in the case when β -≥ n 2 . Note that since β -+ β + = n -α, we have that β + -β -> α when γ < γ crit (α) and β + -β -= α if γ = γ crit (α).

We start with the following: Proposition 6.2. Let 0 ≤ s < α < 2, 0 ≤ γ ≤ γ crit (α) and n ≥ 2α. Then, as → 0, Ω a(ηu

) 2 dx = α R n U 2 dx a(0) + o( α ) if 0 ≤ γ < γ crit (α) ω n-1 a(0) α ln( -1 ) + o( α ln ) if γ = γ crit (α).
Proof of Proposition 6.2. We write

Ω a(ηu ) 2 dx = B δ au 2 dx + B 2δ \B δ a(ηu ) 2 dx = α B -1 δ a( x)U 2 dx + O( β+-β-).
Assume that γ < γ crit (α). Since β + > n 2 and U ∈ C 1 (R n \ {0}) satisfies ( 6), we get that U ∈ L 2 (R n ) and therefore, Lebesgue's convergence theorem and the assumption β

+ (γ) -β -(γ) > α yield Ω a(ηu ) 2 dx = α R n U 2 dx a(0) + o( α ) as → 0. If now γ = γ crit (α), then lim |x|→∞ |x| n 2 U (x) = 1 and β + -β -= α. Therefore Ω (ηu ) 2 dx = ω n-1 a(0) α ln( -1 ) + o( α ln ) as → 0.
This proves Proposition 6.2.

Plugging together (47), (48) and Proposition 6.2 then yields, as → 0,

J Ω a (ηu ) = κ R n U 2 α (s) |x| s dx R n U 2 α (s) |x| s dx 2 2 α (s) -a(0) R n U 2 dx R n U 2 α (s) |x| s dx 2 2 α (s) α + o( α ) = J R n 0 (U ) -a(0) R n U 2 dx R n U 2 α (s) |x| s dx 2 2 α (s) α + o( α ), (49)
when γ < γ crit (α), and

J Ω a (ηu ) = J R n 0 (U ) -a(0) ω n-1 R n U 2 α (s) |x| s dx 2 2 α (s) α ln 1 + o( α ln 1 ), (50)
when γ = γ crit (α). 6.3. The test function for the critical case. Here, we assume that γ > γ crit (α). It follows from Theorem 1.4 that there exists H

: Ω \ {0} → R such that              H ∈ C 1 (Ω \ {0}) , ξH ∈ H α 2 0 (Ω) for all ξ ∈ C ∞ c (R n \ {0}), (-∆) α 2 H -γ |x| α + a) H = 0 weakly in Ω \ {0} H > 0 in Ω \ {0} H = 0 in ∂Ω and lim x→0 |x| β+ H(x) = 1.
Here the solution is in the sense of Definition 2.2. In other words, the second identity means that for any ϕ ∈ C ∞ c (Ω \ {0}), we have that

(51) C n,α 2 (R n ) 2 (H(x) -H(y))(ϕ(x) -ϕ(y)) |x -y| n+α dxdy - R n γ |x| α + a Hϕ dx = 0.
Note that this latest identity makes sense since H ∈ L 1 (Ω) (since β + < n). Let now η be as in (36). Following the construction of the singular function H in [START_REF] Servadei | A critical fractional Laplace equation in the resonant case[END_REF], there exists g ∈ H α 2 0 (Ω) such that

H(x) := η(x) |x| β+ + g(x) for x ∈ Ω \ {0}, where (52)
(-∆) α 2 g - γ |x| α + a g = f, with f ∈ L ∞ (Ω) and f ∈ C 1 (B δ (0) \ {0}). It follows from (30) that there exists c > 0 such that (53) |f (x)| ≤ C|x| -β+ for x ∈ Ω \ {0} and |∇f (x)| ≤ C|x| -β+-1 for all x ∈ B δ/2 (0) \ {0}.
We also have that

(54) g(x) = m α γ,a (Ω) |x| β-+ o 1 |x| β-as x → 0, and |g(x)| ≤ C|x| -β-for all x ∈ Ω.
Define the test function as

T (x) = ηu (x) + β + -β - 2 g(x) for all x ∈ Ω \ {0}, where u (x) := -n-α 2 U ( -1 x) for x ∈ R n \ {0}, and U ∈ H α 2 0 (R n ) is such that U > 0, U ∈ C 1 (R n \ {0}
) and satisfies (37) above for some κ > 0 and also (38). It is easy to see that T ∈ H α 2 0 (Ω) for all > 0. This subsection is devoted to computing the expansion of J Ω a (T ) where J Ω a is defined in (39), (40) and (41). For simplicity, we set S(x) := 1 |x| β + for x ∈ R n \ {0}. In particular, it follows from (10) that we have that

(55) (-∆) α 2 S - γ |x| α S = 0 weakly in R n \ {0}, in the sense that S, ϕ = 0 for all ϕ ∈ C ∞ c (R n \ {0}). First note that lim →0 T β + -β - 2 = H in L ∞ loc (Ω \ {0}). Therefore, since | -β + -β - 2 T (x)| ≤ C|x| -β+ for x ∈ Ω \ {0} with 2β + < n, Lebesgue's theorem yields as → 0, Ω aT 2 dx = β+-β- Ω aH 2 dx + o β+-β-, Since T = ηu + β + -β - 2
g, we have that

A(T ) = T , T -β+-β- Ω aH 2 dx + o β+-β- = ηu , ηu + 2 β + -β - 2 ηu , g + β+-β-g, g -β+-β- Ω aH 2 dx + o β+-β-
We are now going to estimate these terms separately. First, Formula (42) and (47) yield, as → 0

A(T ) = κ R n U 2 α (s) |x| s dx + 2 β + -β - 2 u , ηg + β+-β-M + o β+-β-, (56)
Here again, note that B η (S, g) makes sense. Therefore, we get that M = M + o(1) as → 0 where (60) M := B η (S, ηS) + 2B η (S, g) + g, g -Ω aH 2 dx.

We now estimate B(T ). Note first that since p > 2, there exists C(p) > 0 such that |x + y| p -|x| p -p|x| p-2 xy ≤ C(p) |x| p-2 y 2 + |y| p for all x, y ∈ R.

We therefore get that

B(T ) = R n ηu + β + -β - 2 g 2 α (s) |x| s dx = R n (ηu ) 2 α (s) |x| s dx + 2 α (s) β + -β - 2 R n u 2 α (s)-1 η 2 α (s)-1 g |x| s dx +O β+-β- R n u 2 α (s)-2 η 2 α (s)-2 g 2 |x| s dx + 2 α (s) 2 (β+-β-) R n |g| 2 α (s) |x| s dx .
Since η ≡ 1 around 0, we get that

R n u 2 α (s)-1 η 2 α (s)-1 g |x| s dx = R n u 2 α (s)-1 ηg |x| s dx + O Ω\B δ (0) u 2 α (s)-1 g |x| s dx = θ + o β + -β - 2
as → 0. Therefore, in view of (48), we deduce that (61)

B(T ) = R n U 2 α (s) |x| s dx + 2 α (s) β + -β - 2 θ + o β+-β-
as → 0. Plugging (58), (57) and (61) into (39), we get that

J Ω a (T ) = κ R n U 2 α (s) |x| s dx R n U 2 α (s) |x| s dx 2 2 α (s)   1 + M κ R n U 2 α (s) |x| s dx β+-β-+ o β+-β-   = J R n 0 (U )   1 + M κ R n U 2 α (s) |x| s dx β+-β-+ o β+-β-   (62)
as → 0, where M is defined in (60) and J R n 0 is as in (39). We now express M in term of the mass. Note that in the classical (pointwise) setting, an integration by parts yield that B η (ϕ, ψ) defined in (42) is an integral on the boundary of a domain. Hence, the mass appears by simply integrating by part independently the singular function H. The central remark we make here is that the integral on the boundary on a domain (defined in the local setting) can be seen as the limit of an integral on the domain via multiplication by a cut-off function with support converging to the boundary -which happened to be defined in the nonlocal setting. Therefore, despite the nonlocal aspect of our problem, we shall be able to apply the same strategy as in the local setting.

We shall be performing the following computations in the same order as the ones above made to get A(T ). The constant M will therefore appear naturally in the two settings.

Let χ ∈ C ∞ (R n ) such that χ ≡ 0 in B 1 (0) and χ ≡ 1 in R n \ B 2 (0). For k ∈ N \ {0}, define χ k (x) := χ(kx) for x ∈ R n , so that as k → +∞. We now estimate these terms separately. Our first claim is that (63) lim k→+∞ χ k g, g = g, g .

Indeed, Arguing as in the proof of (59), we get the existence of G ∈ L 1 ((R n ) 2 ) such that |G k (x, y)| ≤ G(x, y) for all (x, y) ∈ (R n ) 2 such that |x| < δ/2 or |x| > 3δ. By symmetry, a similar control also holds for (x, y) ∈ (R n ) 2 such that |y| < δ/2 or |y| > 3δ. Moreover, for > 0 small enough, we have that G k (x, y) = 0 for (x, y) ∈ (R n ) 2 such that |x| > δ/2 and |y| > δ/2 (this is due to the definition of χ k ). Therefore, since lim k→+∞ (χ k (x) -χ k (y)) = 0 for a.e. (x, y) ∈ (R n ) 2 , Lebesgue's convergence theorem yields (R n ) 2 G k (x, y) dxdy → 0 as k → +∞. We can then conclude that lim k→+∞ B η (S, χ k ηS) = B η (S, ηS).

χ k g -g 2 H α 2 0 (R n ) = C n,α 2 (R n ) 2 |(1 -χ k )(x)g(x) -(1 -χ k )(y)g(y)|

Similar arguments yield lim

k→+∞ B η (S, χ k g) = B η (S, g).

Therefore, we get that 0 = B η (S, ηS) + B η (S, g) + B χ k η (S, g) + g, g -

R n aH 2 dx + o(1)
as k → +∞. We also have that

B χ k η (S, g) = C n,α 2 (R n ) 2
χ k (x) η(x) -η(y) |x -y| n+α (S(y)g(x) -S(x)g(y)) dxdy

+ C n,α 2 (R n ) 2
η(y) χ k (x) -χ k (y) |x -y| n+α (S(y)g(x) -S(x)g(y)) dxdy.

As above, the first integral of the right-hand-side goes to B η (S, g) as k → +∞. We now deal with the second integral. Using that β + + β -= n -α, the change of variables X = kx and Y = ky yield

(R n ) 2
η(y) χ k (x) -χ k (y) |x -y| n+α (S(y)g(x) -S(x)g(y)) dxdy

= (R n ) 2 F k (X, Y ) dXdY,
where

F k (X, Y ) := η Y k χ(X) -χ(Y ) |X -Y | n+α 1 |Y | β+ g X k k -β-- 1 |X| β+ g Y k k -β-.
Note that there exists C > 0 such that |g(x)| ≤ C|x| -β-for all x ∈ Ω \ {0}. Since χ(X) = 0 for |X| < 1 and χ(X) = 1 for |X| > 2, arguing as in the proof of (59), we get that |F k (X, Y )| is uniformly bounded from above by a function in L 1 ((R n ) 2 ) for (X, Y ) ∈ (R n ) 2 such that X ∈ B 3 (0) \ B 1/2 (0) or Y ∈ B 3 (0) \ B 1/2 (0).

There exists C > 0 such that |η(X)-η(Y )| ≤ C|X -Y | for all (X, Y ) ∈ [B 3 (0)\B 1/2 (0)] 2 . Therefore, for such (X, Y ), we have that

|F k (X, Y )| ≤ C|X -Y | 1-α-n 1 |Y | β+ - 1 |X| β+ g X k k -β- +C|X -Y | 1-α-n 1 |X| β+ g X k k -β--g Y k k -β- ≤ C|X -Y | 2-α-n + C|X -Y | 1-α-n g X k k -β--g Y k k -β-.
Define g k (X) := g X k k -β-for X ∈ kΩ. It follows from ( 52) and ( 53) that (-∆)

α 2 g k - γ |X| α + k -α a(k -1 X) g k = f k weakly in H

Theorem 1 . 2 .H α 2 0

 122 |u(x) -u(y)|2 |x -y| n+α dxdy,where C n,α = Assume 0 ≤ s < α < 2, n > α and 0 ≤ γ < γ H (α). Then any positive extremal u ∈ (R n ) for µ γ,s,α (R n ) satisfies u ∈ C 1 (R n \ {0}) and(6) limx→0 |x| β-(γ) u(x) = λ 0 and lim |x|→∞ |x| β+(γ) u(x) = λ ∞ ,

Proposition 4 . 1 .

 41

2 0 2 0 2 0 2 0

 2222 the fractional Hardy inequality combined with[START_REF] Lei | Asymptotic properties of positive solutions of the Hardy-Sobolev type equations[END_REF] yields that u k H α (Ω)≤ C for all k. It then follows that there exists u ∈ H α (Ω) such that, up to a subsequence, such that (u k ) goes to u weakly in H α (Ω) and strongly in L 2 (Ω) as k → ∞.We first show that Ω|u| 2 α (s) |x| s dx = 1. Define θ k = u k -u for all k ∈ N. Itfollows from the boundedness in H α (Ω) that, up to a subsequence, we have that θ k 0 weakly in H α 2

au 2

 2 dx and B(u) := Ω |u| 2 α (s) |x| s dx with u, v := C n,α

2 0

 2 χ k (x) = 0 for |x| < 1 k and χ k (x) = 1 for |x| > 2 k .In particular, (χk ) k is bounded in L ∞ (R n ) and χ k (x) → 1 as k → +∞ for a.e. x ∈ R n . Since χ k H ∈ H α (Ω), then by the very definition of H (see (51)), we have that0 = H, χ k H -R n aHχ k H dx = ηS + g, χ k ηS + χ k g -R n χ k aH 2 dx = ηS, χ k ηS + ηS, χ k g + χ k ηS, g + g, χ k g -R n χ k aH 2 dx = S, χ k η 2 S + B η (S, χ k ηS) + S, ηχ k g + B η (S, χ k g) + S, χ k ηg +B χ k η (S, g) + g, χ k g -R n χ k aH 2 dx.Since aH 2 ∈ L 1 (Ω) (this is a consequence of 2β + < n) and S is a solution to (55), we get that 0 = B η (S, χ k ηS) + B η (S, χ k g) + B χ k η (S, g) + g, χ k g -R n aH 2 dx + o(1)

2 |x - 2 0 2 0 2 (R n ) 2 χC n,α 2 (R n ) 2 χ

 2-222222 y| n+α dxdy ≤ C n,α (R n ) 2 |1 -χ k (x)| 2 |g(x) -g(y)| 2 |x -y| n+α dxdy +C n,α (R n ) 2 g(y) 2 |χ k (x) -χ k (y)| 2 |x -y| n+α dxdy.The first integral goes to 0 as k → +∞ with Lebesgue's convergence theorme since g ∈ H α (R n ). For the second term, we use the change of variable X = kx, Y = ky and the control of g(x) by |x| -β-. This proves that (χ k g) → g in H α (R n ) as k → +∞. The claim follows and (63) is proved. We now writeB η (S, χ k ηS) = C n,α k (x) F (x, y) dxdy + (R n ) 2 G k (x, y) dxdy , where F (x, y) := η(x) -η(y) |x -y| n+α (S(y)(ηS)(x) -S(x)(ηS)(y)) ,andG k (x, y) := (η(x) -η(y))(χ k (x) -χ k (y))(ηS)(y)S(x) |x -y| n+α .As in the proof of (59) and (46), F ∈ L 1 ((R n ) 2 ) and Lebesgue's convergence theorem yields lim k→+∞ k (x) F (x, y) dxdy = B η (S, ηS).

  X) := k -β--α f (k -1 X) so that |f k (X)| ≤ Ck -(α-(β+-β-)) |X| -β+ for all X ∈ kΩ. Here again, elliptic regularity yields that (g k ) is bounded in C 1 loc (R n \{0}). Therefore, there exists C > 0 such that |g k (X) -g k (Y )| ≤ C|X -Y | for all (X, Y ) ∈ [B 3 (0) \ B 1/2 (0)] 2 . Therefore, we get that |F k (X, Y )| ≤ C|X -Y | 2-α-n for all (X, Y ) ∈ [B 3 (0) \ B 1/2 (0)] 2 .

) Γ 2 ( n-α 4 ) (seeHerbst and Yafaev [20,[START_REF] Yafaev | Sharp constants in the Hardy-Rellich inequalities[END_REF]), and it converges to the best classical Hardy constant γ H (2) = (n-2) 2

where

, g + g, g -Ω aH 2 dx.

As to the second term of (56), we have

We set θ :=

dx. It is easy to check that, since ηg ∈ H

0 (R n ) and goes to 0 weakly as → 0, we have that (57) lim

Therefore we can rewrite (56) as

as → 0. We now estimate M . First, we write

where

.

Remembering that η ≡ 1 in B δ (0) and η ≡ 0 in B 2δ (0) c and using (54), we get that

Similarly, we have a bound on F on {|x| > 3δ}. By symmetry, this yields also a bound on {|y| < δ/2} ∪ {|y| > 3δ}. We are then left with getting a bound on

For (x, y) ∈ A, we have that

As noticed in the proof of Theorem 1.4, it follows from elliptic theory that g ∈ C 1 (Ω\{0}). Therefore, there exists C > 0 such that |g(x) -g(y)| ≤ C|x -y| for all (x, y) ∈ A.

It then follows from (45) and arguments similar to the Proof of Theorem 1.2 (see Remark 2.5 and Theorem 2.1 of Jian-Li-Xiong [START_REF] Jin | On a fractional Nirenberg problem, part I: blow up analysis and compactness of solutions[END_REF]) that (ũ ) is bounded in C 1 loc (R n \ {0}). Therefore, there exists C > 0 such that |ũ (x) -ũ (y)| ≤ C|x -y| for all (x, y) ∈ A. Then, we get

Therefore, (F ) is uniformly dominated on (R n ) 2 . Noting that

Therefore, since α < 2, (F k ) is also dominated on this domain, and then on (R n ) 2 . Finally, it follows from the definition (54) of the mass that lim k→+∞

where

Without loss of generality, we can assume that χ is radially symetrical and nondecreasing. Therefore, we get that K > 0. With (60), we then get that

Ω) with K > 0. Plugging this identity in (62) yields


Fei Xie

Zhan Jiang
email: jiangzhan@zju.edu.cn

Lanfen Lin

Research on Semantic Interoperability for Business Collaboration

Keywords:

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Software systems introduced by enterprises during the construction of enterprise informatization improve business efficiency. However, due to the heterogeneity caused by different software vendors and programming language, the business collaboration via system interoperation is hard to run smoothly. Web Service adopts a serial of standard technologies and protocols such as WSDL, UDDI, XML and SOAP [START_REF] Van Der Alsta W M P | Business process management: where business processes and web services meet[END_REF]. Relevant business functions of enterprise systems are encapsulated into welldefined web services and invoked to shield system-level heterogeneity. Besides, it breaks through the restriction of territory and becomes a convenient and effective method to achieve long-distance business collaboration. Since enterprises differ in management pattern, business process and product model and so on, and employees in different departments or enterprises describe the business concept, data and model in dissimilar ways, semantic conflicts emerge throughout the process of business collaboration, it's necessary to find a method to resolve the semantic-level conflicts.

The emergence of ontology technology provides a new method for solving semantic problems of system interoperability. Ontology, as "an explicit specification of a conceptualization" [START_REF] Uschold | Methodology for Building Ontologies.Workshop on Basic Ontologies.Issues in Knowledge Sharing[END_REF], is able to represent the knowledge of specific domain in semantic level. It has been widely applied in many fields of computer science, such as knowledge sharing, software reusing and information retrieval. In order to achieve system interoperability among enterprises, ontology has been introduced by researchers to model heterogeneous systems to maintain semantic consistency among different systems. Bichindaritz proposes an ontology-based framework for semantic interoperability of case-based reasoning systems in biology and medicine [START_REF] Bichindaritz | Mé moire: A framework for semantic interoperability of case-based reasoning systems in biology and medicine[END_REF]. Hu and Li etc. presents an ontology-based method of semantic interoperability to solve the interoperability problems in CAD and GIS [START_REF] Hu | Ontology-based product knowledge representation[END_REF]. Yang and Zhang gives a method of construction models interoperability by building construction domain ontology and modeling CAD models [START_REF] Yang | Semantic interoperability in building design : methods and tools[END_REF]. Park and Ram develop a comprehensive framework and methodology to facilitate semantic interoperability among distributed and heterogeneous information systems [START_REF] Park | Information systems interoperability: What lies beneath[END_REF]. And lots of projects, likes STASIS, ONTOGOV, TERREGOV [START_REF] Guglielmina | The COIN Project -Vision, Objectives, Results, COIN IP Workshop[END_REF] also achieve semantic business collaboration in specific domain. Some researches concentrate in certain field and have domain limitations, and some are proposed in the general perspective and testified effectiveness. But it's hard to apply them to the SMEs collaborations in China. The informatization level of SMEs in China is backward, and there not exists available business model in these SMEs, not to mention ontology construction and maintenance. In this paper, taking SMEs in China as subject, considering the real situation encountered throughout business collaboration, a practical framework of semantic interoperability (SI) is proposed to realize business collaboration (BC).

The rest of the paper is organized as follows. Section 2 describes the characteristics and obstacles of SMEs cooperation. The semantic interoperability framework is given in section 3. Section 4 elaborates on the key technologies involved during the process of business collaboration. Section 5 discusses the enterprise interoperation process and the implementation of the framework. Finally, section 6 concludes the paper and discusses some future work.

Analysis of SMEs Collaboration

There are five typical characteristics in the industry cluster of SMEs: 1)the large number of enterprises in the region, more than 90 percent of them are small and medium-sized enterprise(SME) . 2) The business collaboration in this area is extreme complex because of the large number of enterprises and the frequent business cooperation among them. The multiple collaborations among the enterprises include supply, manufacture, sales, logistics and service. 3) Among these SMEs, disparities of levels of IT are extraordinary. Not only the various software vendors, but also the incompatibility of information technologies like system architecture, infrastructure, programming languages and so on, leads to the high heterogeneity of systems, which means it is infeasible to realize system interaction by means of point-to-point system interface. 4) Enterprises benefit from the variety of product range whereas suffering due to the various product naming and coding rules. 5) The business process and marketing strategy are often changed to response to market fleetly.

Every enterprise cooperates with elses in one or more business fields by means of particular system. In the region, usually more than two enterprises participate in intercollaboration process. Taking account of product performance, price and delivery, the manufacturer could spill the order and purchase from a variety of sources. In this situation these systems of enterprises involved co-operate with others to realize the procurement. There are three obstacles existing in the business collaboration between enterprises:1) the business object model differs from system to system, and employee of different enterprises and departments differs in many aspects, like domain engaged, post occupied, education level, expression ability and so on, contributing to the different understanding and description of single concept. When enterprise cooperates with others, there might exist something ambiguous or even conflicting. 2) service adopts an open and standard method to shield the system-level heterogeneity, but the method only stipulate the specific grammar format of the interface, like service operations, inputs and outputs, lacking of semantic description of service features and functionality. Since the services aren't described accurately, it's difficult to meet the requirements of service discovery and combination. 3) the cooperative enterprise is chosen until the business collaboration begins, thus the inter-enterprise business process can't be preset. And enterprises differ in many aspects like business processes, organizations and management decisions, endowing business collaboration with the characteristics of high dynamic and randomness.

3

The Framework of SI for BC

In order to solve the problems of semantic inconsistency throughout the process of system interoperation, a multi-facet ontology system supporting business collaboration is introduced. The ontology system consists of shard top-level ontology, domain ontology, local ontology and service ontology. Shard top-level ontology, describing the general concept and the relationship between concepts, is domain independent. It coordinates the relationship among else ontologies to maintain the concept consistency. Domain ontology and local ontology are both introduced to express the business concepts and relationships between concepts related in business collaboration. They differ in different objects, one for a specific domain and the other for concrete enterprise application. Service ontology is used to describe the concepts related with service and relationships. It gives the description of service attribute, functionality and interactive information. OWL-S [8] is adopted to describe the ontology information, since the model OWL-S of semantic description mainly focuses on the service functionality and is lack of service non-functional description. So the ServiceProfile ontoloty model of OWL-S is extended to express non-functional features like quality, expense, valuation. ServiceModel and ServiceGrouding are inherited to express basic attribute set and interface set.

Based on the building of ontology system, a framework of semantic interoperability for business collaboration is proposed, as shown in Fig. 1. There are three parts in the framework: the ontology system and the rules used for semantic conflicts resolution, the service repository based on the encapsulation of enterprise system and semantic enhancement, the dynamic combination of inter-enterprise process and the rules used for enterprise process conflicts resolution. The three parts dissolve the semantic inconsistency gradually in data level, system level and business process level corresponding to the obstacles met in the process of system interoperation. The three levels are progressive and the lower level of interoperability is the foundation of the upper level. The Key Technologies of SI for BC

The construction of ontology for Business Collaboration

The common method of building ontology [START_REF] Noy | Ontology development 101: A guide to creating your first ontology[END_REF] theoretically is correct, but the heavy workload and long cycle become prohibitive to SMEs in China, and it requires the ontology constructors having a good understanding of this domain. Consequently, a practical method for quickly building ontology is necessary.

(1) Local ontology Construction In the process of ontology construction on basis of structured data, the relational model (such as ER chart) can be obtained from database structure. The concepts, relationship, attributes, attribute type, primary key and foreign key can be extracted from the relational model and be transformed into corresponding rules, such as relationship are mapped to concept in the ontology, namely owl:Class, attribute are mapping to owl:Datatypeproperty, etc. These mapping rules are stored in the database for each enterprise. Under the guidance of the mapping rules, local ontology can be quickly built from databases of every enterprise.

Since there are a great number Semi-structured data, like XML files, which can be obtained from enterprise systems. Considering the regular corresponding relations between XML Schema and the description language of OWL [START_REF] Rahm | A survey of approaches to automatic schema matching[END_REF][START_REF] Bohring | Mapping XML to OWL ontologies[END_REF], utilizing these XML files to construct local ontology appears feasible. Firstly, by parsing the XML document, extracting structural information and its data content, generate the corresponding XML Schema and XML Data file respectively. Based on the detailed mapping relations between the elements of XML Schema and OWL [START_REF] Bohring | Mapping XML to OWL ontologies[END_REF], XSLT template is defined to transform the XML Schema elements to OWL elements. After the XML Data file is transformed to ontology instances, the ontology model and ontology instances are merged together to generate final ontology files.

Unstructured data is data without fixed structure and the method is similar with the common ontology construction method, it's too time-consuming and workload-heavy to build a local ontology in this way for an enterprise.

(2) Domain ontology building Since the creation of domain ontology demands a mass of workload, and it's practically impossible for domain experts to finish it solely. The essence of Web 2.0 is using the internet to publish and share the information. Inspired by this, we utilize Web 2.0 technology to realize self-building of domain knowledge, only if a valid mechanism for information organization and management is adopted. In such case, the domain ontology is more effectual because it comes from the public and takes full advantage of the wisdom of them. This paper introduces a mature business domainindependent ontology as the basis, and the users in various regions can create and maintain the domain ontology by extending ontology content continuously, then domain experts or professionals review and normalize the concepts and knowledge enriched by different enterprise users, consequently forming a domain ontology shared by all users building.

The resolution of semantic conflicts in enterprise interoperability

Semantic conflict can be divided into data-layer conflict and scheme-layer conflict. The former is caused by the different perception of the same concept and data value by different people, such as in naming, precision and representation, while the latter is caused by the differences of logic structure employed to express a same concept and data, including attribute set conflict, summarized conflict, polymerization conflict, associated conflict, etc. The process of semantic conflicts resolution is described below. First, the conflict list is generated by judging whether there are semantic conflicts exists between the collaborators. Then the mapping rules between the domain ontology and local ontologies are established for every collaborator respectively. After connecting the mapping rules of two collaborators together, the mapping specification is obtained. Thus, after the completion of the enterprise local ontology and domain ontology construction, how to establish the mapping between enterprise local ontology and domain ontology became a critical problem for the semantic conflicts resolution.

The multi-strategy mapping method first uses the enterprise local ontology and domain ontology as well as their respective data instances to train the learning classifier. According to classification results, a similarity measure algorithm is used to calculate the similarity of each pair of concept nodes, to determine the relative position of the concept in the other ontology. Then, based on HowNet, after using the method proposed by Ni, Wang etc. [START_REF] Ni | A Heterogeneous System Integration Framework for Business Collaboration[END_REF] to calculate the similarity of concept definition and utilizing similarity evaluation function and to assess the similarity, the similarity matrix is gained. Finally knowledge of domain axiomatic constraints is applied to assess the similarity and the mapping rules are generated. The mapping relations, higher than threshold value settled are stored in the mapping rules library. Most of the data-layer and scheme-layer conflicts can be resolved through the mapping rules, also permitted to be created manually for others conflicts.

On-demand inter-enterprise business processes construction

Since the inter-enterprise business process has the characteristics of high dynamic and randomness, service engine and service mediation are put forward to realize the service combination in two steps. First, requirements are annotated and divided into several structural sub-needs, describing detailed contents like product category, product name, quantity, delivery. The logic and constraint relations among the subneeds are recorded. Secondly, based on the description of sub-need, the service engine searches service repository to generate candidate services sets and evaluates the services combination. A multi-object genetic algorithm is employed to meet specific goals such as quality-preferred or delivery-preferred. After the optimal services portfolio are chosen, the relationship between requirements and service set is established. This process only concentrates on the temporal and logical relationships, hierarchical constraint of services from the functional and non-functional requirements of service, without considering the service provider, the actual business process. Next, service mediation analyze the actual process of every business interactions to generate the inter-enterprise business process. Finally, based on the relationship between requirements and service set and the inter-enterprise business process for each interaction, the on-demand inter-enterprise business processes is constructed by service mediations, as depicted in Fig. 2. Every enterprise needs to define its business processes, and the platform attempts to link the business processes of collaborative enterprise up to detect whether conflict exists. Once conflict is found, it implies related enterprises to resolve the conflicts by creating mapping rules or collaborative business process building. During the process of generating the inter-enterprise business process for each service interaction, owing to the different business process model and the description, the inter-enterprise business process should be constructed based on the business process mapping specification to resolve the inconsistency of business process. Based above, after dividing the requirements into sub-needs and mapping the sub-needs into service sets, the dynamic inter-enterprise business process is achieved by the combination of enterprise business process according to every service interaction of the service set.

Application and Conclusion

Based on the research contents, a platform based on J2EE framework, Web Services and OWL-S 1.1 specification is developed. The system mainly includes modules of ontology building and maintenance, service register center, business process model editing, conflict mapping rules management, workflow engine and so on. Table 1 shows the auto-generated mapping specification between the two enterprises' local ontologies. Fig. 3 shows an instantiation of the cross-enterprise business processes constructed. When manufacturer A publishes the demand as "products X5000 in 2-3 days" and chooses its suppliers, the platform detects the business process conflicts. Based on the mapping rules, both the supplier B and logistics services providers C add a new process as "send order confirmation" in order to comply with the process of A. This platform has been partly applied to the representative Liushi low-voltage apparatus industry cluster in Zhejiang Province of China and proved to be workable. Through this platform, systems of different enterprises can interoperate with others and promote the efficiency of the business cooperation, especially among the core enterprise of the supply chain and the upstream and downstream enterprises surrounding it. But it still needs lots of related prerequisite work and manual intervention during the process of business collaboration, like business process model building, local ontology building, leading to the restriction of the application scope. The future work is to increase automation and reduce the workloads of enterprises.

Fig. 1 .

 1 Fig. 1. The framework of semantic interoperability for business collaboration

Fig. 2 .

 2 Fig. 2. On-demand inter-enterprise business processes construction.

Fig. 3 .

 3 Fig. 3. Instantiation of inter-enterprise business process construction

Table 1 .

 1 The mapping specification between local ontologies

	Local Ontology A	Domain general ontology	Local Ontology B
	Category: low-	Category:	low-voltage	Category: low-
	voltage apparatus		apparatus		voltage apparatus
	Attribute :Pro-	Attribute :Product name	Attribute :Prod-
	duct name						uct description
	Attribute:			Attribute: Product model	Attribute:
	Product code					Product NO.
	Attribute: cost	Attribute: unit cost	Attribute: unit
	per unit						price
	Attribute:			Attribute: number	Attribute:
	number						quantity
	Logistics service provider C					Receive order	… …
	Requester A	Create order	Send order		Receive confirmation	Receive request	Create order	Send freight order
	Supplier B Produced	Receive order	Send confirmation	…	Send shipping request
	product X