O H Miyagaki
email: ohmiyagaki@gmail.com

D Motreanu
email: motreanu@univ-perp.fr

F R Pereira
email: fabio.pereira@ufjf.edu.br

MULTIPLE SOLUTIONS FOR A FRACTIONAL ELLIPTIC PROBLEM WITH CRITICAL GROWTH

Keywords: Nonlocal elliptic equations, variational methods, fractional critical growth, asymmetric nonlinearities, Palais-Smale condition

The paper focuses on a nonlocal Dirichlet problem with asymmetric nonlinearities. The equation is driven by the fractional Laplacian (-∆) s for s ∈ (0, 1) and exhibits a sublinear term containing a parameter λ, a linear term interfering with the spectrum of (-∆) s and a superlinear term with fractional critical growth. The corresponding local problem governed by the standard Laplacian operator was investigated by F. O. de Paiva and A. E. Presoto. It can be recovered by letting s ↑ 1. The statement given here in the nonlocal setting is also related to extensively studied topics for local elliptic operators as the Brezis-Nirenberg problem and asymmetric nonlinearities. We go beyond the case of the standard Laplacian taking advantage of recent contributions on nonlocal fractional equations. Our main result establishes the existence of at least three nontrivial solutions, with one nonnegative and one nonpositive, provided the parameter λ > 0 is sufficiently small. In order to overcome the difficulties in the nonlocal setting we develop new arguments

Problem Statement and Introduction

Let Ω ⊂ R N , with N ≥ 3, be a bounded domain whose boundary ∂Ω is Lipschitz continuous. In the present paper we focus on the nonlocal elliptic problem

(-∆) s u = -λ|u| q-2 u + au + b(u + ) 2 * s -1 in Ω u = 0 in R N \ Ω (1)
driven by the (negative) fractional Laplacian (-∆) s with some s ∈ (0, 1). We recall that (-∆) s is the integro-differential operator defined pointwise by

(-∆) s u(x) := C(N, s) lim ε↓0 R N \ Bε(x) u(x) -u(y) |x -y| N +2s dy, x ∈ R N ,
where B ε (x) denotes the open ball centered at x and of radius ε > 0, and C(N, s) is the normalization constant

C(N, s) = R N 1 -cos(x 1 ) |x| N +2s dx -1
.

From [START_REF] Di Nezza | Hitchhiker's guide to the fractional Sobolev spaces[END_REF]Proposition 4.4] it is known that lim s→1 (-∆) s u = -∆u for every u ∈ C ∞ 0 (R N ). A comprehensive presentation of nonlinear elliptic equations involving nonlocal diffusion is given in [START_REF] Caffarelli | Non-local diffusions, drifts and games[END_REF] and [START_REF] Bisci | Variational methods for nonlocal fractional problems[END_REF].

In the statement of problem [START_REF] Di Nezza | Hitchhiker's guide to the fractional Sobolev spaces[END_REF], the data have the following meaning: λ > 0 is a parameter, q ∈ (1, 2), a > 0 and b > 0 are given constants, and 2 * s = 2N/(N -2s) (the fractional critical exponent). The notation u + stands for the positive part of u, that is, u + = max{u, 0}, while the negative part of u will be denoted u -= max{-u, 0}.

We recall that the Gagliardo seminorm of a measurable function u : R N → R is defined by

[u] s := R 2N |u(x) -u(y)| 2
|x -y| N +2s dx dy

1/2
provided the integral is finite. The fractional Sobolev space H s (R N ) is introduced as H s (R N ) := {u ∈ L 2 (R N ) : [u] s < ∞} endowed with the norm u H s = ( u 2 L 2 + [u] 2 s ) 1/2 making it a Hilbert space. Since statement (1) is subject to the homogeneous Dirichlet boundary condition on Ω, the relevant space associated to this problem is the closed vector subspace of H s (R N ) given by

X s 0 (Ω) := {u ∈ H s (R N ) : u = 0 a.e. in R N \ Ω}, (2)
which is a Hilbert space with the scalar product (u, v)

X s 0 := R 2N (u(x) -u(y))(v(x) -v(y)) |x -y| N +2s dx dy (3)
inducing the equivalent norm

• X s 0 = [ • ] s .
In the present paper, the essential condition that we suppose is λ k,s < a < λ k+1,s , where {λ j,s } j≥1 represents the sequence of eigenvalues of (-∆) s on the space X s 0 (Ω). Consequently, there is interaction with the spectrum of ((-∆) s , X s 0 (Ω)) leading eventually to multiple solutions for problem [START_REF] Di Nezza | Hitchhiker's guide to the fractional Sobolev spaces[END_REF].

The local problem corresponding to (1) was studied by de Paiva-Presoto [START_REF] De Paiva | Semilinear elliptic problems with asymmetric nonlinearities[END_REF], where the driving operator is the standard Laplacian on the Sobolev space H 1 0 (Ω). We prove for problem [START_REF] Di Nezza | Hitchhiker's guide to the fractional Sobolev spaces[END_REF] the nonlocal counterpart of the main result in [START_REF] De Paiva | Semilinear elliptic problems with asymmetric nonlinearities[END_REF] that is recovered letting s ↑ 1. In order to go beyond the case of the ordinary Laplacian on H 1 0 (Ω), here we develop an approach that is to a large extent completely different. This is natural in view of the essential differences in the functional setting related to the underlying spaces (H 1 0 (Ω) versus X s 0 (Ω)) and driving operators (the standard negative Laplacian -∆ versus the integro-differential operator (-∆) s ). Actually, the entire calculus involved in the local and nonlocal arguments are, in most cases, distinct. We illustrate how far are the two frameworks with the following simple hint. A basic tool in [START_REF] De Paiva | Semilinear elliptic problems with asymmetric nonlinearities[END_REF], as well as in other papers in the same area (see, e.g., [START_REF] Calanchi | Elliptic equations with one-sided critical growth[END_REF]), is that any two functions, say u and v, having disjoint supports are orthogonal in H 1 0 (Ω), that is (u, v)

H 1 0 = Ω ∇u • ∇v dx = 0.
This is no longer true in X s 0 (Ω) because, for the functions u and v with supp(u) ∩ supp(v) = ∅, a straightforward calculation of their scalar product in X s 0 (Ω) shows (u, v)

X s 0 = R 2N
(u(x) -u(y))(v(x) -v(y)) |x -y| N +2s dx dy = -Ω×Ω u(x)v(y) |x -y| N +2s dx dy.

We note that in [START_REF] De Paiva | Semilinear elliptic problems with asymmetric nonlinearities[END_REF] was used the approximate eigenfunctions technique developed by Gazzola-Ruf [START_REF] Gazzola | Lower-order perturbations of critical growth nonlinearities in semilinear elliptic equations[END_REF] for problems with local operators. In our cases it is not possible to employ any more the same idea, so it was necessary to build new methods that are suitable for the nonlocal framework.

We describe a few more connections of our work with the existing literature. First, we mention some relevant references addressing elliptic equations with critical nonlinearities as occur in problem [START_REF] Di Nezza | Hitchhiker's guide to the fractional Sobolev spaces[END_REF]. A prototype for such equations is the problem of Brezis-Nirenberg [START_REF] Brezis | Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents[END_REF]:

-∆u = µu + u|u| 2 * -2 in Ω u = 0 on ∂Ω, (4)
with a real parameter µ and the critical Sobolev exponent 2 * = 2N/(N -2) if N ≥ 3. The nontrivial solvability of problem (4) for every µ > 0 and N ≥ 5 (N ≥ 4 when µ does not belong to the spectrum of -∆ on H 1 0 (Ω)) was shown in Capozzi-Fortunato-Palmieri [START_REF] Capozzi | An existence result for nonlinear elliptic problems involving critical Sobolev exponent[END_REF]. An existence theorem for a version of (4) on the whole space R N can be found in Chabrowski-Yang [START_REF] Chabrowski | Existence theorems for the Schrödinger equation involving a critical Sobolev exponent[END_REF]. Observe that the critical nonlinearity in our problem (1) is asymmetric being expressed through the positive part u + of the solution u. We mention that, in the local setting, the study of equations with critical exponent and asymmetric nonlinearities was initiated by De Figueiredo-Yang [START_REF] De Figueiredo | Critical superlinear Ambrosetti-Prodi problems[END_REF] to investigate Ambrosetti-Prodi type problems involving critical growth (see also [START_REF] Calanchi | Elliptic equations with one-sided critical growth[END_REF]). The nonlocal counterpart of problem ( 4) is obtained by replacing -∆ with the fractional Laplacian (-∆) s for s ∈ (0, 1) (or likewise nonlocal operator) and substituting the boundary condition u = 0 on ∂Ω with u = 0 on R N \Ω. This was done in Servadei-Valdinoci [START_REF] Servadei | The Brezis-Nirenberg result for the fractional Laplacian[END_REF], and the existence result corresponding to [START_REF] Capozzi | An existence result for nonlinear elliptic problems involving critical Sobolev exponent[END_REF] in the nonlocal setting is due to Servadei [START_REF] Servadei | The Yamabe equation in a non-local setting[END_REF].

Since equation ( 1) contains a sublinear term |u| q-2 u and a (critical) superlinear term (u + ) 2 * s -1 , it belongs to the class of problems with competing nonlinearities, for instance sublinear-superlinear, whose study started with Ambrosetti-Brezis-Cerami [START_REF] Ambrosetti | Combined effects of concave and convex nonlinearities in some elliptic problems[END_REF]. In this direction, for the nonlocal setting we cite Dipierro-Medina-Valdinoci [START_REF] Dipierro | Fractional elliptic problems with critical growth in the whole of R n[END_REF] whose approach relies on Caffarelli-Silvestre [START_REF] Caffarelli | An extension problem related to the fractional Laplacian[END_REF], where it is shown that the fractional Laplacian (-∆) s on R N can be constructed through the ordinary Laplacian on R N × [0, +∞).

We point out that our assumption λ k,s < a < λ k+1,s forces eigenvalue crossing for equation [START_REF] Di Nezza | Hitchhiker's guide to the fractional Sobolev spaces[END_REF] with respect to the spectrum of the fractional Laplacian ((-∆) s , X s 0 (Ω)). Eventually, this gives rise to the existence of multiple solutions. In [START_REF] De Paiva | Semilinear elliptic problems with asymmetric nonlinearities[END_REF], the eigenvalue crossing holds with respect to the spectrum of the ordinary Laplacian (-∆, H 1 0 (Ω)). Our results cover those in [START_REF] De Paiva | Semilinear elliptic problems with asymmetric nonlinearities[END_REF] taking into account that (-∆, H 1 0 (Ω)) is retrieved from ((-∆) s , X s 0 (Ω)) as s ↑ 1. The main result in the paper, which is stated as Theorem 13, establishes under the assumptions N ≥ 4s and λ k,s < a < λ k+1,s that the nonlocal problem (1) admits at least three nontrivial distinct solutions provided the parameter λ > 0 is sufficiently small. The hypothesis N ≥ 4s is in line with the specific character of the critical problems in the local case for low dimensions (see [START_REF] Brezis | Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents[END_REF][START_REF] Capozzi | An existence result for nonlinear elliptic problems involving critical Sobolev exponent[END_REF][START_REF] De Figueiredo | Critical superlinear Ambrosetti-Prodi problems[END_REF]). The conclusion can still be obtained for 2s < N < 4s if in addition the integer k in hypothesis λ k,s < a < λ k+1,s is sufficiently large. Moreover, we prove in Theorem 6 that assuming N > 2s and λ k,s < a < λ k+1,s , there exists a nontrivial nonpositive solution for every λ > 0, and if λ > 0 is below a threshold determined by the Palais-Smale condition there exists also a nontrivial nonnegative solution. The existence of a third nontrivial solution different from the two solutions of opposite constant sign arises by further diminishing the parameter λ > 0.

The approach in our results is variational in accordance with the variational structure of problem [START_REF] Di Nezza | Hitchhiker's guide to the fractional Sobolev spaces[END_REF]. As it is common for problems with critical nonlinearities, the lack of compactness implies that the Palais-Smale condition is not generally satisfied. For the Euler functional I λ,s : X s 0 (Ω) → R associated to problem (1) (in the notation it is highlighted the dependence on the parameter λ > 0), we are able to prove that every Palais-Smale sequence is bounded and to find precisely the level below which the Palais-Smale condition is fulfilled. Two solutions of opposite constant sign are obtained by applying Mountain Pass Theorem to appropriately truncated functionals related to I λ,s . The existence of local minimizers needed to have the mountain pas geometry is derived by using the property of H s versus C 0 s minimizers proved in Iannizzotto-Mosconi-Squassina [START_REF] Iannizzotto | H s versus C 0 -weighted minimizers[END_REF].

The third nontrivial solution is shown through a minimax result of Generalized Mountain Pass Theorem type complying with the hypothesis λ k,s < a < λ k+1,s . The underlying minimax structure is based on the finite dimensional space V s k spanned by the eigenfunctions of ((-∆) s , X s 0 (Ω)) corresponding to the the first k eigenvalues {λ 1,s , . . . , λ k,s } and on the orthogonal projection e ε on (V s k ) ⊥ of an element u ε ∈ X s 0 (Ω) depending on ε > 0, which comes from the realization of the best constant for the embedding

H s (R N ) → L 2 *
s (R N ) as obtained by Cotsiolis-Tavoularis [START_REF] Cotsiolis | Best constants for Sobolev inequalities for higher order fractional derivatives[END_REF]. Corresponding to numbers ε, R 1 , R 2 > 0, we produce precise estimates of the values I λ,s (u 1 + re ε ) for u 1 ∈ V s k with u 1 X s 0 ≤ R 1 and for 0 ≤ r ≤ R 2 . The estimates enable us to deduce that the desired linking configuration is achieved provided λ > 0 is sufficiently small (see Proposition 8). These estimates are the result of taking into account the interaction of ε > 0 sufficiently small with R 1 > 0 and R 2 > 0 sufficiently large in the relevant expressions.

Here the condition N ≥ 4s is necessary to be imposed. The remaining case 2s < N < 4s cannot be handled with just ε > 0 sufficiently small as can be managed when N ≥ 4s being necessary to add the condition that the integer k is sufficiently large. For the needs of a rigorous proof, we carefully reduce the analysis to discuss situations as ε < 1/R 1 and ε < R 1 /R 2 . An important tool that we use in this part is a device set up by De Figueiredo-Yang [START_REF] De Figueiredo | Critical superlinear Ambrosetti-Prodi problems[END_REF] for the asymmetric critical nonlinearity settled with the positive part of the solution (see also Chabrowski-Yang [START_REF] Chabrowski | Multiple solutions of a nonlinear elliptic equation involving Neumann conditions and a critical Sobolev exponent[END_REF] for its utilization).

The next step in the proof is to ensure that the minimax value of I λ,s obtained along the linking theorem is smaller than the admissible threshold for the Palais-Smale condition. It is achieved in Lemma 11 by taking both λ > 0 and ε > 0 sufficiently small. New ideas were necessary to estimate from above the integral

Ω ((u 1 + re ε ) + ) 2 * s dx,
amongst which is the use of an adequate equivalent expression of the vector u 1 + re ε ∈ X s 0 (Ω). In this process, the crucial point is to show a sufficiently small upper bound for the quotient

C(N,s) 2 e ε 2 X s 0 -a e ε 2 L 2 u ε 2 L 2 * s (5)
whenever ε > 0 is sufficiently small. We emphasize that contrary to all the former works on similar topics, the expression of (5) contains the two vectors u ε and e ε . The study of the dependence of the quotient (5) with respect to ε guaranteeing the desired upper bound for ε > 0 small enough is worked out on the basis of recent formulas in [START_REF] Servadei | The Brezis-Nirenberg result for the fractional Laplacian[END_REF][START_REF] Servadei | The Yamabe equation in a non-local setting[END_REF] that are analogous to the ones proved in the local case in [START_REF] Brezis | Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents[END_REF]. Finally, by comparing the minimax values, we conclude that the three nontrivial solutions are distinct in the case where the parameter λ > 0 is sufficiently small.

The rest of the paper is organized as follows. Section 2 contains preliminary material necessary in the sequel. Section 3 sets forth the results regarding the Palais-Smale condition. Section 4 presents the existence of two nontrivial constant sign solutions of opposite sign. Section 5 describes the linking geometry to cast the third nontrivial solution. Section 6 is concerned with the estimate of the minimax value determined by the linking theorem. Section 7 provides our multiplicity result for problem (1).

Preliminaries

For easy reference, we recall a few notions and results that will be used later on. We refer to [1, Proposition 3.6]) for the equality

[u] s = 2C(N, s) -1 R N |(-∆) s 2 u| 2 dx 1/2 , ∀ u ∈ H s (R N ).
Based on it, (2) and (3) imply that for all u, v ∈ X s 0 (Ω) there holds

2C(N, s) -1 R N u(x)(-∆) s v(x) dx = R 2N (u(x) -u(y))(v(x) -v(y)) |x -y| N +2s dx dy. (6)
In particular, it follows that the linear operator (-∆) s is self-adjoint on X s 0 (Ω). The spectrum of (-∆) s on X s 0 (Ω) consists of a sequence of eigenvalues (λ j,s ) satisfying 0 < λ 1,s < λ 2,s ≤ λ 3,s ≤ . . . ≤ λ j,s ≤ λ j+1,s ≤ . . . , λ j,s → ∞ as j → ∞, where each eigenvalue λ j,s is repeated with its multiplicity. Every eigenfunction ϕ j,s has the regularity ϕ j,s ∈ C 0,σ (Ω) with some σ ∈ (0, 1). For the rest of the paper we fix a sequence (ϕ j,s ) of eigenfunctions forming an orthonormal basis in L 2 (Ω) and an orthogonal basis in X s 0 (Ω). Furthermore, we can choose ϕ 1,s > 0.

The embedding

H s (R N ) → L r (R N ) is continuous for r ∈ [2, 2 * s ]
(see, e.g., [START_REF] Di Nezza | Hitchhiker's guide to the fractional Sobolev spaces[END_REF]Theorem 6.5] or [START_REF] Demengel | Functional spaces for the theory of elliptic partial differential equations[END_REF]Chapter 4]), so the constant

S s := inf u∈H s (R N ), u L 2 * s =1 R 2N |u(x) -u(y)| 2 |x -y| N +2s dx dy (7) is finite. It is known from [17, Theorem 1.1] that S s in (7) is attained by the function (1/ ũ L 2 * s )ũ with ũ(x) = (1 + |x| 2 ) -N -2s 2 , x ∈ R N . (8)
On the basis of (8), we shall use the family of functions {U ε } ε>0 introduced in [11, p. 91] as

U ε (x) = ε -N -2s 2 1 ũ L 2 * s ũ x εS 1/(2s) s , x ∈ R N . (9)
The lemma below will be important in our analysis of problem (1).

Lemma 1. Let K > 0 and β ∈ [0, (N -2s)/2) be constants. Then, for every ε > 0, the following inclusion holds

D ε := x ∈ Ω : U ε (x) > K ε β ⊃ B ρ(ε) (0) with ρ(ε) = C(K)ε N -2s+2β 2(N -2s) ,
where C(K) is a positive constant depending on K. Moreover, one has the estimate

Dε U 2 * s ε dx ≥ B ρ(ε) (0) U 2 * s ε dx ≥ S N 2s s 2
for ε > 0 sufficiently small.

Proof. By ( 8) and ( 9)

, if x ∈ D ε , it is seen that x εS 1/(2s) s N -2s ≤ 1 + x εS 1/(2s) s 2 N -2s 2 ≤ C 0 K ε β-N -2s 2 ,
with a constant C 0 > 0. It follows that

D ε ⊂ B ρ(ε) (0) with ρ(ε) = C(K)ε N -2s+2β 2(N -2s) .
On the other hand, D ε is open, and by the choice of β we have that 0 ∈ D ε with ε > 0 small enough. Hence, for a sufficiently small constant C(K) > 0, we achieve the inclusion for D ε given in the statement of the lemma. The first part of the lemma, ( 8), ( 9), formula (4.13) in [START_REF] Servadei | The Brezis-Nirenberg result for the fractional Laplacian[END_REF] and change of variable with polar coordinates imply

Dε U 2 * s ε dx ≥ B ρ(ε) (0) U 2 * s ε dx = R N U 2 * s ε dx - R N \B ρ(ε) (0) U 2 * s ε dx = S N 2s s -C 1 +∞ cε 2(β+s)-N 2(N -2s) r N -1 (1 + r 2 ) N dr → S N 2s
s as ε → 0, with constants C 1 > 0 and c > 0. This proves the second part of the lemma.

In our development we need an auxiliary result that appears in [10, Lemma 2.5] when s = 1. The proof is similar for s ∈ (0, 1).

Lemma 2. Given u, v ∈ L p (Ω) with 2 ≤ p ≤ 2 * s and u + v > 0 a.e. on a measurable subset Σ ⊂ Ω, it holds Σ (u + v) p dx - Σ |u| p dx - Σ |v| p dx ≤ C Σ (|u| p-1 |v| + |u||v| p-1 ) dx,
with a constant C > 0 depending only on p.

Our variational approach for studying problem (1) relies on the linking minimax result in [START_REF] Rabinowitz | Minimax methods in critical point theory with applications to differential equations[END_REF]Theorem 5.3] (see also [START_REF] Willem | Minimax Theorems[END_REF]Theorem 2.12]), which is a generalization of Mountain Pass Theorem. For clarity, we state it.

Theorem 3. Let Y be a real Banach space satisfying Y = V ⊕ W with V finite dimensional. Suppose that the functional I ∈ C 1 (Y, R) fulfills the conditions: (i) there are constants ρ, α > 0 such that I | ∂Bρ∩W ≥ α, where B ρ denotes
the open ball centered at zero and of radius ρ;

(ii) there are constants R 1 , R 2 > ρ, β < α and a nonzero vector e ∈ W such that I | ∂Q ≤ β, with

Q = {v + re : v ∈ B R 1 ∩ V, 0 < r < R 2 }.
Then I possesses a Palais-Smale sequence at the level c ≥ α determined by

c = inf h∈Γ max u∈Q I(h(u)),
where

Γ = {h ∈ C(Q, Y ) : h = id on ∂Q}.
We recall that (u n ) ⊂ Y is a Palais-Smale (in short, (P S)) sequence for the functional

I ∈ C 1 (Y, R) at the level r ∈ R if I(u n ) → r as n → ∞, and | I (u n ), ϕ Y | ≤ ε n ϕ Y for all ϕ ∈ Y , where ε n → 0 as n → ∞.

The (PS) Condition for the Euler Functional

As mentioned in Introduction, the underlying space for problem (1) is X s 0 (Ω). The corresponding Euler functional I λ,s : X s 0 (Ω) → R is given by

I λ,s (u) = C(N, s) 4 R 2N (u(x) -u(y)) 2 |x -y| N +2s dx dy + λ q Ω |u| q dx - a 2 Ω u 2 dx - b 2 * s Ω (u + ) 2 * s dx, (10)
which is of class C 1 . In this section we study the (P S) sequences of the functional I λ,s and the related (P S) condition.

Lemma 4. Assume that λ > 0, 1 < q < 2, a = λ 1,s , and b > 0. Then every (P S) sequence of I λ,s is bounded.

Proof. If a < λ 1,s , the proof is easier and we omit it. Suppose that a > λ 1,s . Let (u n ) ⊂ X s 0 (Ω) be a (P S) sequence for I λ,s , which means that

|I λ,s (u n )| ≤ C, with a constant C > 0, and | (I λ,s ) (u n ), ϕ X s 0 | ≤ ε n ϕ X s 0 for all ϕ ∈ X s 0 (Ω)
, where ε n → 0 as n → ∞. We have that

I λ,s (u n ) - 1 2 (I λ,s ) (u n ), u n X s 0 = λ 1 q - 1 2 Ω |u n | q dx + b 1 2 - 1 2 * s Ω (u + n ) 2 * s dx, from which we can derive b 1 2 - 1 2 * s Ω (u + n ) 2 * s dx ≤ C + n u n X s 0 . (11)
We note that

(I λ,s ) (u n ), u - n X s 0 = - C(N, s) 2 u - n 2 X s 0 - C(N, s) 2 R 2N u + n (x)u - n (y) + u + n (y)u - n (x) |x -y| N +2s dx dy -λ Ω (u - n ) q dx + a Ω (u - n ) 2 dx. ( 12
) Using that |I λ,s (u n )| ≤ C, it holds C(N, s) 4 u n 2 X s 0 + λ q Ω |u n | q dx - a 2 Ω u 2 n dx - b 2 * s Ω (u + n ) 2 * s dx ≤ C + λ 2 Ω |u n | q dx - Ω (u - n ) q dx .
Hence we find that

C(N, s) 2 R 2N u + n (x)u - n (y) + u + n (y)u - n (x) |x -y| N +2s dx dy + C(N, s) 4 u + n 2 X s 0 + C(N, s) 4 u - n 2 X s 0 (Ω) - b 2 * s Ω (u + n ) 2 * s dx ≤ λ 2 Ω |u n | q dx - λ q Ω |u n | q dx - λ 2 Ω (u - n ) q dx + a 2 Ω (u + n ) 2 dx + a 2 Ω (u - n ) 2 dx + C.
Taking into account that λ > 0, 1 < q < 2, as well as ( 12), we get

C(N, s) 4 u + n 2 X s 0 ≤ a 2 Ω (u + n ) 2 dx + b 2 * s Ω (u + n ) 2 * s dx + 1 2 | I λ,s (u n ), u - n X s 0 | + C. Since 2 < 2 * s , there exists a constant C 1 > 0 such that Ω (u + n ) 2 dx ≤ C 1 1 + Ω (u + n ) 2 * s dx , ∀ n.
We infer that

C(N, s) 4 u + n 2 X s 0 ≤ C 2 Ω (u + n ) 2 * s dx + 1 2 | I λ,s (u n ), u - n X s 0 | + C 3 , (13)
with positive constants C 2 and C 3 . Then [START_REF] Servadei | The Brezis-Nirenberg result for the fractional Laplacian[END_REF] and ( 13) ensure that

1 2 u + n 2 X s 0 ≤ K 1 u n X s 0 + K 2 , (14)
with constants K 1 > 0 and K 2 > 0.

We are going to prove that the sequence (u

+ n ) is bounded in X s 0 (Ω). Admit by contradiction that up to a subsequence u + n X s 0 → ∞. From (14) it turns out that u - n X s 0 → ∞. Knowing that u n X s 0 → ∞, we can set v n = u n / u n X s 0 . Along a relabeled subsequence, we may assume that v n v in X s 0 (Ω), v n → v a.e x ∈ Ω, v n → v in L 2 (Ω)
, with some v ∈ X s 0 (Ω). We can express [START_REF] Dipierro | Fractional elliptic problems with critical growth in the whole of R n[END_REF] in the form

1 2 ≤ K 1 u + n X s 0 + K 1 u - n X s 0 u + n 2 X s 0 + K 2 u + n 2 X s 0 . Then, since u + n X s 0 → ∞, there exists n 0 ∈ N such that u - n X s 0 ≥ 1 4K 1 u + n 2 X s 0 , ∀ n ≥ n 0 , (15)
whence, combining with u -

n X s 0 ≥ 1 for n large enough (note u - n X s 0 → ∞), it follows v + n X s 0 = u + n X s 0 u n X s 0 ≤ u + n X s 0 u + n 2 X s 0 + (1/(4K 1 )) 2 u + n 4 X s 0 1/2 ≤ 4K 1 u + n X s 0 . This implies that v + n → 0 in X s 0 (Ω) as n → ∞, so v = -v -≤ 0 because v + n → v + in L 2 (Ω). Furthermore, by (15) one obtains 4K 1 u - n X s 0 + 1 + 2 u - n 2 X s 0 R 2N (u + n (x)u - n (y) + u + n (y)u - n (x)) |x -y| N +2s dx dy -1/2 ≤ v - n X s 0 = u - n X s 0 u n X s 0 ≤ 1. (16)
Observe from [START_REF] Caffarelli | An extension problem related to the fractional Laplacian[END_REF] and the Cauchy-Schwarz inequality that

1 u - n 2 X s 0 R 2N (u + n (x)u - n (y) + u + n (y)u - n (x)) |x -y| N +2s dx dy = 1 u - n 2 X s 0 | u + n , u - n X s 0 | ≤ 4K 1 u + n X s 0 → 0 as n → ∞. (17)
On the basis of ( 17) and that u - n X s 0 → ∞, passing to the limit in ( 16) yields

v - n X s 0 → 1 as n → ∞. ( 18
)
Notice that ( 12) and ( 18) lead to

v - n 2 L 2 → C(N, s) 2a , which results in v L 2 = C(N, s) 2a = 0.
From ( 6) and the inequality (

I λ,s ) (u n ), ϕ 1,s X s 0 ≤ ε n ϕ 1,s X s 0 we can find (λ 1,s -a) Ω v n ϕ 1,s dx + λ u n 2-q X s 0 Ω |v n | q-2 v n ϕ 1,s dx - b u n X s 0 Ω (u + n ) 2 * s -1 ϕ 1,s dx ≤ ε n ϕ 1,s X s 0 u n X s 0 . ( 19
)
Using v n → v in L q (Ω) (note that q < 2), u n X s 0 → ∞, and estimate [START_REF] Servadei | The Brezis-Nirenberg result for the fractional Laplacian[END_REF], it is seen that ( 19) entails Ω vϕ 1,s dx = 0. Recalling that v ≤ 0 and ϕ 1,s > 0, we obtain v = 0, which is a contradiction. Therefore the sequence (u

+ n ) is bounded in X s 0 (Ω). Now we show that (u n ) is bounded in X s 0 (Ω). Arguing indirectly, sup- pose that u n X s 0 → ∞ as n → ∞.
On the basis of the boundedness of the sequence (u + n ) in X s 0 (Ω) and thanks to the continuity of the embedding

X s 0 (Ω) → L 2 * s (Ω), we get 1 u n X s 0 Ω (u + n ) 2 * s dx → 0 as n → ∞. (20)
On account of inequality [START_REF] Rabinowitz | Minimax methods in critical point theory with applications to differential equations[END_REF] and v n v in X s 0 (Ω) we have

| (I λ,s ) (u n ), ϕ | ≤ ε n ϕ X s 0 , through
C(N, s) 2 R 2N (v(x) -v(y))(ϕ(x) -ϕ(y)) |x -y| N +2s
dx dy = a Ω vϕ dx dy for all ϕ ∈ X s 0 (Ω).

According to [START_REF] Gazzola | Lower-order perturbations of critical growth nonlinearities in semilinear elliptic equations[END_REF] and since v = 0, we deduce that v is an eigenfunction of (-∆) s corresponding to the eigenvalue a, so a = λ 1,s because v ≤ 0 and the only eigenvalue with constant-sign eigenfunctions is λ 1,s . This comes to a contradiction with the hypothesis a > λ 1,s , which completes the proof.

We are able to indicate precisely the level where the (P S) condition (i.e., every (P S) sequence contains a strongly convergent subsequence) holds true for the functional I λ,s . Proposition 5. Assume that λ > 0, 1 < q < 2, a = λ 1,s , and b > 0. Then the functional I λ,s satisfies the (P S) condition at any level c <

s N C(N,s) 2 S s N 2s b 2s-N 2s .
Proof. Let (u n ) ⊂ X s 0 (Ω) be a (P S) sequence of I λ,s at the level c, thus

I λ,s (u n ) → c and | (I λ,s ) (u n ), ϕ X s 0 | ≤ ε n ϕ X s 0 for all ϕ ∈ X s 0 (Ω), with ε n → 0 as n → ∞.
By Lemma 4 we know that (u n ) is bounded in X s 0 (Ω). Consequently, passing to a subsequence if necessary, we may assume that

u n u in X s 0 (Ω), u n → u in L 2 (Ω), u n → u in L q (Ω) and u n → u for a.e x ∈ Ω, (21)
with u ∈ X s 0 (Ω). Therefore, along a relabeled subsequence, we have that

u + n u + in L 2 * s (Ω).
Then the linearity of the operator (-∆) s enables us to conclude that u is a weak solution of problem (1), thus u satisfies

I λ,s (u) = λ 1 q - 1 2 Ω |u| q dx + b 1 2 - 1 2 * s Ω (u + ) 2 * s dx ≥ 0. ( 22
)
Denoting

v n = u n -u, from [22, Theorem 1] we derive lim n→∞ u + n 2 * s L 2 * s -v + n 2 * s L 2 * s = lim n→∞ u + n 2 * s L 2 * s -u + n -u + 2 * s L 2 * s = u + 2 * s L 2 * s .
We note that

I λ,s (u) + I λ,s (v n ) = C(N, s) 2 u 2 X s 0 - C(N, s) 2 R 2N ((u n (x) -u n (y))(u(x) -u(y)) |x -y| N +2s dx dy + C(N, s) 4 u n 2 X s 0 + λ q u q L q + v n q L q - a 2 u 2 L 2 + v n 2 L 2 - b 2 * s u + 2 * s L 2 * s + v + n 2 * s L 2 * s .
Hence the convergence properties in [START_REF] Willem | Minimax Theorems[END_REF] imply

lim n→∞ [I λ,s (u) + I λ,s (v n )] = lim n→∞ I λ,s (u n ) = c. (23)
From ( 21), [22, Theorem 1] and since u is a solution of problem (1), we get

lim n→∞ C(N, s) 2 v n 2 X s 0 + λ v n q L q -a v n 2 L 2 -b v + n 2 * s L 2 * s = C(N, s) lim n→∞ u 2 X s 0 - R 2N ((u n (x) -u n (y))(u(x) -u(y)) |x -y| N +2s dx dy + lim n→∞ C(N, s) 2 u n 2 X s 0 + λ u n q L q -a u n 2 L 2 -b u + n 2 * s L 2 * s - C(N, s) 2 u 2 X s 0 + λ u q L q -a u 2 L 2 -b u + 2 * s L 2 * s = lim n→∞ (I λ,s ) (u n ), u n X s 0 -(I λ,s ) (u), u X s 0 = 0, whence lim n→∞ C(N, s) 2 v n 2 X s 0 -b v + n 2 * s L 2 * s = 0. ( 24
)
Fix a relabeled subsequence such that

lim n→∞ v n 2 X s 0 := l. Then (24) ensures lim n→∞ v + n 2 L 2 * s = C(N, s)l 2b 2 2 * s . ( 25
)
Suppose that l > 0. From [START_REF] Brezis | Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents[END_REF] we infer that v n

2 X s 0 ≥ S s v + n 2 L 2 * s , thereby through (25) it turns out l ≥ S N 2s s C(N, s) 2b N -2s 2s . (26)
Now ( 23), ( 22), ( 21) and (25) provide

c = lim n→∞ [I λ,s (u) + I λ,s (v n )] ≥ lim n→∞ I λ,s (v n ) = lim n→∞ C(N, s) 4 v n 2 X s 0 - b 2 * s v + n 2 * s L 2 * s = C(N, s)sl 2N . (27)
Then ( 26), ( 27) and the assumption on c in the statement generate the contradiction

s N C(N, s) 2 S s N 2s b 2s-N 2s ≤ C(N, s)sl 2N ≤ c < s N C(N, s) 2 S s N 2s b 2s-N 2s .
Therefore we must have l = 0, which proves the strong convergence of a subsequence of (u n ) in X s 0 (Ω). The proof is thus complete.

Two Constant-Sign Solutions

The goal of this section is to investigate the existence of constant-sign solutions to problem (1) with λ > 0. Our result provides the existence of two nontrivial solutions of opposite constant sign. Theorem 6. There exists λ 0 > 0 such that if

0 < λ < λ 0 , (28)
problem (1) with λ as above possesses a nontrivial nonnegative solution u 1 ∈ X s 0 (Ω). Moreover, for every λ > 0, problem (1) has also a nontrivial nonpositive solution u 2 ∈ X s 0 (Ω). Proof. First, we focus on the existence of a nontrivial nonnegative solution for equation [START_REF] Di Nezza | Hitchhiker's guide to the fractional Sobolev spaces[END_REF]. To this end we formulate the auxiliary problem:

(-∆) s u = -λ(u + ) q-1 + au + + b(u + ) 2 * s -1 in Ω u = 0 in R N \ Ω. (29)
The Euler functional I + λ,s : X s 0 (Ω) → R associated to problem (29) is

I + λ,s (u) : = C(N, s) 4 R 2N (u(x) -u(y)) 2 |x -y| N +2s dx dy + λ q Ω (u + ) q dx - a 2 Ω (u + ) 2 dx - b 2 * s Ω (u + ) 2 * s dx.
It is of class C 1 with the derivative

(I + λ,s ) (u), φ X s 0 = C(N, s) 2 R 2N (u(x) -u(y))(φ(x) -φ(y)) |x -y| N +2s dx dy +λ Ω (u + ) q-1 φ dx -a Ω u + φ dx -b Ω (u + ) 2 * s -1 φ dx (30)
for every φ ∈ X s 0 (Ω). The critical points of I + λ,s coincide with the weak solutions to problem (29). Moreover, every critical point of the functional I + λ,s is a nonnegative solution of problem [START_REF] Di Nezza | Hitchhiker's guide to the fractional Sobolev spaces[END_REF]. Indeed, let u ∈ X s 0 (Ω) satisfy (I + λ,s ) (u), φ X s 0 = 0 for all φ ∈ X s 0 (Ω). Acting with φ = -u -in (30) gives

0 = (I + λ,s ) (u), -u - X s 0 = C(N, s) 2 R 2N (u(x) -u(y))(u -(y) -u -(x)) |x -y| N +2s dx dy ≥ C(N, s) 2 R 2N (u -(x) -u -(y)) 2 |x -y| N +2s dx dy = C(N, s) 2 u -2 X s 0 , so u = u + ≥ 0.
Let us show that there exist positive constants α and ρ > 0 such that

I + λ,s (u) ≥ α whenever u ∈ X s 0 (Ω) with u X s 0 = ρ. (31)
Towards this, for every u ∈ X s 0 (Ω), we set

J λ,s (u) := I + λ,s (u) - C(N, s) 4 u 2 X s 0 = λ q Ω (u + ) q dx - a 2 Ω (u + ) 2 dx - b 2 * s Ω (u + ) 2 * s dx.
We claim that u ≡ 0 is a strict local minimizer of the functional J λ,s . By [16, Theorem 1.1], it suffices to prove this assertion in the space C 0 s (Ω) ∩ X s 0 (Ω), where

C 0 s (Ω) = w ∈ C 0 (Ω) : w C 0 s := w δ s L ∞ < ∞ , with δ(x) := dist(x, ∂Ω). For any u ∈ C 0 s (Ω) ∩ X s 0 (Ω) we find that Ω (u + ) 2 dx = Ω u + δ s 2-q (δ s ) 2-q (u + ) q dx ≤ C 1 u 2-q C 0 s Ω (u + ) q dx (32)
and

Ω (u + ) 2 * s dx = Ω u + δ s 2 * s -q (δ s ) 2 * s -q (u + ) q dx ≤ C 2 u 2 * s -q C 0 s Ω (u + ) q dx, (33)
with positive constants C 1 and C 2 . From (32) and (33) we obtain

J λ,s (u) ≥ λ q - aC 1 2 u 2-q C 0 s - bC 2 2 * s u 2 * s -q C 0 s Ω (u + ) q dx.
Since q < 2 < 2 * s and taking into account that

I + λ,s (u) = J λ,s (u) + C(N, s) 4 u 2 X s 0 , the claim in (31) ensues.
Fix Λ > 0. We point out that there exists t 0 = t 0 (Λ) > 0 such that I + λ,s (tϕ 1,s ) ≤ 0 for all t ≥ t 0 and λ < Λ. This follows readily because q < 2 < 2 * s and noticing that, for any large t > 0 and any λ < Λ, it holds

I + λ,s (tϕ 1,s ) ≤ t 2 * s (λ 1,s -a) 2 1 t 2 * s -2 + λ q 1 t 2 * s -q Ω ϕ q 1,s dx - b 2 * s Ω ϕ 2 * s 1,s dx ≤ 0.
The proof of Proposition 5 ensures that the functional I + λ,s satisfies the (P S) condition at any level c < s

I + λ,s (h(t)),
where

Γ = {h ∈ C([0, 1], X s 0 (Ω)) : h(0) = 0 and h(1) = t 0 ϕ 1,s }.
We infer from the inequalities

c λ ≤ max t∈[0,t 0 ] I + λ,s (tϕ 1,s ) ≤ λt 0 (Λ) q q Ω ϕ q 1,s dx, ( 34
) that c λ < s N C(N,s) 2 S s N 2s b 2s-N 2s
provided λ ∈ (0, Λ) is chosen even smaller, say λ < λ 0 as indicated in (28). Hence, with such a λ, the functional I + λ,s verifies the (P S) condition at the level c λ . Altogether, the functional I + λ,s satisfies the conditions required in Mountain Pass Theorem, through which we conclude that c λ is a critical value of I + λ,s . According to (31), we have c λ ≥ α > 0, whence there is a nontrivial solution u 1 ∈ X s 0 (Ω) of problem (29). Then, as shown before, u 1 is a nonnegative solution of problem [START_REF] Di Nezza | Hitchhiker's guide to the fractional Sobolev spaces[END_REF].

We pass to the existence of a nontrivial nonpositive solution of problem [START_REF] Di Nezza | Hitchhiker's guide to the fractional Sobolev spaces[END_REF]. Consider the auxiliary problem

(-∆) s u = λ(u -) q-1 -au -in Ω u = 0 in R N \ Ω. ( 35
)
Its associated Euler functional I - λ,s : X s 0 (Ω) → R is given by

I - λ,s (u) := C(N, s) 4 R 2N (u(x) -u(y)) 2 |x -y| N +2s dx dy + λ q Ω (u -) q dx - a 2 Ω (u -) 2 dx
and is of class C 1 with the derivative

(I - λ,s ) (u), φ = C(N, s) 2 R 2N (u(x) -u(y))(φ(x) -φ(y)) |x -y| N +2s dx dy -λ Ω (u -) q-1 φdx + a Ω u -φ dx
for all φ ∈ X s 0 (Ω). The critical points of the functional I - λ,s are exactly the weak solutions of problem (35). In fact, it is straightforward to check that every critical point of I - λ,s is a nonpositive solution of problem (1). We claim that the functional I - λ,s satisfies the (P S) condition (at any level). To this end, let (u n ) ⊂ X s 0 (Ω) be a (P S) sequence for I - λ,s , thus

|I - λ,s (u n )| ≤ C, with a constant C > 0, and | (I - λ,s ) (u n ), ϕ X s 0 | ≤ ε n ϕ X s 0 for all ϕ ∈ X s 0 (Ω), where ε n → 0 as n → ∞. From | (I - λ,s ) (u n ), u + n X s 0 | ≤ ε n u + n X s 0 we deduce that the sequence (u + n ) is bounded in X s 0 (Ω). Using I - λ,s (u n ) - 1 2 (I - λ,s ) (u n ), u - n X s 0 ≤ C + n u - n X s
0 , the boundedness of (u + n ) in X s 0 (Ω) and that q < 2, we are able to get that the sequence (u n ) is bounded in X s 0 (Ω). From here it is standard to confirm that (u n ) contains a strongly convergent subsequence, thus proving the claim.

Next we show that u ≡ 0 is a strict local minimizer of the functional I - λ,s for all λ > 0, which suffices for the application of Mountain Pass Theorem owing to the fact that the (P S) condition holds true (see [START_REF] De Figueiredo | Lectures on the Ekeland variational principle with applications and detours[END_REF]Theorem 5.10]). Moreover, it is sufficient to justify that u ≡ 0 is a strict local minimizer for I - λ,s in the C 0 s topology (see [START_REF] Iannizzotto | H s versus C 0 -weighted minimizers[END_REF]Theorem 1.1]). As in (33), for any u ∈ C 0 s (Ω) ∩ X s 0 (Ω), the following estimate is valid

Ω (u -) 2 dx ≤ C u 2-q C 0 s Ω (u -) q dx,
with a constant C > 0. Thus we obtain

I - λ,s (u) ≥ λ q Ω (u -) q dx - a 2 Ω (u -) 2 dx ≥ λ q - aC 2 u 2-q C 0 s Ω (u -) q dx.
On account of q < 2 and λ > 0, we can assert that u ≡ 0 is a strict local minimizer.

For every t > 0, using that ϕ 1,s is an eigenfunction associated to λ 1,s , we have

I - λ,s (-tϕ 1,s ) = t 2 (λ 1,s -a) 2 Ω ϕ 2 1,s dx + λ q t q Ω ϕ q 1,s dx.
Since a > λ 1,s and q < 2, it holds that I - λ,s (-tϕ 1,s ) ≤ 0 for all t ≥ t 0 , with t 0 = t 0 (λ) > 0 sufficiently large. The preceding facts enable us to apply Mountain Pass Theorem getting that the minimax value

c λ := inf h∈Γ max t∈[0,1] I - λ,s (h(t)),
where

Γ = {h ∈ C([0, 1], X s 0 (Ω)) : h(0) = 0 and h(1) = -t 0 ϕ 1,s }, is a critical value for the functional I - λ,s satisfying c λ ≤ max t∈[0,t 0 ] I - λ,s (-tϕ 1,s ) ≤ λ q Ω ϕ q 1,s dx. ( 36
)
There is thus a nontrivial critical point u 2 ∈ X s 0 (Ω) for I - λ,s with I - λ,s (u 2 ) = c λ , so u 2 is nontrivial nonpositive solution of (1), completing the proof.

Linking Geometry for Minimax Result

In this section we assume that 0 < s < 1, N > 2s, λ > 0, λ k,s < a < λ k+1,s and b > 0. Recall that I λ,s is the Euler functional associated to problem (1) (see Section 3). Corresponding to the positive integer k postulated above, we consider the following orthogonal decomposition of the Hilbert space X s 0 (Ω):

X s 0 (Ω) = V s k ⊕ W s k , (37)
where V s k = span{ϕ 1,s , ϕ 2,s . . . , ϕ k,s } and

W s k = (V s k ) ⊥ = {u ∈ X s 0 (Ω) :
(u, ϕ j,s ) X s 0 = 0, j = 1, 2, . . . , k}. Denote by P s -and P s + the orthogonal projections of X s 0 (Ω) onto V s k and W s k , respectively. Proposition 7. There exist constants α, ρ > 0 independent of λ > 0 such that

I λ,s (u) ≥ α for all u ∈ W s k with u X s 0 = ρ. Proof. Let u ∈ W s k . Using that u 2 X s 0 ≥ λ k+1,s u 2 L 2 and the continuity of the embedding X s 0 (Ω) → L 2 * (Ω) results in I λ,s (u) ≥ C(N, s) 4 u 2 X s 0 - a 2 u 2 L 2 - b 2 * s Ω |u| 2 * s dx ≥ C(N, s) 4 1 - a λ k+1,s u 2 X s 0 - b 2 * s C u 2 * s X s 0 ,
with a constant C > 0. The conclusion follows because a < λ k+1,s and 2 * s > 2.

In order to apply Theorem 3 relative to the orthogonal decomposition (37) it is necessary to construct a vector e ∈ W s k with the stated properties. To accomplish this we rely on the family of functions {U } ε>0 introduced in (9). Without loss of generality, we suppose that 0 ∈ Ω. Let us define u ∈ X s 0 (Ω) by

u (x) = η(x)U (x), x ∈ R N , (38)
with some η ∈ C ∞ c (R N ) whose support is a compact set contained in Ω and satisfying 0 ≤ η ≤ 1 and η = 1 in the ball B δ for a δ > 0. Thanks to (38), the function u is an element of X s 0 (Ω). We pose

e := P s + u ∈ W s k , (39)
which is a continuous function because e ε = u ε -P s -u ε and P s -u ε is a linear combination of eigenfunctions of (-∆) s

p . An essential fact that will be employed is that for every K > 0 there exists ε(K) > 0 such that 0 ∈ x ∈ Ω : e ε (x) > K (40) whenever 0 < ε ≤ ε(K). This assertion is based on the estimate

P s -u ε L ∞ ≤ Cε N -2s 2
(41) for all ε > 0 sufficiently small, with a constant C > 0 independent of ε, that can be proved as in [9, p. 286]. By ( 8), ( 9), ( 39) and (41) we find that

e (0) = u (0) -P s -u (0) ≥ 1 u L 2 * s ε -N -2s 2 -Cε N -2s 2 → +∞ (42)
as ε → 0 because N > 2s, which establishes the claim in (40). In particular, (40) yields an ε 0 > 0 such that

e ε = 0 for all 0 < ε ≤ ε 0 . ( 43
)
Due to (43), we can set e = e ε in Theorem 3.

Corresponding to any numbers ε ∈ (0, ε 0 ] and R 1 , R 2 > 0, by means of the function e in (39) we introduce the set

Q ε,R 1 ,R 2 := {u ∈ X s 0 (Ω) : u = u 1 + re ε , u 1 ∈ V s k ∩ B R 1 (0), 0 ≤ r ≤ R 2 }. (44)
By ( 43), the construction in (44) is meaningful. Denote by

∂Q ε,R 1 ,R 2 the (relative) boundary of Q ε,R 1 ,R 2 in the underlying finite dimensional space V s k ⊕ Re ε .
The result below is fundamental for the existence of a third nontrivial solution to problem (1). In the sequel, the symbol O(ε ω ) for some ω ≥ 0 designates a function satisfying |O(ε ω )| ≤ Cε ω with a constant C > 0 independent of ε > 0. Such a symbol is not always positive. Its exact value may change from place to place. Proposition 8. There exist R 1 > 0 and R 2 > 0 sufficiently large such that

I λ,s (u) ≤ λ q Ω |u| q dx, ∀ u ∈ ∂Q ε,R 1 ,R 2 , (45)
for all ε > 0 sufficiently small and all λ > 0.

Proof. We split the boundary ∂Q ε,R 1 ,R 2 into three parts:

∂Q ε,R 1 ,R 2 = Γ 1 ∪ Γ 2 ∪ Γ 3 , (46) with
Γ 1 = B R 1 ∩ V s k , (47)
Γ 2 = {u ∈ X s 0 (Ω) : u = u 1 + re ε , u 1 ∈ V s k , u 1 X s 0 = R 1 , 0 ≤ r ≤ R 2 }, (48)
Γ 3 = {u ∈ X s 0 (Ω); u = u 1 + R 2 e ε u 1 ∈ V s k , u 1 X s 0 ≤ R 1 }. (49)
We discuss separately the behavior of the functional I λ,s on every part in (46). For u ∈ Γ 1 , by (47) we know that u ∈ V s k . Since λ k,s < a, we infer that

I λ,s (u) ≤ 1 2 C(N, s) 2 u 2 X s 0 -a u 2 L 2 + λ q Ω |u| q dx ≤ λ q Ω |u| q dx.
Let u ∈ Γ 2 , which according to (48) reads as

u = u 1 + re ε with u 1 ∈ V s k and u 1 X s 0 = R 1 .
By virtue of (39) and [START_REF] Servadei | The Brezis-Nirenberg result for the fractional Laplacian[END_REF]Proposition 21] there exists a constant C > 0 such that

e ε 2 X s 0 ≤ u ε 2 X s 0 ≤ S N 2s s + Cε N -2s , (50)
thus δ := sup 0< ≤ε 0 e X s 0 is finite. In order to comply with the linking condition R 2 > ρ in Theorem 3 we must have R 2 e X s 0 > ρ for ρ > 0 in Proposition 7, whenever 0 < ≤ ε 0 . Consequently, ρ/δ is a lower bound of the admissible R 2 . Set r 0 := max{ρ/δ, 1}. In our analysis we distinguish two cases.

First case: 0 ≤ r ≤ r 0 .

The expression of u provides the estimate

I λ,s (u) ≤ 1 2 1 - a λ k,s R 2 1 + r 2 0 2 δ 2 + λ q Ω |u| q dx.
Taking R 1 > 0 large enough we reach the conclusion because λ k,s < a.

Second case: r > r 0 .

Without loss of generality we may suppose that

R 1 ≥ 1. Choose γ > 0 such that (N -2s) 2 2(N + 2s) > γ (51)
and denote

K(R 1 ) := 1 r 0 sup u 1 L ∞ : u 1 ∈ V s k , u 1 X s 0 = R 1 ∈ [c 0 R 1 , c 1 R 1 ],
with some constants c 0 > 0 and c 1 > 0, which holds true because the space V s k is finite dimensional. We introduce the open set

Ω ε := x ∈ Ω : e ε (x) > c 1 ε γ . (52)
From ( 42) and (51) we notice that

ε γ e (0) ≥ 1 u L 2 * s ε -N -2s-2γ 2 -Cε N -2s+2γ 2 → +∞, which guarantees that 0 ∈ Ω ε provided ε ∈ (0, ε 0 ], with some ε 0 > 0. Now we suppose that 0 < ε γ ≤ 1/R 1 in addition to ε ∈ (0, ε 0 ]. Then from (52) there holds Ω ε,R 1 := x ∈ Ω : e ε (x) > K(R 1 ) ⊃ Ω ε . (53)
Observe that

u 1 (x) r + e ε (x) > 0 (54)
for all x ∈ Ω ε,R 1 , r > r 0 , and u 1 ∈ V s k with u 1 X s 0 = R 1 . Using (50) and (54), with a constant C 1 > 0 we can derive

I λ,s (u) ≤ 1 2 1 - a λ k,s R 2 1 + r 2 2 (S N 2s s + C 1 ) + λ q Ω |u| q dx (55) - b 2 * s r 2 * s Ω ε,R 1 u 1 r + e ε 2 * s dx.
Then (54) and (53) allow us to apply Lemma 2 choosing u = u 1 /r, v = e ε and Σ = Ω ε . We obtain from ( 55) and (53) that

I λ,s (u) ≤ 1 2 1 - a λ k,s R 2 1 + r 2 2 (S N 2s s + C 1 ) + λ q Ω |u| q dx - b 2 * s r 2 * s Ωε u 1 r + e ε 2 * s dx ≤ 1 2 1 - a λ k,s R 2 1 + r 2 2 (S N 2s s + C 1 ) + λ q Ω |u| q dx - b 2 * s r 2 * s Ωε u 1 r 2 * s dx + Ωε |e ε | 2 * s dx -C 2 Ωε u 1 r 2 * s -1 |e ε | dx + Ωε u 1 r |e ε | 2 * s -1 dx (56)
with a constant C 2 > 0. By means of r > r 0 and R 1 ≤ ε -γ , (56) provides the estimate

I λ,s (u) ≤ 1 2 1 - a λ k,s R 2 1 + r 2 2 (S N 2s s + C 1 ) + λ q Ω |u| q dx - b 2 * s r 2 * s e ε 2 * s L 2 * s (Ωε) -C 3 e ε L 1 (Ωε) R 2 * s -1 1 + e ε 2 * s -1 L 2 * s -1 (Ωε) R 1 ≤ 1 2 1 - a λ k,s R 2 1 + r 2 2 (S N 2s s + C 1 ) + λ q Ω |u| q dx - b 2 * s r 2 * s e ε 2 * s L 2 * s (Ωε) -C 3 e ε L 1 (Ω) ε -γ(2 * s -1) + e ε 2 * s -1 L 2 * s -1 (Ω) ε -γ
with a constant C 3 > 0. At this point we make use of Lemma 1 (note that (51) ensures γ ∈ [0, (N -2s)/2)) in conjunction with the estimates for e ε L 1 (Ω) and e ε in the classical local case as appeared in [START_REF] Brezis | Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents[END_REF], to deduce for any sufficiently small ε > 0 that

I λ,s (u) ≤ 1 2 1 - a λ k,s R 2 1 + r 2 2 (S N 2s s + C 1 ) + λ q Ω |u| q dx - b 2 * s r 2 * s S N 2s s 2 + O(ε N ) -O(ε N -2s 2 )ε -γ(2 * s -1) -O(ε N -2s 2 )ε -γ
In view of (51) we have

N -2s 2 -γ > N -2s 2 -γ(2 * s -1) = N -2s 2 -γ N + 2s N -2s > 0.
Consequently, since 2 * s > 2, there is a constant C 4 > 0 independent of ε > 0 sufficiently small such that

I λ,s (u) ≤ 1 2 1 - a λ k,s R 2 1 + λ q Ω |u| q dx + C 4
(see also [START_REF] De Figueiredo | Critical superlinear Ambrosetti-Prodi problems[END_REF]Lemmas 2.4 and 2.6] for an alternate treatment). Therefore, owing to the fact that ε > 0 can be arbitrarily small, R 1 > 0 is allowed by the condition 0 < ε γ ≤ 1/R 1 to be taken arbitrarily large in the inequality above. This leads to the conclusion in the statement of the proposition for u ∈ Γ 2 . Let u ∈ Γ 3 . It can be expressed by (49) as u = u 1 + R 2 e ε with u 1 ∈ V s k and u 1 X s 0 ≤ R 1 . We can write

I λ,s (u) ≤ 1 2 1 - a λ k,s u 1 2 X s 0 (Ω) + R 2 2 2 e 2 X s 0 (Ω) - b 2 * s R 2 * s 2 Ω u 1 R 2 + e + 2 * s dx + λ q Ω |u| q dx. (57)
Suppose that R 1 and R 2 > 0 are arbitrarily large with 0 < ε ≤ R 1 /R 2 ≤ 1. Then from (57) and (50) we get

I λ,s (u) ≤ R 2 2 2 S N 2s s + C 1 R 1 R 2 N -2s - b 2 * s R 2 * s 2 Ω u 1 R 2 + e ε + 2 * s dx + λ q Ω |u| q dx. (58)
Taking into account that the space V s k is finite dimensional, there is a constant c 2 > 0 such that

u 1 L ∞ ≤ c 2 u 1 X s 0 ≤ c 2 R 1 for all u 1 ∈ V s k with u 1 X s 0 ≤ R 1 . (59)
From (40) it is apparent that 59) and (60) we find

Ω ε,R 1 ,R 2 := x ∈ Ω : e ε (x) > c 2 R 1 R 2 + 1 ⊃ x ∈ Ω : e ε (x) > c 2 + 1 =: D ε (60) with the Lebesgue measure |D ε | > 0 provided ε > 0 is sufficiently small, say ε ≤ ε 0 . We emphasize that |D ε | is independent of R 1 and R 2 . For every x ∈ Ω ε,R 1 ,R 2 , by (
u 1 (x) R 2 + e ε (x) > u 1 (x) R 2 + c 2 R 1 R 2 + 1 ≥ u 1 (x) R 2 + u 1 L ∞ R 2 + 1 ≥ 1. ( 61
)
Returning to (58), we infer through (60) and (61) that

I λ,s (u) ≤ R 2 2 2 S N 2s s + C 1 R 1 R 2 N -2s - b 2 * s R 2 * s 2 Ω ε,R 1 ,R 2 u 1 R 2 + e ε 2 * s dx + λ q Ω |u| q dx ≤ R 2 2 2 S N 2s s + C 1 - b 2 * s R 2 * s 2 |D ε | + λ q Ω |u| q dx (62)
for all u ∈ Γ 3 . From (62) we arrive at the conclusion of the proposition for R 2 > 0 large enough keeping R 1 > 0 to preserve the relation 0 < ε ≤ R 1 /R 2 ≤ 1. This completes the proof.

Remark 9. It is worth noting that R 1 , R 2 and ε in Proposition 8 are independent of λ.

Estimate of Minimax Value

For every ε > 0 sufficiently small and with R 1 , R 2 > 0 sufficiently large complying with Propositions 7 and 8, we focus on the set Q ε,R 1 ,R 2 constructed in (44). Our purpose in the present section is to show that we can maintain the values of the functional I λ,s along Q ε,R 1 ,R 2 below the admissible level for the (P S) condition exhibited in Proposition 5. The idea is to take ε > 0 and λ > 0 as small as necessary. The lemma below is the main step in this direction.

Lemma 10. Assume that N > 2s. If ε > 0 is sufficiently small, there holds the estimate

I λ,s (u) ≤ s N 1 b N -2s 2s C(N,s) 2 e ε 2 X s 0 -a e ε 2 L 2 u ε 2 L 2 * s N 2s + Cε N N -2s + λ q Ω |u| q dx. ( 63
)
for all u ∈ Q ε,R 1 ,R 2 , with a constant C > 0 independent of ε. Proof. Let u = u 1 + re ε ∈ Q ε,R 1 ,R 2 , hence u 1 ∈ V s k with u 1 X s 0 (Ω) ≤ R 1 and 0 ≤ r ≤ R 2 .
The orthogonality between u 1 and e ε in X s 0 (Ω) and L 2 (Ω) ensures

I λ,s (u) = 1 2 C(N, s) 2 u 1 2 X s 0 -a u 1 2 L 2 + r 2 2 C(N, s) 2 e ε 2 X s 0 -a e ε 2 L 2 - b 2 * s Ω (u + ) 2 * s dx + λ q Ω |u| q dx. (64)
Since a > λ k , we have 1 2

C(N, s) 2 u 1 2 X s 0 -a u 1 2 L 2 ≤ 0. ( 65
)
In order to estimate the term

-(b/2 * s ) u + 2 * s L 2 * s , it is convenient to express u ∈ Q ε,R 1 ,R 2 as follows u = u 1 + re ε = ũ1 + ru ε , with ũ1 = u 1 -rP s -u ε . (66)
This is a consequence of (39). Then (66), the absolute continuity of the Lipschitz continuous function t → (tu 1 (x) + re ε (x)) + ) 2 * s on [0, 1] with a fixed x ∈ Ω and Fubini theorem on Ω × (0, 1) render

Ω ((ũ 1 + ru ε ) + ) 2 * s dx - Ω (ũ + 1 ) 2 * s dx - Ω (ru ε ) 2 * s dx = Ω 1 0 d dt ((tũ 1 + ru ε ) + ) 2 * s -(tũ + 1 ) 2 * s dt dx = 2 * s Ω 1 0 ((tũ 1 + ru ε ) + ) 2 * s -1 d dt ((tũ 1 + ru ε ) + ) -(ũ + 1 ) 2 * s t 2 * s -1 dt dx = 2 * s 1 0 Ω ((tũ 1 + ru ε ) + ) 2 * s -1 -(tũ + 1 ) 2 * s -1 ũ1 dx dt. (67)
Here we have also used that u ε ≥ 0 and the expression of the derivative in t a.e. d dt ((tũ

1 (x) + ru ε (x)) + ) = ũ1 (x) if tũ 1 (x) + ru ε (x) > 0 0 if tũ 1 (x) + ru ε (x) < 0.
By the mean value theorem applied to the function ξ → ξ 2 * s -1 on (0, +∞), the last equality in (67) takes the form

Ω ((ũ 1 + ru ε ) + ) 2 * s dx - Ω (ũ + 1 ) 2 * s dx - Ω (ru ε ) 2 * s dx = 2 * s (2 * s -1) 1 0 Ω tũ + 1 (x) + θ(t, x) (tũ 1 (x) + ru ε (x)) + -tũ + 1 (x) 2 * s -2 × (tũ 1 (x) + ru ε (x)) + -tũ + 1 (x) ũ1 (x) dx dt, (68)
with θ(t, x) ∈ (0, 1). We note that

|tũ + 1 (x) + θ(t, x)((tũ 1 (x) + ru ε (x)) + -tũ + 1 (x))| ≤ max{tũ + 1 (x), (tũ 1 (x) + ru ε (x)) + } ≤ tũ + 1 (x) + ru ε (x).
Taking into account that the function ξ → ξ + on R is a contraction, we also have

|(tũ 1 (x) + ru ε (x)) + -tũ + 1 (x)| ≤ ru ε (x). Consequently, (68) implies Ω ((ũ 1 + ru ε ) + ) 2 * s dx - Ω (ũ + 1 ) 2 * s dx - Ω (ru ε ) 2 * s dx ≤ 2 * s (2 * s -1) 1 0 Ω (tũ + 1 (x) + ru ε (x)) 2 * s -2 ru ε (x)|ũ 1 (x)| dx dt ≤ C Ω ru ε (x)(ũ + 1 (x)) 2 * s -1 + (ru ε (x)) 2 * s -1 |ũ 1 (x)| dx, (69)
with a constant C > 0. The equivalence of the norms on the finite dimensional space V k s , the boundedness of ũ1 (that is due to the boundedness of u 1 and r) and Young's inequality enable us to deduce from (69) that

Ω ((ũ 1 + ru ε ) + ) 2 * s dx - Ω (ũ + 1 ) 2 * s dx - Ω (ru ε ) 2 * s dx ≤ C u ε L 1 r ũ+ 1 2 * s -1 L 2 * s + ũ1 L ∞ u ε 2 * s -1 L 2 * s -1 r 2 * s -1 ≤ 1 2 ũ+ 1 2 * s L 2 * s + c 0 u ε 2 * s L 1 r 2 * s + C ũ1 L ∞ u ε 2 * s -1 L 2 * s -1 r 2 * s -1 , (70)
The next estimate of the minimax value of the functional I λ,s on Q ε,R 1 ,R 2 in connection with the threshold level in Proposition 5 for the (P S) condition is crucial to apply the abstract linking result in Theorem 3.

Lemma 11. If N ≥ 4s, there holds the estimate

c s := inf h∈Γ max u∈Q ε,R 1 ,R 2 I λ,s (h(u)) < s N 1 b N -2s 2s C(N, s) 2 S s N 2s , (73)
with

Γ = {h ∈ C(Q ε,R 1 ,R 2 , X s 0 (Ω)) : h = id on ∂Q ε,R 1 ,R 2 },
provided ε > 0 and λ > 0 are sufficiently small. If 2s < N < 4s, estimate (73) holds true provided ε > 0 and λ > 0 are sufficiently small, and k given in the hypothesis λ k,s < a < λ k+1,s is sufficiently large.

Proof. It is clear that

h = id Q ε,R 1 ,R 2 ∈ Γ, so the definition of c s in (73) ensures c s ≤ max u∈Q ε,R 1 ,R 2 I λ,s (u).
Therefore, by the compactness of Q ε,R 1 ,R 2 , for proving the strict inequality (73) it suffices to show that

I λ,s (u) < s N 1 b N -2s 2s C(N, s) 2 S s N 2s , ∀ u ∈ Q ε,R 1 ,R 2 . (74)
Suppose that N > 4s. This implies through (39), [START_REF] Servadei | The Brezis-Nirenberg result for the fractional Laplacian[END_REF]Proposition 22] and (41) that

e ε 2 L 2 = u ε 2 L 2 -P s -u ε 2 L 2 ≥ C s 2s + O(ε N -2s ), (75)
with a constant C s > 0, as well as

u ε 2 * s L 2 * s = S N 2s s + O(ε N ). ( 76
)
In view of (75), ( 76), (50), and by using the mean value theorem for the function (1 + t)

N -2s

N , we find that

C(N,s) 2 e ε 2 X s 0 -a e ε 2 L 2 u ε 2 L 2 * s ≤ C(N,s) 2 (S N 2s s + O(ε N -2s )) -a(C s 2s + O(ε N -2s )) (S N 2s s + O(ε N )) N -2s N = C(N, s) 2 S s + C(N,s) 2 S N 2s s 1 -1 + S -N 2s s O(ε N ) N -2s N + C(N,s) 2 O(ε N -2s ) (S N 2s s + O(ε N )) N -2s N -ε 2s a(C s + O(ε N -4s )) (S N 2s s + O(ε N )) N -2s N = C(N, s) 2 S s + C(N,s) 2 O(ε N ) + C(N,s) 2 O(ε N -2s ) (S N 2s s + O(ε N )) N -2s N -ε 2s a(C s + O(ε N -4s )) (S N 2s s + O(ε N )) N -2s N < C(N, s) 2 S s (77)
with ε > 0 sufficiently small. This is fulfilled as we supposed that N > 4s.

Here we invoke Lemma 10. Specifically, from (63), (77) and the boundedness of the set Q ε,R 1 ,R 2 in X s 0 (Ω), we infer that (74) holds true provided that both ε > 0 and λ > 0 are sufficiently small. Now we suppose that N = 4s. In this case, by (39), [START_REF] Servadei | The Brezis-Nirenberg result for the fractional Laplacian[END_REF]Proposition 22] and (41), it results

e ε 2 L 2 (Ω) = u ε 2 L 2 (Ω) -P s -u ε 2 L 2 (Ω) ≥ C s 2s | log ε| + O(ε 2s ). (78)
We can proceed along the same lines as in the case N > 2s, this time on the basis of (50), ( 76) and (78), to obtain under the requirement that k (related to the hypothesis λ k,s < a < λ k+1,s ) is sufficiently large. The proof is thus complete.

C(4s,s) 2 e ε 2 X s 0 (Ω) -a e ε 2 L 2 (Ω) u ε 2
Remark 12. It is seen from the strict inequality in (80) that k postulated in the statement of Lemma 11 does not depend on ε > 0 and λ > 0 small enough. Taking advantage that λ n,s → +∞ as n → ∞ it is meaningful to consider k sufficiently large combined with the assumption λ k,s < a < λ k+1,s .

Multiplicity Result

Our main result establishes the existence of three distinct nontrivial solutions to problem (1).

Theorem 13. Assume that 1 < q < 2, λ k,s < a < λ k+1,s , b > 0, s ∈ (0, 1) and N ≥ 4s. Then there exists λ > 0 such that if 0 < λ < λ, problem (1) possesses at least three nontrivial solutions with one nonnegative and one nonpositive. The same conclusion holds for 2s < N < 4s too if the hypothesis λ k,s < a < λ k+1,s is met with an integer k so large as required in Lemma 11.

Proof. Theorem 6 provides a nontrivial nonnegative solution u 1 ∈ X s 0 (Ω) of problem (1) for every λ > 0 sufficiently small (see (28)) and a nontrivial nonpositive solution u 2 ∈ X s 0 (Ω) of problem (1) for each λ > 0. Obviously, these two solutions are distinct.

In order to find a third nontrivial solution of problem (1) whose existence will depend on a threshold λ, we apply the abstract minimax result in 32 Theorem 3 to the functional I λ,s : X s 0 (Ω) → R defined in [START_REF] De Figueiredo | Critical superlinear Ambrosetti-Prodi problems[END_REF] and with the orthogonal decomposition (37). Proposition 7 renders that assumption (i) in Theorem 3 is fulfilled with some α > 0 independent of λ. Regarding the verification of condition (ii) in Theorem 3, set Q := Q ε,R 1 ,R 2 as introduced in (44) and c := c s given in (73). With some R 1 > 0 and R 2 > 0 sufficiently large and for all ε > 0 sufficiently small, Proposition 8 guarantees the estimate (45), from which it is seen that there is λ > 0 such that if 0 < λ < λ one has that condition (ii) required in Theorem 3 is satisfied. This can be done addressing Remark 9. Therefore Theorem 3 may be applied supplying a (P S) sequence for the functional I λ,s , with 0 < λ < λ, at the level c s . By Lemma 11 with every positive λ < λ, for a possibly smaller λ > 0, and by Proposition 5, we know that I λ,s satisfies the (P S) condition at the level c s . Then there is a limit point of the obtained (P S) sequence at the level c s , which is a nontrivial critical point of the functional I λ,s , thus a nontrivial solution u 3 ∈ X s 0 (Ω) of problem [START_REF] Di Nezza | Hitchhiker's guide to the fractional Sobolev spaces[END_REF]. It remains to show that u 3 = u 1 and u 3 = u 2 , so resulting in the existence of three distinct nontrivial solutions. This will be done by proving that I λ,s (u 3 ) = I λ,s (u 1 ) and I λ,s (u 3 ) = I λ,s (u 2 ) with any λ > 0 small enough. Using that the solution u 1 is nonnegative, the definitions of the minimax values c λ and c s in (34) and (73), respectively, in conjunction with Proposition 7, entail I λ,s (u 1 ) = I + λ,s (u 1 ) = c λ ≤ λ q Ω ϕ q 1,s dx < α ≤ c s = I λ,s (u 3 ) (81) provided λ > 0 is so small to satisfy the strict inequality in (81). Proceeding analogously with the nonpositive solution u 2 , we find from (36) and Proposition 7 that I λ,s (u 2 ) = I - λ,s (u 2 ) = c λ ≤ λ q Ω ϕ q 1,s dx < α ≤ c s = I λ,s (u 3 ) (82) provided λ > 0 is taken so small to fulfill the strict unequality in (82). This completes the proof.

Remark 14. From (81) and (82) it is clear that, if 0 < λ < qα/ Ω ϕ q 1,s dx, the solutions u i , with i = 1, 2, 3, obtained in Theorem 13 are distinct, where α > 0 is given in Proposition 7 and does not depend on λ.

L 2 *C(4s,s) 2 (S 2 s 2 (ε 2 L 2 (Ω) = u ε 2 L 2 (Ω) -P s -u ε 2 L 2 ( 2 O 2 O

 222222222222 s (Ω) ≤ + O(ε 2s )) -a(C s 2s | log ε| + O(ε 2s )) O(ε 2s ) + O(1)) -a(C s | log ε| + O(1)) 0 is small enough, since | log ε| → +∞ as ε → 0. Likewise in the case N > 4s we can conclude that (74) is verified when ε > 0 and λ > 0 are sufficiently small. If 2s < N < 4s, from (39),[START_REF] Servadei | The Brezis-Nirenberg result for the fractional Laplacian[END_REF] Proposition 22] and (41) it holdse Ω) ≥ C s N -2s + O(ε 2s ). (ε N ) + C(N,s) (ε N -2s ) (S N 2s s + O(ε N )) N -2s N -ε N -2s a(C s + O(ε 4s-N )) (S N 2s s + O(ε N ))

* s -1 L 2 * s -1 (Ω) in [12,Proposition 12], which are similar to those

Acknowledgments O. H. Miyagaki was partially supported by CNPq/Brazil PROC. 304015/2014-8 and FAPEMIG/Brazil CEX APQ 0063/15. D. Motreanu received a Visiting

Professor fellowship from CNPq/Brazil PV-400633/2017-5. He thanks Departamento de Matemática of Universidade Federal de Juiz de Fora for kind hospitality.

The authors thank the Referee for meaningful comments and kind advice.


D M Smadja
email: david.smadja@aphp.fr

N Gendron

L Mauge

Quand et quel bilan de thrombophilie réaliser ? When and what thrombophilia testing should be performed?

La maladie veineuse thromboembolique (

Chez qui et quand réaliser un bilan de thrombophilie ?

Le bilan de thrombophilie peut être utile pour le propositus (appréciation du rapport bénéfice/risque dans la durée du traitement anticoagulant, nécessité d'augmenter parfois les doses d'héparine pour obtenir une héparinothérapie efficace en cas de déficit en antithrombine) et pour les apparentés asymptomatiques, en particulier les femmes en âge de procréer (prévention de la thrombose pendant la grossesse et le post-partum).

Il est recommandé de ne pas réaliser de bilan de thrombophilie constitutionnelle systématique après un premier épisode de MVTE [START_REF] Sanchez | Recommendations of good practice for the management of thromboembolic venous disease in adults. Short version][END_REF] notamment après 50 ans que la thrombose soit provoquée ou non. Ainsi, il est suggéré de réaliser un bilan de thrombophilie constitutionnelle chez les patients ayant un premier épisode de thrombose veineuse profonde (TVP) proximale ou d'embolie pulmonaire (EP), non provoqué, avant l'âge de 50 ans et avec une histoire familiale au premier degré de thromboses ainsi que chez les patients avec un évènement thromboembolique veineux récidivant (au moins un épisode de TVP proximale ou d'EP et au moins un épisode non provoqué, avant 50 ans) ou encore chez les patients ayant une thrombose veineuse non provoquée dans des sites atypiques. Dans les autres situations, compte tenu de la complexité d'analyse des dossiers et des conséquences thérapeutiques éventuelles, il est suggéré d'avoir un avis d'un centre expert en thrombose multidisciplinaire.

Lorsqu'un bilan de thrombophilie constitutionnelle est indiqué, il est suggéré de rechercher, dans un laboratoire spécialisé, les anomalies suivantes : déficits en antithrombine (AT), (protéine C) PC, (protéine S) PS, mutations Leiden du FV et G20210A du FII, entre le 3ème et le 6ème mois après le diagnostic de thrombose. En cas de déficit en inhibiteur (AT, PC, PS), il est suggéré de déterminer le phénotype de manière précise, voire le génotype. Il est suggéré de rechercher une thrombophilie constitutionnelle chez des apparentés asymptomatiques en cas de thrombophilie constitutionnelle sévère chez le propositus (déficit en AT, PC, PS, double hétérozygote et homozygote FV et FII). Lorsqu'un bilan de thrombophilie constitutionnelle est indiqué chez les apparentés asymptomatiques, il est recommandé d'adresser le sujet dans un centre expert en thrombose agréé. D'autres anomalies biologiques acquises sont associées à un risque accru de thrombose. Ainsi, il est suggéré de rechercher un syndrome des antiphospholipides (SAPL) chez les patients de moins de 50 ans en cas de premier épisode de TVP proximale ou d'embolie pulmonaire non provoqué ou de thromboses de siège inhabituel ou de MVTE récidivante. Il est suggéré de rechercher un SAPL chez les patients en cas de TVP (proximale ou distale) ou d'embolie pulmonaire, quel que soit le contexte de survenue, en cas d'association à une nécrose cutanée ou en cas d'antécédent de pathologie vasculaire placentaire. Dans le cas d'un premier épisode non provoqué de MVTE avec une numération formule sanguine normale, il est suggéré de ne pas réaliser de recherche des mutations associées aux syndromes myéloprolifératifs, sauf en cas de thromboses de sites atypiques. De plus, il est suggéré de rechercher une hémoglobinurie paroxystique nocturne (HPN) en cas de thrombose de sites atypiques associée à des signes d'hémolyse et/ou à une cytopénie Il n'y a généralement pas d'urgence à réaliser un bilan de thrombophilie. Il n'est pas recommandé de le faire à la phase aiguë de la thrombose (risque d'anomalies acquises), ni sous traitement anticoagulant (interférences avec les tests). Par ailleurs, dans certaines conditions physiologiques ou pathologiques, les activités des inhibiteurs de la coagulation peuvent être impactées (Tableau 1). Cependant, la mise en évidence d'un SAPL ou d'un déficit en AT peut modifier l'attitude thérapeutique et nécessiter d'être identifiée rapidement. On peut donc être amené dans certains contextes à rechercher une thrombophilie chez un patient traité. Il est alors capital de renseigner le traitement en cours. La recherche d'un anticoagulant circulant de type lupique reste cependant délicate sous traitement anticoagulant : recherche possible sous héparine selon les réactifs, perte de sensibilité des tests sous AVK, risque élevé de faux positifs sous anticoagulant oral direct (AOD). Il est alors conseillé de réaliser les analyses auprès d'un centre expert.

Quel bilan de thrombophilie réaliser pour rechercher un facteur de risque constitutionnel ou acquis ?

La fréquence et le risque thromboembolique de première thrombose ou de récidive dépendent du type de thrombophilie (sévère ou non sévère) et sont détaillés dans le tableau 2. Les risques de thrombose étant cumulables, le bilan de thrombophilie comprend la recherche de plusieurs facteurs de risque biologiques, qu'ils soient associés à un haut ou un faible risque de récidive.

Il comprend :

-une numération et formule sanguine qui permettra de déceler un syndrome myéloprolifératif ou une anémie, -un bilan de coagulation de routine avec un TP et un TCA, sauf patient sous AOD, -le dosage d'activité des inhibiteurs de la coagulation : AT, PC et PS. En cas de déficit, il faut exclure une cause acquise et contrôler l'anomalie. Si une suspicion de déficit constitutionnel est confirmée, un typage plasmatique doit être réalisé. L'intérêt du génotypage a été montré récemment pour l'AT notamment car il apporte des informations supplémentaires sur le caractère thrombogène des mutations, notamment dans le groupe HBS [START_REF] Alhenc-Gelas | GFHT study group on Genetic Thrombophilia. Thrombotic risk according to SERPINC1 genotype in a large cohort of subjects with antithrombin inherited deficiency[END_REF]. Il peut également permettre de diagnostiquer des mutations thrombogènes non associées à des déficits francs en inhibiteurs par les tests plasmatiques et d'aider au conseil génétique dans les formes récessives par exemple, -la recherche des mutations FVL et F2 G20210A (consentement nécessaire), -la recherche d'un anticorps antiphospholipides (pour le diagnostic de SAPL) qui inclus la recherche d'un anticoagulant circulant de type lupique, d'anticorps IgG anticardiolipine et anti-β2-glycoprotéine-I (β2GP1) La présence d'un anticoagulant circulant de type lupique e/ou d'anticorps anticardiolipine et/ou d'anti-β2GP1 sur 2 prélèvements réalisés à au moins 12 semaines d'intervalle dans les 5 ans suivant une thrombose veineuse ou artérielle ou des complications obstétricales définit le SAPL. En cas de thrombose veineuse non provoquée de sites atypiques, par exemple, une recherche de la mutation JAK2 (V617F) et/ou une recherche d'HPN (quantification de l'expression membranaire de protéines GPl-dépendantes des leucocytes et/ou des globules rouges par cytométrie en flux) peuvent être réalisées.

Conclusion

Le risque de thrombose étant multifactoriel, la réalisation d'un bilan de thrombophilie ne doit pas être systématique et il n'y a généralement pas d'urgence à le réaliser. Malgré la recherche de facteurs de risque biologiques, une grande partie des thrombophilies cliniques reste cependant inexpliquée. La découverte récente de nouveaux polymorphismes associés au risque de thrombose par des approches de type GWAS (étude d'association pangénomique) ouvre de nouvelles pistes physiopathologiques à explorer. Nous rappelons qu'il est recommandé d'adresser les patients dans un centre expert en thrombose agréé.

Tableau 1 .

 1 Fréquence et risque thromboembolique des thrombophilies constitutionnelles. Dosages des inhibiteurs de la coagulation : antithrombine, protéine C, protéine S. Pour le dosage de la protéine S, la méthode théoriquement la plus pertinente, à utiliser en première intention parce qu'elle est susceptible de dépister tous les types de déficits, est celle qui évalue l'activité de la protéine. Il a cependant pendant longtemps été recommandé d'utiliser la mesure de la PSL (protéine S libre) en première intention à cause du manque de spécificité des mesures d'activité. Actuellement, ces problèmes sont largement atténués et cette mesure peut être plus largement proposée c selon les réactifs utilisés d HNF : héparines non fractionnées

		Prévalence dans la population générale	Prévalence dans la population avec MVTE	Risque relatif (RR) de thrombose*	RR de récidive de MVTE
		Thrombophilie sévère
	Déficit en AT	0.02 %	1 %	≈ 15	1.9 -2.6
	FVL (homozygote)	0.02 %	< 2 %	≈ 10	1.2 (0.5 -2.6)
	Double hétérozygote (FVL et FII G20210A)	0	< 2 %	≈ 10	(0.6 -1.9)
	Déficit en PC	0,2 -0,4 %	3 %	4 -6	1.4 -1.8
	Déficit en PS	0.03 -0.5%	1 % -2 %	1 -10	1.0 -1.4
		Thrombophilie non sévère
	FVL (hétérozygote)	3 -7 %	20 %	3 -5	1.2 -1.4
	F2 G20210A (hétérozygote)	0.7 -4 %	6 %	2 -3	0.7 -1.4
	*A noter que les premiers évènements thromboemboliques veineux peuvent se manifester après
	50 ans chez les patients ayant une thrombophilie sévère.

a Méthode recommandée pour dépister tous les types de déficits. b


Laetitia Debry

Joël Luu

Loïc Boulanger

Marie-Cécile Le Deley

Claudia Regis
email: c-regis@o-lambret.fr

Claudia Régis

Keywords: Quilting suturing for preventing seroma formation Breast reconstruction, Latissimus Dorsi Flap, Seroma prevention, Quilting suture Reconstruction mammaire, Lambeau de grand dorsal, Prévention des séromes, capitonnage

Reducing the seroma volume by quilting suture after breast reconstruction with a latissimus dorsi flap: single institutional experience.

INTRODUCTION

Breast reconstruction using the latissimus dorsi flap (LDF), described for the first time by Transini in 1896 [START_REF] Ismaïl | Sequelae after harvesting latissimus dorsi flap and derivates--review[END_REF], is currently widely used in France. This technique allows for a natural breast reconstruction that is associated to a high level of patient satisfaction and a low complication rate.

The main disadvantage of this technique lies in the formation of dorsal seroma, which occurs in over 60% of cases [START_REF] Gisquet | Prevention of seroma by quilting suture after harvesting latissimus dorsi flap. The "Chippendale" technic][END_REF]. Seromas are characterized by a fluctuant swelling in the dead space that forms in the compartment of latissimus dorsi muscle after its removal. Several patient-related factors can contribute to seroma formation such as advanced age, elevated body mass index (BMI) and intervention-related factors such as axillary surgery (sentinel lymph node or axillary node dissection), the presence of a positive lymph node, the extent of flap dissection and the type of adjuvant treatments [START_REF] Delay | Autologous latissimus breast reconstruction: a 3-year clinical experience with 100 patients[END_REF][START_REF] Tomita | Postoperative seroma formation in breast reconstruction with latissimus dorsi flaps: a retrospective study of 174 consecutive cases[END_REF][START_REF] Titley | Preventing seroma in the latissimus dorsi flap donor site[END_REF][START_REF] Yan | Natural History of Seroma Following the Immediate Latissimus Dorsi Flap Method of Breast Reconstruction[END_REF]. Seroma causes functional discomfort and back pain, and can lead to complications such as delayed dorsal healing or infection. Although it can be gradually reabsorbed, it often requires repeated punctures.

Several techniques to reduce the dorsal dead space and the subsequent seroma formation have been described, including muscle-sparing latissimus dorsi [START_REF] Saint-Cyr | The Pedicled Descending Branch Muscle-Sparing Latissimus Dorsi Flap for Breast Reconstruction[END_REF][START_REF] Kim | Comparison of morbidity of donor site following pedicled muscle-sparing latissimus dorsi flap versus extended latissimus dorsi flap breast reconstruction[END_REF].

The efficacy of quilting suture after breast reconstruction using LDF has been extensively reported in the literature [START_REF] Daltrey | Randomized clinical trial of the effect of quilting latissimus dorsi flap donor site on seroma formation[END_REF][START_REF] Sajid | Prevention of postoperative seroma-related morbidity by quilting of latissimus dorsi flap donor site: a systematic review[END_REF][START_REF] Lee | Effects of Quilting Suture Interval on Donor Site Seromas After Breast Reconstruction With Latissimus Dorsi Muscle Flap: A Randomized Trial[END_REF][START_REF] Hart | Borderline epithelial tumors of the ovary[END_REF]. In our surgery unit, this technique is used since 2016. However, since no recommendations exist regarding indications, the procedure is done according to surgeon's discretion and experience.

The aim of this study was to report the early experience of our surgical team in in the seroma prevention by quilting suture of LDF in BR.

MATERIALS AND METHODS

This retrospective monocenter study was conducted in a French Cancer centre. The study was conducted in compliance with the MR003 reference methodology from the National Commission on Informatics and Liberty (CNIL). All patients meeting the eligibility criteria were included consecutively in the study provided that they signed a non-opposition form allowing to use their data for research purposes.

Eligibility criteria

A patient was eligible if she had an autologous breast reconstruction (immediate or delayed) with LDF between 1st January 2016 and 31st December 2017. Patients were identified through the French Computerized Medical Information Systems Program (PMSI) (surgical procedure codes: QEFA013 or QEMA008). Demographic, clinical, surgical and follow-up data were collected retrospectively from computerized medical records.

Treatment and monitoring

Following general anesthesia, patients received a locoregional anesthesia by « PECS bloc » (infiltration of local anesthetic between the pectoralis major muscle and the pectoralis minor muscle under ultrasound guidance). No locoregional anesthesia was used in the LDF compartment. The procedure was carried out using either a classical LDF or muscle-sparing LDF [START_REF] Saint-Cyr | The Pedicled Descending Branch Muscle-Sparing Latissimus Dorsi Flap for Breast Reconstruction[END_REF][START_REF] Kim | Comparison of morbidity of donor site following pedicled muscle-sparing latissimus dorsi flap versus extended latissimus dorsi flap breast reconstruction[END_REF]. An implant could be used for reconstruction if necessary. Patients were placed in lateral decubitus position for the flap removal and placed back in supine position for breast reconstruction. During the intervention, a monopolar or bipolar cautery was used. Quilting suture was done with simple or overedge stitches, depending on the surgeon. Two drains were fitted, one in the latissimus dorsi compartment and one in the breast reconstruction compartment. After the intervention, drains were removed wen less than 100 mL/24 hours of fluid was drained. Discharge was allowed after drain removal.

Patients were seen in consultation 5 days after discharge and a nurse punctured the seroma if considered symptomatic with pain or discomfort. Punctures were repeated if necessary.

Endpoints

The primary endpoint was the total seroma volume, which corresponded to the sum of the volume drained during hospitalization and the volume punctured during consultation, in patients with quilting suture and those without.

The secondary endpoints were the postoperative drainage volume, the formation of a seroma (defined by the occurrence of a symptomatic collection of fluid in the LDF compartment requiring a puncture after discharge), the volume punctured after discharge, duration of postoperative drainage, total drainage time (defined as the time from the date of surgery to the date of the last puncture), the occurrence and type of post-operative complications, graded according to the Clavien-Dindo classification, [START_REF] Dindo | Classification of surgical complications: a new proposal with evaluation in a cohort of 6336 patients and results of a survey[END_REF] the occurrence of functional sequelae (evaluated at 4 weeks and at 3 months) and post-operative pain. As the seroma was punctured if the patient was symptomatic, we didn't studied the number of puncture considered highly variable and preferred to study the volume of fluid drained.

Pain during hospitalization was evaluated using the mean scores of the numerical scale (NS) collected daily. Long-lasting pain was assessed by the surgeon in consultation at 4 weeks and then at 3 months.

Statistical analysis

Data were analyzed using descriptive statistics. Quantitative variables were expressed as mean and standard deviation (SD), and qualitative variables as numbers and percentages. In order to identify a potential indication bias and account for it in the analyzes, we first assessed the association between placing quilting sutures in the latissimus dorsi compartment and the characteristics of patients or those of the surgical procedure.

Univariate analyses were performed using the Chi 2 or the Fisher exact tests for categorical data, and the Student or the Welch test for continuous variables. A multivariate logistic regression was used to estimate odds ratios (OR) and their 95% confidence intervals (CI). Details are provided in Appendix 1.

The variables of interest (drainage volumes, puncture volume, total drainage time, durations, complications) were described according to quilting suture (yes/no) and the between-group (with vs without quilting suture) differences were analyzed using the Chi2 or Fisher exact test for categorical variables and by Student or Welch tests for quantitative variables.

The total volume of drainage according to whether quilting suturing was done or not was analyzed by a multivariate linear regression adjusted for the variables potentially associated with quilting suture and those associated with seroma risk. The adjustment variables considered were: age, BMI, surgeon's level of experience (confirmed or not: defined by more than 10 LDF with quilting performed over the study period), the timing of breast reconstruction (immediate vs delayed), axillary surgery during the intervention (axillary node or sentinel lymph node dissection), and the surgical technique (classical LDF vs muscle-sparing LDF).

Continuous variables were included in the analysis without transformation, after verifying the absence of deviation from the linearity assumption using a spline function (detail in Appendix 2 for BMI).

In order to evaluate the effect of quilting suture according to BMI, an interaction term between the quilting suture variable and the BMI, categorized in three classes (BMI "normal" <25, overweight 25-30, obese ≥30), was included in the multivariate model.

A secondary analysis was performed considering the type of quilting suture as an explanatory variable. For this analysis, three groups were considered: no quilting suture, quilting suture with stitches, quilting suture with overedge stitches.

The comparison between groups -with vs without quilting suture -for each of the other endpoints was carried out using a multivariate model (linear regression with continuous variables, logistic regression with qualitative binary variables) considering the same co-variables as those listed for the analysis of the primary endpoint.

The analysis of risk factors (drainage volumes, durations, complications) was carried out by flap given that a patient could have a bilateral reconstruction.

All statistical analyzes were performed using STATA version 15.0 (StataCorp. 2017.

Stata Statistical Software: Release 15. College Station, TX: StataCorp LLC).

Significance level was set at a p value of 0.05 for all analyses.

RESULTS

Description of the population and surgical procedures

Between 1st January 2016 and 31st December 2017, 110 patients who underwent breast reconstruction (immediate or delayed) using LDF were eligible to participate in the study. Of these, 108 were included; one patient declined and one was lost to follow-up. A total of 110 reconstructions were carried out: two patients underwent a bilateral mastectomy with immediate breast reconstruction. The primary endpoint analysis included 105 patients and 107 flaps (data were missing for 3 patients) (Figure 1).

During the study period, nine surgeons performed breast reconstructions using LDF, the number of interventions ranging from 3 to 30 per surgeon. were done with overedge stitches and 37 (59%) with simple stitches.

Evaluation of the association between quilting suture and total drained volume

The mean total volume drained was 1238 mL (1111), with a median of 910 mL (range 30-7545). As shown in Figure 2, interventions with quilting suture led to a smaller total volume of drained lymph, but the difference was not significant in univariate analysis (1100 mL vs 1461 mL, corresponding to a difference of -361 mL, p=0.15). However, 3 patients had a total volume > 4000 mL, all three after LDF without quilting suture.

In order to account for potential confounding factors, the difference between groups was estimated by a multivariate analysis adjusted for age, patient BMI, number of acts performed by the surgeon, timing of reconstruction (immediate vs delayed), axillary surgery during the intervention and the surgical technique. Further details are provided in Appendix 3. After adjusting for these covariates, the analysis revealed a significant difference of -502 mL between groups (SD = 231, p=0.03).

Compared with the group without quilting suture, a significantly smaller total volume of drained fluid was collected in patients in which quilting suture was used (-372 mL) and, in particular, with overedge stitches (-795 mL) (p=0.02) in multivariate analysis).

The difference in total drainage volume between the groups with and without quilting suture varies significantly according to the patient's BMI (interaction test, p=0.01): while the difference associated with quilting suture is negligible for nonobese patients (-161 mL for women with BMI<25 and -42 mL for overweight women with BMI 25-30), it was -1819 mL for obese women (BMI ≥30).

Assessment of the association between quilting suture and other variables of interest

As summarized in Table 2, quilting suture was significantly associated with punctured volume after discharge (p=0.03) and with a lower incidence of pain 4 weeks after the operation (p=0.04). However, we did not observe any association with drainage volume or drainage duration during hospitalization, the occurrence of punctured seroma, the total duration of drainage, the complications or sequelae after the intervention.

DISCUSSION

Results on our series of 108 patients undergoing breast reconstruction with LDF shows a benefit of quilting suture which significantly decrease the total volume of fluid drained. In addition, the benefit of quilting suture for reducing drained fluid volume was significantly greater in obese women (BMI> 30) compared with nonobese women.

Performance of a quilting suture of the LD flap compartment in breast reconstruction was associated with a significant reduction in total fluid volume (with vs without quilting suture, -502 mL, p = 0.03).

Our results are consistent with data from retrospective studies that also show a decrease in volume of post-operative seroma, seroma formation and duration of drainage, the latters not being significant in our study [START_REF] Titley | Preventing seroma in the latissimus dorsi flap donor site[END_REF][START_REF] Yan | Natural History of Seroma Following the Immediate Latissimus Dorsi Flap Method of Breast Reconstruction[END_REF][START_REF] Saint-Cyr | The Pedicled Descending Branch Muscle-Sparing Latissimus Dorsi Flap for Breast Reconstruction[END_REF][START_REF] Kim | Comparison of morbidity of donor site following pedicled muscle-sparing latissimus dorsi flap versus extended latissimus dorsi flap breast reconstruction[END_REF][START_REF] Daltrey | Randomized clinical trial of the effect of quilting latissimus dorsi flap donor site on seroma formation[END_REF][START_REF] Sajid | Prevention of postoperative seroma-related morbidity by quilting of latissimus dorsi flap donor site: a systematic review[END_REF][START_REF] Lee | Effects of Quilting Suture Interval on Donor Site Seromas After Breast Reconstruction With Latissimus Dorsi Muscle Flap: A Randomized Trial[END_REF][START_REF] Hart | Borderline epithelial tumors of the ovary[END_REF][START_REF] Dindo | Classification of surgical complications: a new proposal with evaluation in a cohort of 6336 patients and results of a survey[END_REF].

The surgical technique used to perform quilting suture can have a great impact on its efficacy. In our study, overedge stitching was superior to simple stitches in reducing the volume of the postoperative seroma leading to a significant decrease in the drained and punctured volume. In a retrospective study including 70 patients, the occurrence of seroma decreased in the overedge stitching group (0/47 in the quilting suture group vs 7/23 (30%) in the non quilting suture group) [START_REF] Rios | Progressive tension sutures to prevent seroma formation after latissimus dorsi harvest[END_REF].

Other techniques to reduce the dead space in the latissimus dorsal compartment exist, and can also be associated with the quilting suturing [START_REF] Bailey | Latissimus dorsi donor-site morbidity: the combination of quilting and fibrin sealant reduce length of drain placement and seroma rate[END_REF][START_REF] Cheng | Is it necessary to perform routine appendectomy for mucinous ovarian neoplasms? A retrospective study and metaanalysis[END_REF][START_REF] Hart | A Prospective Randomized Trial of the Efficacy of Fibrin Glue, Triamcinolone Acetonide, and Quilting Sutures in Seroma Prevention after Latissimus Dorsi Breast Reconstruction[END_REF]. In 2017, Hart et al. compared quilting suture vs other techniques (biological glue, triamcinolone acetonide or placebo) in a randomized monocenter study involving 96 patients [START_REF] Hart | A Prospective Randomized Trial of the Efficacy of Fibrin Glue, Triamcinolone Acetonide, and Quilting Sutures in Seroma Prevention after Latissimus Dorsi Breast Reconstruction[END_REF].

Quilting suture was superior to other techniques in reducing the incidence of lymphocele and allowing earlier removal of the drains. A meta-analysis by Lee et al.

in 2015, including 14 studies, showed a reduction in post-operative seroma formation, which was a result of the combined efficacy of quilting suture and fibrin adjunction [START_REF] Lee | Fibrin Sealants and Quilting Suture for Prevention of Seroma Formation Following Latissimus Dorsi Muscle Harvest: A Systematic Review and Meta-analysis[END_REF].

Muscle-sparing LDF should also be favored because it decreases in our experience as reported the literature post-operative seroma. Indeed the limited dorsal dissection reduce the dead space, regardless of the use of quilting suture [START_REF] Saint-Cyr | The Pedicled Descending Branch Muscle-Sparing Latissimus Dorsi Flap for Breast Reconstruction[END_REF][START_REF] Kim | Comparison of morbidity of donor site following pedicled muscle-sparing latissimus dorsi flap versus extended latissimus dorsi flap breast reconstruction[END_REF].

Our study differs from most in that it involved a high proportion of obese women (22%) [START_REF] Daltrey | Randomized clinical trial of the effect of quilting latissimus dorsi flap donor site on seroma formation[END_REF][START_REF] Yan | Natural History of Seroma Following the Immediate Latissimus Dorsi Flap Method of Breast Reconstruction[END_REF][START_REF] Cheng | Is it necessary to perform routine appendectomy for mucinous ovarian neoplasms? A retrospective study and metaanalysis[END_REF]. In obese patients, fluid production is greater than in patients with normal BMI due to a larger dorsal dead space (see Appendices 2 and 3) [START_REF] Delay | Autologous latissimus breast reconstruction: a 3-year clinical experience with 100 patients[END_REF][START_REF] Tomita | Postoperative seroma formation in breast reconstruction with latissimus dorsi flaps: a retrospective study of 174 consecutive cases[END_REF][START_REF] Randolph | Prediction of postoperative seroma after latissimus dorsi breast reconstruction[END_REF].

In our study, the mean BMI was 26.9 kg/m² and the mean seroma volume was 1238mL (1111). In the study by Yan et al. carried out in 24 patients with a lower mean BMI of 21.3 kg/m 2 , had lower seroma volume: 524 mL [START_REF] Yan | Natural History of Seroma Following the Immediate Latissimus Dorsi Flap Method of Breast Reconstruction[END_REF]. It is also for obese women that quilting suture is the most difficult to carry out, due to the lack of exposure and the difficulty to reach the limits of the cavity. We show that, obese patients benefited more from quilting suture than patients with a "normal" BMI in terms of reduced lymph volume.

In our series, quilting suturing did not increase the complication rate, which was low and comparable to that reported in the literature (9% of Grade 3 complications according to the Clavien Dindo classification) [START_REF] Daltrey | Randomized clinical trial of the effect of quilting latissimus dorsi flap donor site on seroma formation[END_REF][START_REF] Sajid | Prevention of postoperative seroma-related morbidity by quilting of latissimus dorsi flap donor site: a systematic review[END_REF][START_REF] Lee | Effects of Quilting Suture Interval on Donor Site Seromas After Breast Reconstruction With Latissimus Dorsi Muscle Flap: A Randomized Trial[END_REF][START_REF] Yan | Natural History of Seroma Following the Immediate Latissimus Dorsi Flap Method of Breast Reconstruction[END_REF].

The impact of quilting suture on back pain and the occurrence of functional sequelae is debatable. Firstly in our experience it didn't increase the pain either immediately after surgery or at 3 months. However, it seems to have decreased the numerical scale reported 4 weeks after surgery (p=0.04). Only one study by Daltrey et al. reports similar results, with a decrease in the amount of level 1 analgesic on the first post-operative week in the group of patients with quilting suture [START_REF] Daltrey | Randomized clinical trial of the effect of quilting latissimus dorsi flap donor site on seroma formation[END_REF].

On the other hand this result differs from previous reports in a prospective study, comparing from a functional point of view, a population having benefited or not from quilting suture [START_REF] Button | Shoulder function following autologous latissimus dorsi breast reconstruction. A prospective three year observational study comparing quilting and non-quilting donor site techniques[END_REF]. Button et al showed a significant increase in the DASH score (clinical score based on a self-questionnaire on the ability of patients to perform daily life activities) translating a functional degradation immediately after intervention, at 3 and 6 weeks for patients with quilting suture but this difference was no longer significant on the long term.

The limitations of our study are related to its retrospective nature. Some patient data are missing, such as the quilting technique used (simple stitches or overedge stitches) or the pain and functional sequelae evaluation. Quilting suturing was carried out by 9 surgeons without a standardized approach. Surgical procedures such as quilting suture, as well as the choice of simple stitches or overedge stitches, depend on the surgeon. The dissection of the LDF can also be more or less extensive depending on the practice of the surgeon. The heterogeneity in practice limits the interpretation and comparison of the differences observed.

The main strength of our study is the large number of patients included (108) and their representativeness of the population having LDF breast reconstruction. This allowed to carry out statistical analyses with a rigorous methodology of 69 flaps in the quilting group and 41 in the group without quilting.

CONCLUSION

According to our experience, LDF compartment quilting reduces significantly the total volume of drained lymph. The study also shows that this intervention is essential in obese women, who benefit more from quilting suture than BMI patients <25.

Conflit d'intérêt : aucun.

FIGURES LEGENDS

Figure 1 :Figure 2 :Figure 1 :

 121 Figure 1: Patient flow chart

Table 1

 1 details the general characteristics of the patients and the surgical technique, overall and based on whether a quilting suturing was carried out or not.

	performed in patients with a BMI<25, 45 (41%) in overweight patients and 24 (22%)
	in obese patients.
	Quilting suture was performed in 69 cases (63%), and was significantly more
	frequent in immediate breast reconstruction than in delayed reconstruction (48/62
	[77%] versus 21/48 [44%], p<0.001). Muscle-sparing LDF was used in 8 cases,
	including 3 with quilting suture, while a conventional latissimus dorsi flap was used in
	102 cases, including 66 with quilting suture (38% versus 65%, p=0.15). Performing
	quilting suturing was surgeon-dependent (p<0.001) and was not related to the level
	of experience. In addition, there was no difference between cases with or without
	quilting suture in terms of characteristics of patient or disease. In multivariate
	analysis, quilting suture was significantly more frequent in immediate breast
	reconstruction (p<0.001) and classical latissimus dorsi flap compared to muscle-

The characteristics of the patients are usual for this population. Patients had a mean age of 49.7 years (9.3), 19% were smokers and 30% presented with ductal carcinomas in situ. The mean BMI was 26.9 kg/m² (4.1), 41 interventions (37%) were sparing (p=0.02) (detail of logistic regression in Appendix 1).

Out of the 63 reconstruction cases with informative quilting suture, 26 (41%)

Table 1 :

 1 General description of the population and the surgical technique, overall and according to the realization or not of a quilting suture (description per flaps, N = 110)

	Characteristics	Total N=110		Without quilting N=41	With quilting N=69	P
	Age in years, mean (standard deviation, SD)	49.7	(9.3)	49.	(8.9)	49.8	(9.6)	0.82

Comorbidities

The sum of percentages in column may differ from 100% due to rounding.

(2) Welch test for mean comparison, performed due to rejection of the variance equality hypothesis.

(3) The evaluation includes all acts, with a volume equal to 0 in case there was no puncture after hospital discharge.

(4) Fisher's exact test performed due to limited numbers in some categories. [START_REF] Titley | Preventing seroma in the latissimus dorsi flap donor site[END_REF] Comparison of the distribution between the absence of a complication and the occurrence of a complication. [START_REF] Yan | Natural History of Seroma Following the Immediate Latissimus Dorsi Flap Method of Breast Reconstruction[END_REF] Complication graded according to the Clavien -Dindo classification.