Oumarou Asso

Mabel Cuesta

Jonas T Êl É Doumat È

Liamidi Leadi

J T Doumat

PRINCIPAL EIGENVALUES FOR THE FRACTIONAL p-LAPLACIAN WITH UNBOUNDED SIGN-CHANGING WEIGHTS

Keywords: 2020 Mathematics Subject Classification. 35J70, 35P30 Fractional p-Laplacian, fractional Sobolev space, indefinite weight, principal eigenvalues. ©2023

Let Ω be a bounded regular domain of R N , N 1, p ∈ (1, +∞), and s ∈ (0, 1). We consider the eigenvalue problem

where the potential V and the weight m are possibly unbounded and are signchanging. After establishing the boundedness and regularity of weak solutions, we prove that this problem admits principal eigenvalues under certain conditions. We also show that when such eigenvalues exist, they are simple and isolated in the spectrum of the operator.

Introduction

For p ∈ (1, +∞) and s ∈ (0, 1), the fractional (s, p)-Laplacian is an extension of the s-fractional Laplacian and it is defined, for a regular function u : R N → R, as (-∆ p ) s u(x) := 2K(1 -s) P. V.

R N |u(x) -u(y)| p-2 u(x) -u(y)
|x -y| N +sp dy for all x ∈ R N with

K = p S N -1 | ω, e | p dH N -1 (ω) -1
, e ∈ S N -1 , where H N -1 denotes the (N -1)-dimensional Hausdorff measure of the unit sphere S N -1 of R N . Let us recall that for all measurable function on a subset D of R N and for all x ∈ R N , the principal value function on the integral D Ψ (x, y)dy is denoted by P. V. Ψ (x, y)dx, where B ε (x) is a ball centered at x ∈ R N with radius ε > 0.

In this article, we study the conditions under which the principal eigenvalues of the following homogeneous Dirichlet problem exist (-∆ p ) s u + V |u| p-2 u = λm(x)|u| p-2 u in Ω,

u = 0 in R N \ Ω, (1.1)
where Ω is a bounded regular domain of R N , V and m are indefinite sign-changing functions and satisfying the following conditions: (C1) V , m ∈ L r (Ω) with r ∈ (1, +∞) ∩ ( N sp , +∞), (C2) m + = max(m, 0) ≡ 0. Our aim is to extend some results obtained by Del-Pezzo et al. in [START_REF] Del Pezzo | An optimization problem for the first eigenvalue of the p-fractional Laplacian[END_REF] for the eigenvalue problem (1.1). These authors studied, among other issues, the existence of eigenvalues, the positivity of the eigenfunctions associated with the first eigenvalue of (1.1) with m ≡ 1 and V satisfying (C1). We want here to address the question of existence of principal eigenvalue in a wide range of weights, precisely when m and V changing sign. The presence of such weights in problem (1.1) brings us to proceed by a considerably different approach called "eigencurve arguments" which requires the construction of some equivalent problem.

To illustrate this eigencurve argument, let us mention the work of Fleckinger et al. [START_REF] Fleckinger | Existence of Multiple Principal Eigenvalues for some Indefinite Linear Eigenvalue problems[END_REF], where the following eigenvalue problem is considered.

-∆u + a 0 (x)u = λm(x)u, in Ω, u = 0 on ∂Ω (

with Ω a bounded smooth domain, a 0 , m ∈ L r (Ω), r > N 2 are indefinite and m is unbounded. After separating the positive and negative parts of a 0 and m one find equation (1.2) as -∆u + a + 0 (x)u + λm -(x)u = λm + (x)u + a - 0 (x)u.

(1.3) So, for any fixed λ, they were led to study the following eigenvalue problem of eigenvalue parameter σ(λ),

-∆u + (a + 0 (x) + 1)u + λm -(x)u = σ(λ) m + (x) + a - 0 (x) + 1 λ u in Ω, u = 0 on ∂Ω .

It is clear that λ > 0 is an eigenvalue of (1.2) if and only if σ(λ) = λ. For this purpose, they studied the properties of continuity, concavity and monotonicity of the curve λ → σ(λ) and they proved that, under certain conditions, the existence of λ > 0 satisfies σ(λ) = λ. For more details see [START_REF] Fleckinger | Existence of Multiple Principal Eigenvalues for some Indefinite Linear Eigenvalue problems[END_REF]. Our construction of the equivalent problem is different from the one made in [START_REF] Fleckinger | Existence of Multiple Principal Eigenvalues for some Indefinite Linear Eigenvalue problems[END_REF] and it is closer to the one used by Binding and Huang [START_REF] Binding | Existence and nonexistence of positive eigenfunctions for the p-Laplacian[END_REF]. These authors considered, for bounded potential V and bounded weight m, the principal eigencurve µ 1 (λ), that is, µ 1 (λ) is the principal eigenvalue of -∆ p u + (V (x) -λm(x))|u| p-2 u = µ 1 (λ)|u| p-2 u in Ω, u = 0 on ∂Ω and deduced the existence of λ ∈ R such that µ 1 (λ) = 0 under some conditions on V and m. This technique has generated several results which have enriched the scientific literature (see for example [START_REF] Binding | The principal eigencurve for the p-Laplacian[END_REF][START_REF] Binding | Existence and nonexistence of positive eigenfunctions for the p-Laplacian[END_REF][START_REF] Cuesta | Weighted eigenvalue problems for quasilinear elliptic operators with mixed Robin-Dirichlet boundary conditions[END_REF][START_REF] Hess | On some linear and nonlinear eigenvalue problems with an indefinite weight function[END_REF][START_REF] Leadi | A weighted eigencurve for Steklov problems with a potential[END_REF]). For instance, recently [START_REF] Cuesta | A weighted eigenvalue problem for the p-Laplacian plus a potential[END_REF] made use of such arguments when solving the above problem for a potential V and a weight function m that may change sign and may be unbounded. They looked and established additional conditions on V and m that guarantee the existence of principal eigenvalues. In this work, our main results extend those of [START_REF] Cuesta | A weighted eigenvalue problem for the p-Laplacian plus a potential[END_REF] and references therein to the fractional p-Laplacian.

This article is organized as follows. We start by recalling some basic properties of essential the fractional Sobolev spaces in Section 2. In Section 3 we prove the boundedness and regularity of the weak solutions. Section 4 is devoted to the existence of principal eigenvalues. In Section 5, we show that when principal eigenvalues exist, they are isolated in the spectrum and we give a lower bound of the measure of the nodal domains for changing sign eigenfunctions. Finally in Section 6 we prove some sort of continuity of the principal eigenvalues when varying s. We collect in appendix the proof of a discrete version of some well known identity as well as a regularity result for more general equations involving the fractional p-Laplacian with unbounded terms.

Preliminaries

The Lebesgue measure of a Lebesgue measurable set Z ⊂ R N is denoted by |Z|. • The (s, p)-fractional Sobolev space, denoted by W s,p (Ω), is defined by

W s,p (Ω) := u ∈ L p (Ω) : Ω Ω |u(x) -u(y)| p |x -y| N +sp dx dy < +∞ .
The space W s,p (Ω) is a separable Banach space endowed with the norm

u W s,p (Ω) := Ω Ω |u(x) -u(y)| p |x -y| N +sp dx dy + Ω |u| p dx 1/p . W s,p (Ω) is reflexive if p > 1.
• For any function u of W s,p (Ω) we denote the Gagliardo semi-norm by

u W s,p (Ω) := Ω Ω |u(x) -u(y)| p |x -y| N +sp dx dy 1/p .
• The space W s,p (Ω) is defined as the space of all u ∈ W s,p (Ω) such that ũ ∈ W s,p (R N ), where ũ is the extension by zero of u, outside of Ω. W s,p (Ω) is a Banach space endowed with the norm

u W s,p (Ω) := ũ W s,p (R N )
and it is a reflexive space if p > 1.

Let us quote some properties of these spaces that will be used later. Here we will denote by C(N, p) any positive constant depending only on N and p. (1) There exists C(N, p) such that, for any u ∈ W s,p (Ω), it holds

u p L p (Ω) ≤ C(N, p)(diam(Ω)) sp (1 -s) u p W s,p (R N ) . (2.1)
Thus, the Gagliardo semi-norm

• W s,p (R N ) is a norm in W s,p (Ω) equivalent to the previous norm • W s,p (Ω) (c.f. [10, Lemma 2.5]).
(2) Let 0 < s ≤ s < 1. Then there exists a positive constant C(N, p) such that [START_REF] Bourgain | Another look at Sobolev spaces[END_REF][START_REF] Del Pezzo | Eigenvalues for a nonlocal pseudo p-Laplacian[END_REF][START_REF] Grisvard | Elliptic problems in nonsmooth domains[END_REF]). Let Ω ⊂ R N be a bounded open set with Lipschitz boundary. Then

u p W s,p (R N ) ≤ u p W s ,p (R N ) + C(N, p) 1 sp - 1 s p u p L p (R N ) for any u ∈ W s ,p (R N ) (cf. [10, Lemma 2.3]). Proposition 2.2 ([
(1) C ∞ 0 (Ω) is dense in W s,p (Ω) (c.f. [17, Theorem 1.4.2.2]). (2) If u ∈ W s,p (Ω) and f is a Lipschitz function then f (u) ∈ W 1,p (Ω). (3) Let 0 < s ≤ s < 1.
Then there exists a positive constant C(N, p) such that

(1 -s)[u] p W s,p (Ω) ≤ 2 (1-s)p diam(Ω) (s -s)p (1 -s )[u] p W s ,p (Ω)
for any u ∈ W s ,p (Ω) (c.f. [7, Lemma 2]; [START_REF] Del Pezzo | Eigenvalues for a nonlocal pseudo p-Laplacian[END_REF]Lemmas 4.3 and 4.4]). ( 4)

For any u ∈ W 1,p (Ω), lim s→1 - (1 -s)[u] p W s,p (Ω = Ω |∇u| p dx. (c.f. [7, Corollary 2 ]).
2.2. Embeddings. Let the fractional critical exponent of Sobolev be defined by

p * s = N p N -sp if sp < N, +∞ if sp N.
The following results are versions of the classical Sobolev injection theorem in the case of fractional Sobolev spaces (c.f. [14, pages 218 and 219]).

Theorem 2.3. [START_REF] Demengel | Functional spaces for the theory of elliptic partial differential equations[END_REF] Let Ω be an open set with a Lipschitz boundary. We have the following continuous injections:

(1) If sp < N, W s,p (Ω) → L q (Ω) for all q ∈ [p, p * s ]. (2) If sp = N, W s,p (Ω) → L q (Ω) for all q ∈ [p, +∞). (3) If sp > N , W s,p (Ω) → C 0, α ( Ω) with α ∈ 0, s -N
p . Furthermore we have the following compact injections when Ω is an open bounded domain of R N with a Lipschitz boundary:

4. If sp N , then W s,p (Ω) → c L q (Ω) for all q ∈ [1, p * s ). 5. If sp > N , then W s,p (Ω) → c C 0, α ( Ω) with α ∈ (0, s -N p ). 6. W s,p (Ω) → c L pq (Ω) with max{1, N
sp } < q < +∞ and 1 q + 1 q = 1. Throughout this work we will assume that Ω is a bounded domain of R N with a Lipschitz boundary.

Weak solutions of the eigenvalue problem and regularity results

For simplicity, from now on we will denote by u, instead of ũ, the extension by 0 of any function u ∈ W s,p (Ω). Definition 3.1.

(1) We will say that a function u ∈ W s,p (Ω) is a weak solution of (1.1) if

H(u, v) + Ω V (x)|u| p-2 uvdx = λ Ω m(x)|u| p-2 uv dx (3.1)
for all v ∈ W s,p (Ω), where

H(u, v) := K(1 -s) R N R N |u(x) -u(y)| p-2 u(x) -u(y) |x -y| N +sp v(x) -v(y) dx dy. (3.
2) It should be noted that for all u ∈ W s,p (Ω), we have

H(u, u) = K(1 -s) u p W s,p (R N ) .
(2) We will say that a real number λ is an eigenvalue of (1.1) if there exist u ≡ 0 satisfying (3.1). In this case, we say that u is an eigenfunction associated with λ.

(3) Moreover, if the eigenfunction u has a constant sign on Ω, then λ is called a principal eigenvalue of the problem (1.1). ( 4) Finally, the eigenvalue λ is said to be simple if any two eigenfunctions u and v associated with λ are such that u = cv for some real constant c.

Definition 3.2. For each u ∈ W s,p (Ω), let the energy associated with the problem (1.1) be

E V (u) := H(u, u) + Ω V (x)|u| p dx = K(1 -s) u p W s,p (R N ) + Ω V (x)|u| p dx. (3.3) It is clear that E V is of class C 1 on W s,p (Ω) with E V (u), v = pH(u, v) + p Ω V (x)|u| p-2 uv dx ∀(u, v) ∈ W s,p (Ω) × W s,p (Ω).
Let us now state the main result of this section. Let us consider the homogeneous problem (-∆ p ) s u + V |u| p-2 u = 0 in Ω,

u = 0 in R N \ Ω, (3.4)
where V satisfies condition (C1).

Theorem 3.3. If u ∈ W s,p (Ω) is a weak solution of (3.4), then u ∈ L ∞ (Ω)∩C(Ω).
Furthermore, there exists a positive constant C = C(s, p, N, Ω,

V L r (Ω) ) such that u L ∞ (Ω) ≤ C u L r p (Ω) . (3.5)
The proof of this theorem will follow from Lemma 3.4 below, based on the De Giorgi-Stampacchia iteration technique (see for instance [START_REF] Del Pezzo | Non-resonant Fredholm alternative and anti-maximum principle for the fractional p-Laplacian[END_REF][START_REF] Franzina | Fractional p-eigenvalues[END_REF][START_REF] Servadei | Weak and Viscosity Solutions of the Fractional Laplace Equation[END_REF], where the case V ≡ 1 has been considered).

Lemma 3.4. Assume that sp ≤ N . Let u be a weak solution of (3.4) admitting a positive part u + ≡ 0. Let us define the sequence (w k ) k by

w k := u -1 - 1 2 k + .
Then there exists a positive constant σ = σ(s, p, N, Ω,

V L r (Ω) ) such that, if u + L r p (Ω) < σ, then u ≤ 1 a.e. Proof. Let us denote W k = w k p L r p (Ω)
. The conclusion of the lemma will follow from the following results that we prove below:

(1)

lim k→+∞ W k = (u -1) + p L r p (Ω)
.

(2) lim k→+∞ W k = 0.

Notice that, by definition, w k ∈ W s,p (Ω) and w k = 0 a.e. in Ω c . 1. Trivially the sequence (w k ) k is decreasing so, for all k ∈ N we have

|w k | r p ≤ |w 0 | r p = |u + | r p ∈ L 1 (Ω).
u| p-1 w k+1 L r (Ω) ≤ 2 (p-1)(k+1) W k . (3.6) Indeed, first observe that if w k+1 (x) > 0, that is, if u(x) > 2 k+1 -1 2 k+1 , then w k (x) = w k+1 (x) + 1 2 k+1 ≥ 1 2 k+1 , w k (x) ≥ u(x) 2 k+1 -1 and Ω |u| r (p-1) w r k+1 dx = {w k+1 >0} |u| r (p-1) w r k+1 dx ≤ {w k+1 >0} (2 k+1 -1) r (p-1) w r (p-1) k w r k (x)dx ≤ (2 k+1 -1) r (p-1) w k r p L r p (Ω) ≤ 2 r (p-1)(k+1) W r k . (3.7)
Claim 2. There exist D > 1 and β > 0 such that for all k ∈ N,

W k+1 ≤ D k W 1+β k .
To prove this claim, let us quote the following (trivial) inequality:

∀(a, b) ∈ R 2 , |a + -b + | p ≤ |a -b| p-2 (a -b)(a + -b + ). (3.8)
By taking

a = u(x) -1 - 1 2 k+1 , b = u(y) -1 - 1 2 k+1 in (3.8) for all (x, y) ∈ R N , we obtain w k+1 p W s,p (R N ) ≤ R N R N |u(x) -u(y)| p-2 (u(x) -u(y)) |x -y| N +sp (w k+1 (x)-w k+1 (y)) dx dy.
Besides, by taking w k+1 in the weak formulation of (3.4), we obtain from the previous inequality

K(1 -s) w k+1 p W s,p (R N ) ≤ Ω |V (x) u| p-1 w k+1 dx,
and therefore, using Claim 1,

K(1 -s) w k+1 p W s,p (R N ) ≤ V L r (Ω) Ω |u| p-1 w k+1 r dx 1/r ≤ C2 (p-1)(k+1) W k (3.9)
for some positive constant C depending on V L r (Ω) , s, p, N, and Ω. On the other hand, using Hölder's inequality with the exponents q := N r (N -sp) if q < N s (or any q > 1 if N = ps), by Sobolev's embedding we have

W k+1 = w k+1 p L r p (Ω) ≤ w k+1 p L r pq (Ω) |{w k+1 > 0}| q-1 r q ≤ C w k+1 p W s,p (R N ) |{w k+1 > 0}| q-1 r q (3.10) for some 0 < C = C(N, p, s, Ω). Moreover, since w k = w k+1 + 1 2 k+1 , then |{w k+1 > 0}| ≤ |{w k > 2 -k-1 }| ≤ 2 r p(k+1) W r k (3.11)
and hence, using (3.9), (3.10) and (3.11)

W k+1 ≤ C2 (p-1)(k+1) W k × |{w k+1 > 0}| q-1 r q ≤ C2 (p-1)(k+1) W k × 2 r p(k+1) W r k q-1 r q ≤ C(2 p-1 × 2 p(q-1) q ) k+1 W 1+ q-1 q k ≤ D k W 1+β k , with D = [1 + C] 2 p-1 × 2 p(q-1) q 2 > 1 and β = q-1 q > 0. Claim 2 is proved. Now we complete the proof of 2. Let σ = D -q 2 p(q-1) 2 , denote ρ = u + p L r p (Ω)
and assume that ρ 1/p < σ. Choose η ∈ (ρ q-1 q , D -q q-1 ). It should be noted that

η ∈]0, 1[, ρ q-1 q
≤ η, and Dη q-1 q ≤ 1. Let us prove by induction that for all k ∈ N

W k ≤ ρη k . (3.12)
By definition

W 0 := w 0 p L r p (Ω) = u + p L r p (Ω) = ρ ≤ ρη 0 .
Assume that (3.12) holds at order k and let us show that it holds at order k + 1. By Claim 2,

W k+1 ≤ D k W 1+ q-1 q k ≤ D k (ρη k ) 1+ q-1 q = ρ η q-1 q k ρ q-1 q η k ≤ ρη k+1 .
Thus by passing to the limit in (3.12), we finally obtain that W k → 0.

Proof of Theorem 3.3.

Take v = σu 2 u + L r p (Ω)
, where σ = σ(s, p, N, Ω, V L r (Ω) ) is given by Lemma 3.4. Since v is a weak solution of (1.1) and satisfies

v + L r p (Ω) = σ 2 then v ≤ 1 a.e., which gives u ≤ 2 σ u + L r p (Ω) a.e.
If u -is not identically zero, we apply the same argument to -u, which is a weak solution of (1.1), to find that

u ≥ - 2 σ u - L r p (Ω) a.e.
and estimate (3.5) follows.

Finally, the continuity of u results from [6, Theorem 3.13], which derives from [23, Theorem 1.5].

Remark 3.5. To our knowledge, it is not known if the solutions are continuous up to the boundary of Ω or class C 0,α in the case sp ≤ N and V unbounded. Indeed, when V is bounded then f = V |u| p-2 u ∈ L ∞ (Ω) and, by the results by Iannizzotto et al. [START_REF] Iannizzotto | Global Hölder regularity for the fractional p-Laplacian[END_REF], u ∈ C α (Ω) for some α > 0.

Existence of principal eigenvalues with indefinite weights

Let us assume in this section that V and m satisfy conditions (C1) and (C2) and consider the eigenvalue problem

(-∆ p ) s u + V |u| p-2 u = µ|u| p-2 u in Ω, u = 0 in R N \ Ω, ( 4.1)
with V = (V -λm) and µ an eigenvalue parameter depending on the real λ.

According to [START_REF] Del Pezzo | An optimization problem for the first eigenvalue of the p-fractional Laplacian[END_REF], problem (4.1) admits a unique principal eigenvalue which we will denote µ(λ). Moreover, µ(λ) is simple and can be characterized as

µ(λ) = inf K(1 -s)[u] p W s,p (R N ) + Ω (V (x) -λm(x))|u| p dx : u ∈ W s,p (Ω), u L p (Ω) = 1 . (4.2)
Note that λ 0 is a principal eigenvalue of our problem (1.1) if and only if µ(λ 0 ) = 0. Our aim here is to give reasonable assumptions on V and m so that the curve of the function λ → µ(λ) intersects the x-axis. We introduce the sets

G 0 := u ∈ W s,p (Ω) : u L p (Ω) = 1, Ω m(x)|u| p dx = 0 , G := u ∈ W s,p (Ω) : u L p (Ω) = 1 . (4.
3)

The following proposition gives useful properties on the function λ → µ(λ). We will denote here

Ω + := {x ∈ Ω, m(x) > 0}, Ω -:= {x ∈ Ω, m(x) < 0}, Ω 0 := {x ∈ Ω, m(x) = 0},
and ϕ λ the unique positive eigenfunction of L p (Ω)-norm equal to 1 associated with µ(λ).

Proposition 4.1.

(i) µ : R → R is concave and differentiable, with

µ (λ) = - Ω m(x)ϕ p λ dx ∀λ ∈ R. (ii) (a) lim λ→+∞ µ(λ) = -∞. (b) If |Ω -| > 0 then lim λ→-∞ µ(λ) = -∞. (iii) If |Ω -| = 0 then µ is strictly decreasing on R and, if moreover |Ω -∪ Ω 0 | = 0, then lim λ→-∞ µ(λ) = +∞. (iv) sup λ∈R µ(λ) = α(V, m) where α(V, m) := inf {E V (u), u ∈ G 0 } . (4.4) Moreover, α(V, m) is finite if and only if |Ω + | < |Ω|.
Proof. (i) We prove that µ : R → R is concave. Let λ and β be two distinct real numbers. Let t ∈ [0, 1] and set

θ t = tλ + (1 -t)β. Let u ∈ G. Since -E V -θtm (u) = t(-E V -λm (u)) + (1 -t)(-E V -βm (u)), we have -µ(θ t ) t (-µ(λ)) + (1 -t) (-µ(β))
, which means that the function -µ is convex.

Let λ ∈ R and (λ k ) k be a sequence converging to λ. Let ϕ k and ϕ λ be eigenfunctions associated with µ(λ k ) and µ(λ) respectively with L p (Ω)-norm equal to 1. By the results of [12, Theorem 2.9], the eigenfunction ϕ λ is > 0 a.e. in Ω (see Remark 4.3). By Lemma 4.4 below we have, for some C 1 > 0 and C 2 > 0,

ϕ k p W s,p (R N ) ≤ C 1 E V -λ k m (ϕ k ) + C 2 Ω |ϕ k | p dx = C 1 µ(λ k ) + C 2 , so lim sup k→∞ ϕ k p W s,p (R N ) ≤ C 1 µ(λ) + C 2
and therefore the sequence (ϕ k ) k is bounded in W s,p (Ω). Hence there exists ϕ 0 ∈ W s,p (Ω) and some subsequence, written again

(ϕ k ) k , such that ϕ k ϕ 0 in W s,p (Ω), ϕ k → ϕ 0 in L p (Ω) and in L r p (Ω). In particular ϕ 0 L p (Ω) = 1. Since µ(λ) = lim k→+∞ µ(λ k ), it follows that µ(λ) lim k→+∞ E V -λ k m (ϕ k ) E V -λm (ϕ 0 ) µ(λ)
and hence µ(λ) = E V -λm (ϕ 0 ). Using the simplicity of the principal eigenvalue of problem (4.1) and the fact that ϕ 0 L p (Ω) = 1 and ϕ 0 ≥ 0, we conclude that ϕ 0 = ϕ λ . Moreover,

µ(λ k ) = E V -λ k m (ϕ k ) = E V -λm (ϕ k ) + (λ -λ k ) Ω m(x)|ϕ k | p dx µ(λ) + (λ -λ k ) Ω m(x)|ϕ k | p dx
and, by replacing λ k by λ and ϕ k by ϕ λ in the inequality above, we obtain:

µ(λ) µ(λ k ) + (λ k -λ) Ω m(x)|ϕ λ | p dx.
Putting together this two inequalities we obtain

(λ -λ k ) Ω m(x)|ϕ k | p dx µ(λ k ) -µ(λ) (λ -λ k ) Ω m(x)|ϕ λ | p dx from which we conclude that µ (λ) = - Ω m(x)|ϕ λ | p dx. (ii) Since |Ω + | > 0 by (C1), there exists a function ξ ∈ W s,p (Ω) such that Ω m(x)|ξ| p dx > 0, Ω |ξ| p dx = 1 and therefore µ(λ) K(1 -s) R 2N |ξ(x) -ξ(y)| p |x -y| N +sp dx dy + Ω V (x)|ξ| p dx -λ Ω m(x)|ξ| p dx. Thus lim λ→+∞ µ(λ) = -∞. Similarly, if |Ω -| > 0 then lim λ→-∞ µ(λ) = -∞. (iii) If |Ω -| = 0, then µ is strictly decreasing on R because -µ (λ) = Ω m(x)|ϕ λ | p dx > 0 for all λ ∈ R. If |Ω -∪Ω 0 | = 0 assume by contradiction that the function λ → µ(λ) is bounded. Let (λ k ) k be such that λ k → -∞ and write ϕ k = ϕ λ k . Using Lemma 4.4, we have sup λ∈R µ(λ) ≥ µ(λ k ) = E (V -λ k m) (ϕ k ) = E V (ϕ k ) -λ k Ω m(x)|ϕ k | p dx 0 ≥ E V (ϕ k ) 1 C 1 ϕ k p W s,p (R N ) -C 2 so (ϕ k ) k is a bounded sequence in W s,p (Ω)
. Thus, there exist ϕ ∈ W s,p (Ω) and some subsequence (

ϕ k ) k such that ϕ k ϕ in W s,p (Ω) and ϕ k → ϕ in L pr (Ω) and in L p (Ω). As ϕ k is L p -normalized, then ϕ L p (Ω) = 1 and Ω m(x)|ϕ| p dx = lim k→∞ Ω m(x)|ϕ k | p dx > 0. Then -∞ = lim k→∞ λ k Ω m(x)|ϕ k | p dx ≥ E V (ϕ) -sup λ∈R µ(λ) > -∞, a contradiction. (iv). If |Ω -∪ Ω 0 | = 0 then G 0 = ∅ and using (ii) and (iii) we obtain α(V m) = +∞ = lim λ→-∞ µ(λ) = sup λ∈R µ(λ). If |Ω -∪ Ω 0 | > 0, as G 0 ⊂ G and µ(λ) E V -λm (u) = E V (u) -λ Ω m(x)|u| p dx = E V (u) ∀u ∈ G 0 .
then sup λ∈R µ(λ) α(V, m). To obtain the reverse inequality observe that, by (i) and (ii), the function µ possesses a global maximum, that is, sup λ∈R µ(λ) is reached at some λ 0 ∈ R, which in particular implies that

0 = µ (λ 0 ) = Ω m(x)|ϕ λ0 | p dx. Consequently, ϕ λ0 ∈ G 0 and then α(V, m) E V (ϕ λ0 ). But µ(λ 0 ) = E V -λ0m (ϕ λ0 ) = E V (ϕ λ0 ) and µ(λ 0 ) = sup λ∈R µ(λ), then α(V, m) sup λ∈R µ(λ). Thus we obtain α(V, m) = sup λ∈R µ(λ).
The proof that α(V, m) is achieved whenever it is finite, is standard and we omit it.

As a consequence of this proposition we have the following result. (i) If |Ω -| = 0, then (1.1) admits a principal eigenvalue if and only if α(V, m) > 0. In this case the principal eigenvalue is unique and it is characterized by the equation λ 1 (V, m) = min M E V , where M := u ∈ W s,p (Ω) :

Ω m(x)|u| p dx = 1 . (ii) If |Ω -| > 0, then (1.
1) admits a principal eigenvalue if and only if α(V, m) 0. More precisely, (a) if α(V, m) > 0, then (1.1) admits exactly two principal eigenvalues

λ -1 (V, m) = -min M -E V < λ 1 (V, m) = min M E V ,
where

M -:= u ∈ W s,p (Ω) : Ω m|u| p dx = -1 ; (b) if α(V, m) = 0, then (1.1) admits a unique principal eigenvalue λ 1 (V, m) given by λ 1 (V, m) = inf M E V = -inf M -E V . These infima are not achieved. In addition, any function u ∈ W s,p (Ω)\ {0} such that E V (u) = Ω m|u| p dx = 0 is an eigenfunction associated with λ 1 (V, m). (iii) In the case α(V, m) > 0 any function u ∈ M satisfying E V (u) = λ 1 (V, m)
is an eigenfunction associated with λ 1 (V, m) and it is sign definite. Same

result for u ∈ M -satisfying E V (u) = λ -1 (V, m).
Proof. The proof given in [START_REF] Cuesta | A weighted eigenvalue problem for the p-Laplacian plus a potential[END_REF] can be easily adapted here as a corollary of Proposition 4.1. We only give the proof (b) of ii. to show how to use Picone's inequality stated in Lemma 7.1. If α(V, m) = 0, then there exists a real λ 0 such that µ(λ 0 ) = 0 so λ 0 is a principal eigenvalue of (1.1). Let us show that

λ 0 = inf M E V = -inf M -E V .
We only give the proof of the first identity, the proof of the second one is similar. As α(V, m) = sup λ∈R µ(λ) then µ (λ 0 ) = 0 = -Ω m(x)|ϕ λ0 | p dx.

Let u ∈ M be such that u 0. For any T > 0 define u T := min{u, T } and take ϕ λ0 + ε with ε > 0 Let us prove that z :=

u p T ϕ λ 0 +ε p-1 ∈ W s,p (Ω). Indeed, for any (x, y) ∈ R N × R N we have |z(x) -z(y)| ≤ u T (x) p -u T (y) p (ϕ λ0 (x) + ε) p-1 + |u T (y)| p (ϕ λ0 (y) + ε) p-1 -(ϕ λ0 (x) + ε) p-1 (ϕ λ0 (y) + ε) p-1 (ϕ λ0 (x) + ε) p-1 (4.5
) and using for all (a, b) ∈ R + × R + and q > 0 the trivial inequality

|a q -b q | ≤ q |a| q-1 + |b| q-1 |a -b| with q = p or q = p -1 we have |z(x) -z(y)| ≤ 2pT p-1 ε p-1 |u(x) -u(y)| + 2(p -1)T p ε p |ϕ λ0 (x) -ϕ λ0 (y)| (4.6)
and therefore

[z] p W s,p (R N ) ≤ C [u] p W s,p (R N ) + [ϕ λ0 ] p W s,p (R N ) < ∞.
An application of Picone's inequality to functions u T and ϕ λ0 + ε, and the fact that

u p T ϕ λ 0 +ε p-1 ∈ W s,p (Ω), imply that 0 R 2N L(u T , ϕ λ0 + ε)(x, y) |y -x| N +sp dx dy EJDE-2023/38 = u T p W s,p (R N ) - R 2N ϕ λ0 (y) -ϕ λ0 (x) p-2 ϕ λ0 (y) -ϕ λ0 (x) |x -y| N +sp × u p T (y) ϕ λ0 + ε p-1 (y) - u p T (x) ϕ λ0 + ε p-1 (x) dx dy = u T p W s,p (R N ) + 1 K(1 -s) -λ 0 Ω m(x) ϕ λ0 (x) p-1 u p T ϕ λ0 + ε p-1 dx + Ω V (x) ϕ λ0 p-1 u p T ϕ λ0 + ε p-1 dx .
So when ε → 0, by the Lebesgue convergence theorem

0 ≤ K(1 -s)[u T ] p W s,p (R N ) -λ 0 Ω m(x)|u T | p dx + Ω V (x)u p T dx
for all T > 0. Moreover, since as T → +∞ we obtain u T = u. Then by Fatou's lemma,

0 K(1 -s)[u] p W s,p (R N ) -λ 0 + Ω V (x)|u| p dx. (4.7) So λ 0 inf M E V .
To prove the reverse inequality let us show that there exists a sequence of functions of M whose energy

E V converges to λ 0 . Let ψ ∈ C ∞ (Ω) such that ψ > 0, Ω m(x)ψ p dx > 0 and Ω m(x)ϕ p-1 λ0 ψdx > 0. Let the sequence (u k ) k be of the form u k = ϕ λ0 + ψ k Ω m(x) ϕ λ0 + ψ k p dx 1/p .
It is straightforward that all elements of this sequence are in manifold M, and when k is big enough u k > 0. Furthermore, because the functions t → E V ϕ λ0 + tψ and s → ϕ λ0 + sψ p are continuous and at least once differentiable on 0, 1 k , then there exist 0 < t k , s k < 1/k such that

E V ϕ λ0 + ψ k = 1 k E V ϕ λ0 + t k ψ , ψ , Ω m(x) ϕ λ0 + 1 k ψ p dx = p k Ω m(x) ϕ λ0 + s k ψ p-1 ψdx.
As a result,

E V (u k ) = 1 Ω m(x) ϕ λ0 + 1 k ψ p dx E V ϕ λ0 + ψ k = k p Ω m(x) ϕ λ0 + s k ψ p-1 ψdx × E V ϕ λ0 + t k ψ , ψ k .
So when k tends to infinity we find that E V (u k ) -→ λ 0 . Thus we can conclude that

λ 0 = inf M E V . (4.8)
But ϕ λ0 is an eigenfunction associated with µ(λ 0 ) = 0, that means that

E V (ϕ 0 ) = λ 0 Ω m(x) ϕ λ0 p dx = 0 which implies λ 0 is not achieved. Finally, if u ∈ W s,p (Ω) \ {0} satisfies E V (u) = Ω m(x)|u| p dx = 0 we have sup λ∈R µ(λ) = 0 = E V (u) = E V -λ0m (u) µ(λ 0 ) Ω |u| p dx = 0, (4.9)
and therefore u is a function where the infimum µ(λ 0 ) of equation (4.2) is achieved. Then, as the eigenvalue µ(λ 0 ) is simple, there exists c > 0 such that u = cφ λ0 so then λ 0 = λ 1 (V, m).

Remark 4.3. One can prove, as at the beginning of the previous proof, that if 0 ≤ u ∈ W s,p (Ω) ∩ L ∞ (Ω) and v ∈ W s,p (Ω) satisfies v ≥ c > 0 a.e. for some c > 0 then

u p v p-1 ∈ W s,p (Ω) ∩ L ∞ (Ω). Lemma 4.4.
Let ω be a function satisfying (C1) and let Z be a bounded subset of L r (Ω). If ω > 0 a.e. is a function on L r (Ω) for some 1 ≤ r < p * s , then there are two strictly positive constants C 1 and C 2 such that

u p W s,p (R N ) C 1 E V (u) + C 2 Ω ω(x)|u| p dx (4.10)
for all functions V ∈ Z and for all u ∈ W s,p (Ω).

Proof. This proof is a partial adaptation of Lemma 2 of [START_REF] Cuesta | A weighted eigenvalue problem for the p-Laplacian plus a potential[END_REF]. Let T be a positive real such that

V L r (Ω) T for all V ∈ Z. Let ε > 0 fixed such that ε < K(1-s)
T . According to Hölder inequality and the hypothesis (C1), we can write

Ω V (x)|u| p dx V L r (Ω) u p L pr (Ω) . Claim. For all ε > 0, there exists M ε > 0 such that u p L pr (Ω) ε u p W s,p (R N ) + M ε Ω ω(x)|u| p dx (4.11)
for all u ∈ W s,p (Ω). Indeed, suppose by contradiction that there exists ε 0 > 0, and sequence (u k ) k of W s,p (Ω) such that

u k L pr (Ω) = 1 and ε 0 u p W s,p (R N ) + k Ω ω(x)|u k | p dx < 1.
Then (u k ) k is bounded W s,p (Ω), so there exists u 0 ∈ W s,p (Ω) and sub-sequence also denoted by (u k ) k of W s,p (Ω) such that u k u 0 in W s,p (Ω) and u k → u 0 in L pr (Ω) (see [START_REF] Del Pezzo | Global bifurcation for fractional p-Laplacian and application[END_REF]Theorem 2.16]). So, we have on one hand

lim k→+∞ u k L pr (Ω) = u 0 L pr (Ω) = 1,
and therefore u 0 ≡ 0 in Ω. Moreover, using once again the inequality of the hypothesis we have

Ω ω(x)|u k | p dx < 1 k .
Then passing to the limit we find by Fatou's lemma that

Ω ω(x)|u 0 | p dx 0,
which is a contradiction since ω > 0 in Ω and u 0 ≡ 0 in Ω. We have proved the claim. By applying the inequality (4.11) for 0

< ε < K(1-s) T there is a positive real M ε such that Ω V (x)|u| p dx ε V L r (Ω) u p W s,p (R N ) + V L r (Ω) M ε Ω ω(x)|u| p dx εT u p W s,p (R N ) + T M ε Ω ω(x)|u| p dx.
So we obtain

u p W s,p (R N ) 1 K(1 -s) -εT E V (u) + T M ε K(1 -s) -εT Ω ω(x)|u| p dx.
The lemma follows by setting

C 1 = 1 K(1 -s) -εT and C 2 = T M ε K(1 -s) -εT .
As an application of Picone's inequality of Lemma 7.1 we can prove the simplicity and the uniqueness of the principal eigenvalues λ ±1 (V, m). Proposition 4.5. Assume that α(V, m) ≥ 0. Let u > 0 a.e. be an eigenfunction of problem (1.1) associated with λ 1 (V, m) and let v ≥ 0 a.e. be an eigenfunction associated with an eigenvalue λ ≥ λ 1 (V, m). Then there exists c ∈ R such that u = cv a.e. and λ = λ 1 (V, m).

Similarly, if u is an eigenfunction of problem (1.1) associated with λ -1 (V, m) with u > 0 a.e. and v is eigenfunction associated with an eigenvalue λ ≤ λ 1 (V, m) with v > 0 a.e. then there exists c ∈ R such that u = cv a.e. and λ = λ -1 (V, m).

Proof. Let us apply Picone's inequality given in Lemma 7.1 to the functions u and v + ε with ε > 0. By Remark 4.3, 0

R N R N L(u, v + ε)(x, y) |x -y| N +sp dx dy = [u] p W s,p (R N ) - R N R N × |v(y) -v(x)| p-2 (v(y) -v(x)) |x -y| N +sp u p (y) (v(y) + ε) p-1 - u p (x) (v(x) + ε) p-1 dx dy = 1 K(1 -s) λ 1 (V, m) Ω m(x)|u| p dx - Ω V (x)|u| p dx - 1 K(1 -s) λ Ω m(x)|v| p-1 u p (v + ε) p-1 dx - Ω V (x)|v| p-1 u p (v + ε) p-1 dx .
By using the Lebesgue dominated convergence theorem and passing to the limit we have 0

R N R N L(u, v)(x, y) |x -y| N +sp dx dy λ 1 (V, m) -λ K(1 -s) Ω m(x)|u| p dx.
Therefore if α(V, m) > 0 and λ > λ 1 (V, m), as we have Ω m(x)|u| p dx > 0, we conclude from the previous inequality that λ 1 (V, m) = λ and L(u, v) = 0. Hence, by Picone's inequality there is a constant c > 0 such that u = cv. In the case α(V, m) = 0 we have Ω m(x)|u| p dx = 0 and we can conclude from the previous calculation that L(u, v) = 0. Hence, by the conclusions of Picone's inequality, there is a constant c > 0 such that u = cv from which we deduce that E V (v) = Ω m(x)|v| p dx = 0. Thus, according to the result (ii)(b) of Theorem 4.2, v is an eigenfunction associated with λ 1 (V, m), and therefore one must have λ = λ 1 (V, m). Theorem 5.1. Let v be an eigenfunction of (1.1) associated with an eigenvalue λ different from λ 1 (V, m) and λ -1 (V, m). Then there exists constant

C = C(s, p, N, Ω) > 0 such that, if N is a nodal domain of v, then |N | ≥ C V -λm L r (Ω) -γ > 0, (5.1)
for

γ = r q q -r p with      q = ∞ if N < sp q ≥ p if N = sp q = p * s if N > sp.
Proof. Let N be a nodal domain, and assume for instance that v < 0 on N . Let us take ϕ = v -.χ N as test function in (1.1). Notice that trivially ϕ ∈ W (s,p) (Ω). Thus

K(1 -s) R N R N |v(y) -v(x)| p-2 (v(y) -v(x)) (ϕ(y) -ϕ(x)) |x -y| N +sp dx dy = N (λm -V )|v -| p dx so K(1 -s) ϕ p W s,p (R N ) = N (V -λm)|v -| p dx ≤ V -λm L r (Ω) N |v -| pr dx 1/r .
Let us start with the case N > ps. By the previous Sobolev embedding theorem, for some constant c > 0, we have

c ϕ p L p * s (Ω) ≤ ϕ p W s,p (R N ) . Hence cK(1 -s) ϕ p L p * s (Ω) ≤ K(1 -s) ϕ p W s,p (R N ) ≤ V -λm L r (Ω) N |v -| pr dx 1/r ≤ V -λm L r (Ω) v -p L p * s (Ω) |N | 1 r -p p * s ,
and the estimate (5.1) follows.

If N = sp there exists some c > 0 such that for all q ≥ p,

cK(1 -s) ϕ p L q (Ω) ≤ K(1 -s) ϕ p W s,p (R N ) ≤ V -λm L r (Ω) N |v -| pr dx 1/r ≤ V -λm L r (Ω) v -p L q (Ω) |N | 1 r -p q ,
and the estimate (5.1) follows.

In the case N < sp, there exists some c > 0 such that

cK(1 -s) ϕ p L ∞ (Ω) ≤ K(1 -s) ϕ p W s,p (R N ) ≤ V -λm L r (Ω) N |v -| pr dx 1/r ≤ V -λm L r (Ω) v -p L ∞ (Ω) |N | 1/r
, and the estimate (5.1) follows.

The following statement is a straightforward consequence of the above theorem.

Corollary 5.2. Any weak solution of (1.1) has a finite number of nodal domains.

Proof. Let N j be a nodal domain of a certain eigenfunction associated with an eigenvalue. Let us assume by contradiction that there exists an infinity of nodal domains (N j ) j≥1 of this eigenfunction. We know that according to (5.1) there exists a positive constant c > 0 such that we have

|N j | > c ∀j . Thus |Ω| j |N j | > c j 1 ,
which is a contradiction.

5.2.

Isolation of λ 1 (V, m) and λ -1 (V, m). The following theorem states that the eigenvalues λ ±1 (V, m) are isolated provided α(V, m) ≥ 0. Notice that if α(V, m) > 0, there are no eigenvalues in the interval (λ -1 (V, m), λ 1 (V, m)).

Theorem 5.3. Let α(V, m) ≥ 0. The eigenvalues λ ±1 (V, m) are isolated in the spectrum of (1.1), that is to say that there exists δ ± > 0 such that there are no eigenvalues in the intervals (λ 1 (V, m), λ 1 (V, m) + δ + ) and (λ -1 (V, m) -δ -, λ -1 (V, m)).

Proof. We only prove the result for λ 1 (V, m) by arguing by contradiction. Let us assume that there exists a sequence (λ k ) k of eigenvalues such that

λ k > λ 1 (V, m) and lim k→∞ λ k = λ 1 (V, m).
Denote by u k a positive eigenfunction associated with λ k . Replacing

u k by u k /[u k ] W s,p (Ω)
if necessary, we can assume that the sequence (u k ) k is bounded. By the results on compact embeddings, there exists a subsequence (still denoted (u k ) k ) converging to some u ∈ W s,p (Ω) weakly in W s,p (Ω), strongly in L r p (Ω), a.e. and in measure in Ω such that

lim k→∞ Ω V (x)|u k | p dx = Ω V (x)|u| p dx, lim k→∞ Ω m(x)|u k | p dx = Ω m(x)|u| p dx.
Since u k is an eigenfunction associated with λ k we have

E V (u k ) = K(1 -s) u k p W s,p (R N ) + Ω V (x)|u k | p dx = K(1 -s) + Ω V (x)|u k | p dx = λ k Ω m(x)|u k | p dx.
Thus passing to the limit and using that E V is weakly lower semi-continuous we obtain

E V (u) = K(1 -s)[u] p W s,p (R N ) + Ω V (x)|u| p dx ≤ K(1 -s) + Ω V (x)|u| p dx = λ 1 (V, m) Ω m(x)|u| p dx.
In particular u ≡ 0 and

E V (u) ≤ λ 1 (V, m) Ω m(x)|u| p dx. (5.2)
Assume first that α(V, m) > 0. Then (5.2) implies that Ω m(x)|u| p dx = 0. In fact

we have Ω m(x)|u| p dx > 0 otherwise, by taking v = u/ -Ω m(x)|u| p dx 1/p ∈ M -we will have from the definition of λ -1 (V, m) that -λ -1 (V, m) ≤ E V (v) = E V (u) -Ω m(x)|u| p dx =⇒ λ -1 (V, m) Ω m(x)|u| p dx ≤ E V (u)
which, jointly with the inequality (5.2) will give λ -1 (V, m) ≥ λ 1 (v, M ), a contradiction. Since we have proved that Ω m(x) |u| p dx > 0 we then have, by definition of λ 1 (V, m), that

λ 1 (V, m) Ω m(x) |u| p dx ≤ E V (u)
and therefore λ 1 (V, m) Ω m(x) |u| p dx = E V (u). Thus, u is an eigenfunction associated with the principal eigenvalue λ 1 (V, m) and it must be either positive a.e. or negative a.e. in Ω. On the other hand, if for each k we denote

N + k := {x ∈ Ω : u k (x) > 0} and N - k := {x ∈ Ω : u k (x) < 0}, by Theorem 5.1, we obtain the existence of a constant c > 0 such that |N + k | > c and |N - k | > c.
However, if we assume that u > 0 (the case u < 0 is analogous) it follows from the convergence in measure that |N - k | → 0, which is a contradiction. Assume now that α(V, m) = 0. We claim that Ω m(x)|u| p dx = 0. Indeed, if for instance Ω m(x)|u| p dx > 0 then we will have, by definition of λ 1 (V, m), that

λ 1 (V, m) Ω m(x)|u| p dx ≤ E V (u)
that, jointly with equation (5.2) will give that the infimum λ 1 (V, m) is achieved, a contradiction. If Ω m(x)|u| p dx < 0 then we will have instead

λ -1 (V, m) Ω m(x) |u| p dx ≤ E V (u)
and, since λ 1 (V, m) = λ -1 (V, m), we again get a contradiction. We have just proved that Ω m(x)|u| p dx = 0. Hence, by equation (5.2) E V (u) ≤ 0, it must be E V (u) = 0 by the definition of α(V, m) = 0. Thus u is an eigenfunction associated with λ 1 (V, m) so u must be either > 0 a.e. or < 0 a.e. in Ω and we obtain a contradiction as in the previous case.

Regularity of the principal eigenvalues with respect to s

Now we study the behaviour of the first eigenvalues λ ±1 (V, m) with respect to s. As we want to vary s, then to simplify the study, we now impose conditions on V and m which are independent of s. So we assume V, m in L r (Ω) with r > max{1, N p }. We start by proving a lemma in the behaviour of sequences (1 -s)[u s ] p W s,p (R N ) as s varies. Lemma 6.1. Let s 0 ∈ (0, 1] and (s n ) n be a sequence in (0, 1) converging to s 0 . Let (u n ) be a sequence of functions such that, for all n ∈ N, u n ∈ W sn,p (Ω) and

(1 -s n )[u n ] p W sn,p (R N ) ≤ L for some L ≥ 0. Let q ∈ [1, p * ).
Then there exists a function u ∈ W s0,p (Ω) such that, up to a subsequence, (

) [u] p W s 0 ,p (R N ) ≤ lim inf n→∞ [u n ] p W sn ,p (R N ) if s 0 < 1 and Ω |∇u| p dx ≤ lim inf n→∞ (1 -s n )K[u n ] p W sn ,p (R N ) if s 0 = 1. (2) u n → u in L q (Ω) and u n → u in L p (Ω). 1
Proof. First of all, by Poincaré's inequality,

u n p p ≤ C(N, p)(diam(Ω)) snp (1 -s n )[u n ] p W sn ,p (R N ) ≤ C(N, p)(diam(Ω) snp L ≤ C
for some constant depending only N, p, diam(Ω), s 0 and L. Assume first that s 0 < 1 and let ε > 0 be small enough. Observe that since q < p * it follows that q < p * s0-ε if ε is small enough. Hence, if s 0 -ε < s n , using property 2 of Proposition 2.1, and the previous estimate we have

[u n ] p W s 0 -ε,p (R N ) ≤ [u n ] p W sn,p (R N ) + C(N, p) 1 (s 0 -ε)p - 1 s n p ≤ C (6.1)
for some C independent of n. Then there exists u ∈ W s0-ε,p (Ω) and a subsequence, still denoted by (u n ) n , such that

u n u in W s0-ε,p (R N ), u n → u in L q (Ω), u n → u in L p (Ω),
where we have used the compact imbedding of W 1-ε,p (Ω) into L q (Ω) and into L p (Ω). Hence for all ε > 0, using (6.1),

[u] p W s 0 -ε,p (R N ) ≤ lim inf n→∞ [u n ] p W s 0 -ε,p (R N ) ≤ lim inf n→∞ [u n ] p W sn,p (R N ) + C(N, p) 1 (s 0 -ε)p - 1 s 0 p .
Letting ε → 0 and using Fatou's lemma the conclusion 1 is reached. If s 0 = 1, by Lemma 3.10 of [START_REF] Brasco | Stability of variational eigenvalues for the fractional p-Laplacian[END_REF] we infer the existence of u ∈ W 1,p 0 (Ω) such that, up to a subsequence, u n → u in L p (Ω). Moreover, using property 3 of Proposition 2.2 and the hypothesis we obtain that, since 1

-ε < s n if n is large enough, ε[u n ] p W 1-ε,p (Ω) ≤ (1 -s n )2 (1-sn+ε)p diam(Ω) (sn-1+ε)p [u n ] p W sn ,p (Ω) ≤ C (6.2)
for some C independent of n. Thus, the sequence (

u n ) is bounded in W 1-ε,p (Ω) .
Hence there exists u ∈ W 1-ε,p (Ω) and a subsequence, still denoted by (u n ) n , such that

u n u in W 1-ε,p (Ω), u n → u in L q (Ω), u n → u in L p (Ω).
Thus, letting n → ∞ in (6.2) and using that u n u in W 1-ε,p (R N ) we obtain

ε[u] p W 1-ε,p (Ω) ≤ lim inf n→∞ ε[u n ] p W 1-ε,p (Ω) ≤ 2 εp diam(Ω) εp lim inf n→∞ (1 -s n )[u n ] p W sn,p (R N ) .
Finally, letting ε → 0 and using Corollary 2 of [START_REF] Bourgain | Another look at Sobolev spaces[END_REF] we obtain the result of 1.

Our next result concerns the hypothesis on α(V, m) that allow us to have principal eigenvalues. As we want to study the sign of α(V, m) as s varies, for s ∈ (0, 1] let us write

α(s) = inf (1 -s)K[u] p W s,p (R N ) + Ω V (x)|u| p dx : u ∈ W s,p (Ω), u p = 1,
and

Ω m(x)|u| p dx = 0 if s = 1, and
α(s) = inf Ω |∇u| p dx + Ω V (x)|u| p dx : u ∈ W 1,p 0 (Ω), u p = 1 and Ω m(x)|u| p dx = 0 if s = 1.
Proposition 6.2. Let s 0 ∈ (0, 1] and assume that α(s 0 ) > 0. Then there exists ε > 0 such that α(s) > 0 for all s ∈ (s 0 -ε, s 0 + ε) ∩ (

Proof. Assume by contradiction that there exists a sequence s n → s 0 and a function

u n ∈ W sn,p (Ω) such that (1 -s n )K[u n ] p W sn,p (R N ) + Ω V (x)|u n | p dx ≤ 0, u n p = 1, Ω m(x)|u n | p dx = 0.
Let t n = u n r p and distinguish two cases. Case (a): the sequence (t n ) n is bounded. Then the sequence (1-

s n )K[u n ] p W sn ,p (R N )
is bounded. Case (b): the sequence (t n ) n tends to +∞. Then taking v n = u n /t n we have

(1 -s n )K[v n ] p W sn,p (R N ) + Ω V (x)|v n | p dx ≤ 0 and the sequence (1-s n )K[v n ] p W sn,p (R N ) is bounded.
Let us write z n = u n if case (a) occurs and z n = v n if case (b) occurs. Let us now distinguish the cases 0 < s 0 < 1 and the case s 0 = 1. 1. Case 0 < s 0 < 1. It follows from Lemma 6.1 with q = r p that there exists z ∈ W s0,p (Ω) such that, in case (a),

(1 -s 0 )K[z] p W s 0 ,p (R N ) + Ω V (x)|z| p dx ≤ lim inf n→∞ (1 -s n )K[z n ] p W sn,p (R N ) + Ω V (x)|z n | p dx dx ≤ 0, z p = 1,
and the same inequality holds in case (b) with z r p = 1. Since Ω m(x)|z| p = 0 we have a contradiction with α(s 0 ) > 0. 2. Case s 0 = 1. We obtain similarly that the sequence (1

-s n )K[z n ] p W sn ,p (R N )
is bounded, with either z n p = 1 or z n r p = 1. By Lemma 6.1 there exists z ∈ W 1,p 0 (Ω) such that

Ω |∇u| p dx+ Ω V (x)|z| p dx ≤ lim inf n→∞ (1-s n )K[z n ] p W sn ,p (R N ) + Ω V (x)|z n | p dx ≤ 0.
Notice that again z q = 1, with either q = p or q = r p, and Ω m|(x)z| p = 0. Thus α(1) ≤ 0, a contradiction.

Let us now write

λ ±1 (s) := λ ±1 (V, m) = ± inf K(1 -s) R 2N |u(x) -u(y)| p dx dy |x -y| N +sp + Ω V |u| p dx : u ∈ W s,p (Ω), Ω m|u| p dx = ±1
and

λ ±1 := ± inf Ω |∇u| p + V (x)|u| p dx : u ∈ W 1,p 0 (Ω), Ω m(x)|u| p dx = ±1 .
We have the following result that generalizes, for indefinite weights, [10, Lemma 4.12].

Proposition 6.3. Assume that for some s 0 ∈ (0, 1], α(s 0 ) > 0. Then

lim s→s0 λ ±1 (s) = λ ±1 (s 0 ).
Proof. We only give the proof for λ 1 (s). By Proposition 6.2 α(s) > 0 for s close to s 0 , so λ 1 (s) is a principal eigenvalue associated with problem (1.1). Let (s n ) n be a sequence in (0, 1] converging to

s 0 ∈ (0, 1]. Let us show that lim n→+∞ λ 1 (s n ) = λ 1 (s 0 ). (6.3) 1.
Case s 0 ∈ (0, 1). By definition of the first eigenvalue, we know that if ϕ ∈ C ∞ 0 (Ω) and Ω m(x)|ϕ| p = 1, then

λ 1 (s n ) ≤ K(1 -s n ) R 2N |ϕ(x) -ϕ(y)| p |x -y| N +snp dx dy + Ω V (x)|ϕ| p dx
for all n ∈ N. Therefore, by dominated convergence theorem, we obtain lim sup n→+∞ λ 1 (s n ) ≤ λ 1 (s 0 ). To prove the reverse inequality let (s n k ) k be a subsequence of (s n ) n such that lim

k→+∞ λ 1 (s n k ) = lim inf n→+∞ λ 1 (s n ).
Let 0 ≤ u n k ∈ W s k ,p (Ω) be an eigenfunction associated with λ 1 (s n k ) such that

Ω m|u n k | p dx > 0 and [u n k ] p W sn k ,p (R N ) = 1,
then in particular, using u n k as test function in equation (1.1) for λ = λ(s n k ), we have

λ 1 (s n k ) Ω m(x)|u n k | p dx = K(1 -s n k ) + Ω V (x)|u n k | p dx. (6.4)
By Lemma 6.1 there exists u ∈ W s0,p (Ω) such that, up to a subsequence,

[u] p W s 0 ,p (R N ) ≤ lim inf k→∞ [u n k ] p W sn k ,p (R N ) = 1, u n k → u in L r p
(Ω) and u n k → u in L p (Ω). (6.5) Hence, using (6.4) we find on the one hand that lim inf

k→∞ λ 1 (s n k ) Ω m(x)|u| p dx = K(1 -s 0 ) + Ω V (x)|u| p dx (6.6)
and, on the other hand using (6.4) and (6.6),

K(1 -s 0 )[u] p W s 0 ,p (R N ) + Ω V (x)|u| p dx ≤ K(1 -s 0 ) + lim inf k→∞ Ω V (x)|u n k | p dx = lim inf k→∞ λ 1 (s n k ) Ω m(x)u| p dx. (6.7)
It remains to prove that Ω m(x)|u| p dx > 0 to conclude from the previous inequality that λ 1 (s 0 ) ≤ lim inf k→∞ λ(s n k )

(notice that the function v = u/ Ω m(x)|u| p dx 1/p will be then admissible in the definition of λ 1 (s 0 )) and the proof of the proposition is completed.

To prove that Ω m(x)|u| p dx > 0, remember first that Ω m(x)|u n k | p dx > 0 for all k ∈ N and assume by contradiction that Ω m(x)|u| p dx = 0. Using (6.6) we infer that u ≡ 0 and, using (6.7) we obtain

K(1 -s 0 )[u] p W s 0 ,p (R N ) + Ω V (x)|u| p dx ≤ 0, a contradiction with the hypothesis α(s 0 ) > 0. 2. Case s 0 = 1. Let ϕ ∈ C ∞ 0 (Ω) such that Ω m(x)|ϕ| p dx = 1. Then for any n ∈ N, λ 1 (s n ) ≤ K(1 -s n ) R 2N |ϕ(x) -ϕ(y)| p |x -y| N +snp dx dy + Ω V (x)|ϕ| p dx.
Thus, by Proposition 2.2, lim sup

n→+∞ λ 1 (s n ) ≤ Ω |∇ϕ| p dx + Ω V (x)|ϕ| p dx.
As ϕ is arbitrary, we have lim sup

n→+∞ λ 1 (s n ) ≤ λ 1 (1).
As in the previous case, let us prove that lim inf

n→+∞ λ 1 (s n ) ≥ λ 1 (1). Let (s n k ) k be a subsequence of (s n ) n such that lim k→+∞ λ 1 (s n k ) = lim inf n→+∞ λ 1 (s n ). (6.8)
Let u k be an eigenfunction associated with λ 1 (s n k ) such that

K(1 -s k )[u k ] p W s k ,p (R N ) = 1. (6.9)
Then, as u n k is an eigenfunction we have

λ 1 (s n k ) Ω m(x)|u n k | p dx = K(1 -s k )[u n k ] p W s k ,p (R N ) + Ω V (x)|u n k | p dx = 1 + Ω V (x)|u n k | p dx.
(6.10) By Lemma 6.1 there exists u ∈ W 1,p 0 (Ω) such that

Ω |∇u| p dx + Ω V (x)|u| p dx ≤ lim inf k→∞ (1 -s n k )K[u n k ] p W s k ,p (R N ) + Ω V (x)|u n k | p dx = lim inf k→∞ λ 1 (s n k ) Ω m(x)|u| p dx. (6.11)
Thus, if Ω m(x)|u| p dx > 0, we can conclude, rescaling the previous inequality, that

λ 1 ≤ lim inf k→+∞ λ 1 (s n k )
and the proof of the proposition is complete. To prove that Ω m(x)|u| p dx > 0 we argue as before using now equations (6.10), (6.11) and that α(1) > 0 by hypothesis.

Appendix A

The following two results are, essentially, consequence of the convexity of the function t → |t| p-2 t. Lemma 7.1. A discrete version of Picone's inequality [START_REF] Amgibech | On the discrete version of Picone's identity[END_REF] Let p ∈ (1, +∞). For all functions ξ and φ defined on R N such that ξ 0, and φ > 0, we have Proof. For sake of completeness we give the proof of this inequality. It uses the following convexity inequality due to [START_REF] Amgibech | On the discrete version of Picone's identity[END_REF]. Fix x, y in R N and put a = ξ(y), b = ξ(x), t = φ(x) φ(y) and assume that 0 < b < a. It suffices to prove that for any p > 1 and 0 < t < 1, one has

L(ξ, φ) 0 on R N × R N
|a -b| p ≥ a p (1 -t) p-1 -b p 1 t -1 p-1
which is equivalent to say that

(1 -t) |a -b| p (1 -t) p + t b p t p > a p
which follows from the convexity of the function f (x) = |x| p . Notice that the equality on this inequalities arrives if and only if t = b/a, that is, L(ξ, φ)(x, y) = 0 for all x, y ∈ R N if and only if ξ/φ = cte.

Let us quote without proof the following second estimate. for every a, b ∈ R, and every A, B ≥ 0.

Appendix B

For completeness we give the following regularity result for the nonlocal nonhomogeneous problem

(-∆ p ) s u + V (x)|u| p-2 u = f (x) in Ω, u = 0 in R N \ Ω. ( 8.1)
Proposition 8.1. Assume sp < N , f ∈ L q (Ω), V ∈ L q (Ω) for some q ≥ N/sp, and u ∈ W s,p (Ω) is a solution of (8.1). Then for any t ∈ [1, +∞), u ∈ L t (Ω) and there exists a constant C t depending on t and on Ω,

V N/sp , f N/sp , N , s, p such that u L t (Ω) ≤ C t . (8.2)
Proof. We borrow some ideas from [START_REF] Brasco | The second eigenvalue of the fractional p-Laplacian[END_REF][START_REF] Guedda | Quasilinear elliptic equations involving critical Sobolev exponents[END_REF]. For every 0 < ε 1 and any positive function ϕ ∈ C ∞ 0 (Ω) we define the smooth convex Lipschitz function

g ε (t) = (ε 2 + t 2 ) 1/2 ,
and choose the test function ψ = ϕ|g ε (u)| p-2 g ε (u) in the variational formulation of (8.1). Then we obtain

K(1 -s) R 2N u(x) -u(y) p-2 u(x) -u(y) |x -y| N +sp ϕ(x)|g ε u(x) | p-2 g ε u(x) -ϕ(y)|g ε u(y) | p-2 g ε u(y) dx dy ≤ Ω |f (x)ϕ(x)|g ε u(x) | p-2 g ε u(x) |dx + Ω |V (x)|u(x)| p-1 ϕ(x)|g ε u(x) | p-2 g ε u(x) |dx.
By using (7.1) with a = u(x), b = u(y), A = ϕ(x) and B = ϕ(y) we have

K(1 -s) R 2N g ε u(x) -g ε u(y) p-2 g ε u(x) -g ε u(y) |x -y| N +sp ϕ(x) -ϕ(y) dx dy ≤ Ω |f (x)|ϕ(x)|g ε u(x) | p-1 dx + Ω |V (x) u(x)| p-1 ϕ(x)|g ε u(x) | p-1 dx.
By observing that g ε converges to g(t) := |t| as ε → 0, |g ε (t)| ≤ 1 and using Fatou's Lemma, we obtain

K(1 -s) R 2N u(x) -u(y) p-2 u(x) -u(y) |x -y| N +sp ϕ(x) -ϕ(y) dx dy ≤ Ω |f (x)|ϕ(x)dx + Ω |V (x) u(x)| p-1 ϕ(x)dx. (8.3)
By the density of C ∞ 0 (Ω) in W s,p (Ω) (see Proposition 2.2), the same inequality remains true for any positive ϕ ∈ W s,p (Ω).

For k > 0 and t ≥ p we define u k and ϕ k as follows:

u k := min{|u|, k} and ϕ k (u) := t p p p (t -p + 1) u t-p+1 k .
By definition, ϕ k (u) ∈ W s,p (Ω), and then by relation (8.3), we can write

K(1 -s) R 2N u(x) -u(y) p-2 u(x) -u(y) |x -y| N +sp (ϕ k (u)(x) -ϕ k (u)(y)) dx dy ≤ Ω |f (x)|ϕ k (u(x))dx + Ω |V (x) u(x)| p-1 ϕ k (u(x))dx. (8.4) For M > 0, set Ω M := {x ∈ Ω : |V (x)| > M }. We have Ω |V (x) u| p-1 ϕ k (u)dx ≤ M Ω\Ω M |u(x)| p-1 |ϕ k (u)|dx + V L N/sp (Ω M ) Ω u(x)| p-1 ϕ k (u)| N/(N -sp) dx N -sp/N . Moreover, thanks to [4, Lemma C.2] we have u(x) -u(y) p-2 u(x) -u(y) (u t-p+1 k (x) -u t-p+1 k (y)) ≥ (t -p + 1)p p t p u k (x) t/p -u k (y) t/p p . (8.5)
Thus, by (8.4) and (8.5), the relation

(1 -s)K R 2N u k (x) t/p -u k (y) t/p p |x -y| N +sp dx dy ≤ M t p (t -p + 1)p p Ω |u(x)| p-1 |ϕ k (u(x))|dx + t p V L N/sp (Ω M ) (t -p + 1)p p Ω u(x)| p-1 ϕ k (u(x))| p * s /p dx p/p * s + t p (t -p + 1)p p Ω |f (x)|ϕ k (u(x))dx
holds and by the Sobolev's embedding of W s,p (Ω) into L p * s (Ω), there exists a constant S N,s,p such that Then for all t p we obtain

S N,s,p Ω u k (x) tp * s p p/p * s ≤ (1 -s)K R 2N u k (x) t/p -u k (y) t/p p |x -y| N +sp dx dy ≤ M t p (t -p + 1)p p Ω |u| p-1 |ϕ k (u)|dx + t p V L N/sp (Ω M ) (t -p + 1)p p Ω u| p-1 ϕ k (u)| p * s /p dx p/p * s + t p (t -p + 1)p p Ω |f (x)|ϕ k (u)dx. If we choose M such that V L N/sp (Ω M ) ≤ (t-p+1)p p S N
(t -p + 1)p p S N,s,p 2t p Ω |u k (x)| tN/(N -sp) (N -sp)/N ≤ M Ω |u(x)| t dx + f L N/sp (Ω) |Ω| p(p-1)/p * s t Ω |u(x)| tN/(N -sp) (t+1-p)(N -sp) tN . (8.6)
Let us set t 0 = p. Since by definition, u ∈ L t0 (Ω), it follows that u ∈ L t0N/(N -sp) (Ω), and thus, thanks to Fatou's Lemma we have

(t 0 -p + 1)p p S N,s,p 2t p 0 u t0 L t 0 N/(N -sp) ≤ M u t0 L t 0 (Ω) + |Ω| p(p-1)/p * s t0 f L N/sp u t0+1-p L t 0 N/N -sp . Therefore using Young's inequality we obtain u L t 0 ≤ C 1,t0 u L t 0 (Ω) + C 2,t0 f 1/(p-1)
L N/sp , where C 1,t0 and C 2,t0 depend on M , N , s, p, t 0 , and |Ω|. Now if we take t 1 = t0N N -sp ≥ p and since u ∈ L t1 (Ω), it follows that u ∈ L t1N/(N -sp) (Ω) and we let k → +∞, by Fatou's Lemma, and using Young's inequality we obtain

u L t 1 N/(N -sp) ≤ C 1,t1 u L t (Ω) + C 2,t1 f 1/(p-1)
L N/sp , where C 1,t1 and C 2,t1 depend on M , N , s, p, t 1 , and |Ω|.

Thus as a consequence, if we define the sequence (t l ) l∈N by t 0 = p, t l = N N -sp l p, l ∈ N * we find that u ∈ L t l (Ω) for any l ∈ N * . Since 1 < p ≤ t l for all l ∈ N and t l -→ l→+∞ +∞, we conclude that u ∈ L t (Ω) for any t > 1, and (8.2) follows.

When V ∈ L r (Ω) and f ∈ L r (Ω) with r > N sp , a better estimate holds. Thus, by replacing f by f + V |u| p-2 u we can assume that V ≡ 0 in equation (8.1). Let us assume first that u ≥ 0. For any k > 0 take u k defined as above and define now for any α > 0, Email address: leadiare@imsp-uac.org

DΨ

  (x, y)dy := lim ε→0 D\ Bε(x)

2. 1 .

 1 Basic results about fractional Sobolev spaces. Let p ∈ [1, +∞), s ∈ (0, 1) and let Ω ⊂ R N be an open set.

Proposition 2 . 1 (

 21 [START_REF] Del Pezzo | Global bifurcation for fractional p-Laplacian and application[END_REF]). Let Ω be a bounded open set of R N .

Theorem 4 . 2 .

 42 Assume that V and m satisfy the hypotheses (C1) and (C2).

5 . 5 . 1 .

 551 Nodal domains and isolation of the principal eigenvalues Measure of the nodal domains of non principal eigenvalues. By a nodal domain of a function v ∈ W s,p (Ω) ∩ C(Ω), v ≡ 0, we mean a maximal connected open subset of either {x ∈ Ω : v(x) > 0} or {x ∈ Ω : v(x) < 0}.

  with L(ξ, φ)(x, y) := |ξ(y)-ξ(x)| p -|φ(y)-φ(x)| p-2 (φ(y) -φ(x)) ξ p (y) φ(y) p-1 -ξ p (x) φ(x) p-1 , for all (x, y) ∈ R N × R N . Moreover, we have L(ξ, φ) = 0 ⇐⇒ ∃k ∈ R s.t. φ = kξ.

Lemma 7 . 2 ([ 6 ,

 726 Lemma A 1 ]). Let 1 < p < ∞ and g : R → R be a convex function, then |a -b| p-2 (a -b) A|g (a)| p-2 g (a) -B|g (b)| p-2 g (b) ≥ |g(a) -g(b)| p-2 g(a) -g(b) A -B , (7.1)

Proposition 8 . 2 .

 82 Assume that V ∈ L r (Ω) and f ∈ L r (Ω) with r > N sp . Let u ∈ W p,s (Ω) be a solution of (8.1). Then u ∈ L ∞ (Ω) and there existsC = C s, p, N, Ω, f L r (Ω) , V L r (Ω) , u L p * s (Ω) such that u L ∞ (Ω) ≤ C. (8.7)Proof. By Proposition 8.1, u ∈ L t (Ω) for any t > 1 and therefore V |u| p-2 u ∈ L t (Ω) for any t ∈ N sp , r ∩ 1 p-1+ 1 r , +∞ . Moreover, by Holder's inequality, V |u| p-2 u L t (Ω) ≤ V L r (Ω) C (p-1)tr r-t .

∈W 1 (α+1 1 C 2 L

 112 φ α,k := u k αp+1 ∈ W s,p (Ω) ∩ L ∞ (Ω).Using φ α,k as test function, one obtainsK(1 -s) R 2N |u(x) -u(y)| p-2 (u(x) -u(y)) |x -y| N +sp u W s,p (Ω), by the Sobolev's embedding theorem there exists C 1 > 0 such thatu k L p * s (α+1) (Ω) s,p (R N ) .Then, by (8.9), we obtainu k L p * s (α+1) (Ω) α + 1) p K(1 -s)(αp + 1) 1 p(α+1) f L r (Ω) Ω u r (αp+1) 1/r 1 p(α+1) . So, denoting C 2 = f L r (Ω) , we have u k L p * s (α+1) (Ω) ≤ C 1 (α + 1) p K(1 -s)(αp + 1) 1 p(α+1) Ω u r (αp+1) 1 pr (α+1) .On the other hand there exists C 3 > 0 such thatC 2 (α + 1) p K(1 -s)(αp + 1) 1 p √ α+1 ≤ C 3 for all α > 0.Consequently, we obtain that u k L p * s (α+1) (Ω) (α+1)pr (Ω) .

( 8 . 10 ) 1 +1 1 C 1 √ α 1 +1 3 |Ω| p-1 p 2 (L∀ m ≥ 1 . 2 m 1 C βm 3 |Ω| β 4 m

 8101111321214 Choosing α = α 1 in (8.10) such that (α 1 + 1)pr = p * s we obtainu k L p * s (α 1 +1) (Ω) ≤ C 1 α 1+α 1 ) 2 r u α 1 p+1 (α 1 +1)p L (α 1 +1)pr (Ω) .Next we choose α = α 2 in (8.10) such that (1 + α 2 )pr = (1 + α 1 )p * s and obtainu k L p * s (1+α 2 ) (Ω) (α 1 +1)p * s (Ω). By induction, for all m ∈ N * we can show thatu k L p * s (1+αm) (Ω) m ) m∈N isa sequence of positive numbers defined by α 0 = 0 and (1 + α m )pr = (1 + α m-1 )p * s One easily see that for all m ∈ N, 1 + α m = p * s pr m , and then, by hypothesis, α m → +∞ as m → +∞ since r > N ps . Moreover we haveu k L σm (Ω) ≤ C β (p-1) p 2 r u δm L σ m-1 (Ω) , with σ m = p * s (α m + 1), β m = 1 √αm+1 , and δ m = pαm+1 (αm+1)p . Notice that σ m → +∞, β m → 0 and δ m ↑ 1 as m → +∞. Letting k → +∞ and using Fatou's lemma we obtainu L σm (Ω) ≤ C βm 4 u δm L σ m-1 (Ω) ,(8.12)for some constant C 4 > 0. A simple computation givesu L σm (Ω) ≤ C βm+ m-1 i=1 βm-i i-1 k=0 δ m-k 4 u m i=1 δi L σ 0 (Ω) .Using that δ m ↑ 1 and thatβ m = ( u L σm (Ω) ≤ C max 1, u L p * s (Ω)and the conclusion follows. If u changes sign one can use instead, as in Proposition 8.1, u k = max{|u|, k}. Liamidi Leadi Département de Mathématiques, Faculté des Sciences et Techniques, Institut de Mathématiques et de Sciences Physiques, Université d'Abomey-Calavi, Benin

  Moreover the sequence (|w k | r p ) k converges to ((u -1) + ) r p almost everywhere in Ω. Hence W k → (u -1) + p

	L r p (Ω)	by the Lebesgue's
	dominated convergence theorem.
	2. Let us first prove two claims.
	Claim 1. For all k ∈ N,

Acknowledgements. O. Asso was financially supporting by the Deutscher Akademischer Austauschdienst (DAAD) through IMSP. This work was partially carried out while L. Leadi was visiting the LMPA of the Université du Littoral Côte d'Opale (ULCO). We would like to express our gratitude to these institutions.


B Merit

G Lemarquand

Member, IEEE V Lemarquand

In pursuit of increasingly linear loudspeaker motors

Keywords:

In this paper, analytical calculations are used to study existing loudspeaker motors totally made of permanent magnets. When the iron is removed from a loudspeaker, Eddy currents and reluctant effects are also removed, improving the linearity of the loudspeaker. Performances of each proposed magnetic motor structures are compared, from the intensity of their created induction to their ability to respond as an accurate image of the driving current. For a given structure, the authorized excursion length on which the loudspeaker motor is linear is a criterion for limiting its frequency range. The good understanding of the behavior of any magnets assemblies lead to a so called "linear structure", whose volume is optimized and able to create a very uniform induction, along a distance range that can be chosen. When this is achieved, the force that makes the moving part of the loudspeaker oscillate becomes proportional to the current running through its voice-coil. Thus, this loudspeaker motor is linear as well as using a reduced magnets volume, and can be applied to all frequency ranges.

I. INTRODUCTION

HE DRAWBACKS due to the iron in loudspeaker motors have already been identified [START_REF] Vanderkooy | A model of loudspeaker driver impedance incorporating Eddy currents in the pole structure[END_REF][START_REF] Lemarquand | Ironless loudspeakers[END_REF]. Iron is the source of Eddy currents, and creates the additional so called reluctant force, proportional to the square of the current! Moreover, when the magnetic field seen by the voice-coil depends on the axial position in the airgap, the motion force becomes also nonlinear. The iron in loudspeaker motors always leads to this dependence. The linearity of a loudspeaker needs the removal of the iron.

Thanks to a price decrease, motors totally made of permanent magnets become more and more interesting [START_REF] Lemarquand | Ironless loudspeakers[END_REF][START_REF] Ohashi | Magnetic circuit and speaker[END_REF][START_REF] House | Transducer motor assembly[END_REF]. Searching for even more linear and accurate loudspeakers, the last problem to solve when iron is removed is to make uniform the radial magnetic induction seen by the coil along its entire trajectory [START_REF] Klippel | Tutorial "Loudspeaker Nonlinearities-Causes, Parameters, Sysmptoms[END_REF].

This paper studies ironless structures, with the aim to compare them and to understand them, and paying attention to the linearity of the loudspeaker rather than its efficiency. To consider magnets that exist in the commerce, these annular structures only have rectangular cross sections. Then we propose a structure able to create well-controlled uniform magnetic field.

II. ANALYTICAL STUDIES

A. Interest of analytical calculations

Most work on magnetic structures is lead thanks to finite elements analysis. The magnetic field created by permanent magnets can be predicted whatever their shape or magnetization are. But it takes too long when searching for optimizing these structures. Thus, understanding and control of the behavior of such magnets is quite hard. This is very restrictive in the loudspeaker world, where the magnetic field has to be intense and uniform in a limited gap.

That is the reason why we have decided to calculate exactly the magnetic field created by any structures totally made of permanent magnets. Using a coulombian approach [START_REF] Rakotoarison | Using coulombian approach for modeling scalar potential and magnetic field of a permanent magnet with radial polarization[END_REF], the analytical resolution of ( 1 In (1) the index m refers to the coordinates of the considered magnet.

B. Magnetic pole surface density

This section only considers magnetic pole surface density, i.e. the first part of (1):

The three components of the magnetic field created by right angles magnets have already been analytically calculated [START_REF] Rakotoarison | Using coulombian approach for modeling scalar potential and magnetic field of a permanent magnet with radial polarization[END_REF].

But, especially because most loudspeakers are of revolution symmetry, it becomes important to estimate the effects due to the curvature of the magnets, whose magnetization can be axial or either radial. The induction created by annular structures, whatever the shape of their cross section is, can be semi-analytically calculated thanks to the approximations of [START_REF] Bancel | Three-dimensional Analytical Optimization of Permanent Magnets Alternated Structure[END_REF] and [START_REF] Yonnet | Rare-Earth Iron Permanent Magnets[END_REF]. But for annular magnets with rectangular cross sections, a complete analytical expression of this magnetic field can now be used [START_REF] Ravaud | Analytical calculation of magnetic field created by permanent magnet rings[END_REF].

C. Magnetic pole volume density

The calculations presented in reference [START_REF] Ravaud | Analytical calculation of magnetic field created by permanent magnet rings[END_REF] do not take into account the magnetic pole volume density. But we have shown [START_REF] Ravaud | Influence of the magnetic pole volume density on the magnetic field created by ring permanent magnets whose polarization is radial[END_REF] that this magnetic pole volume density can be omitted if the magnetic field is observed close to the faces of the ring. In loudspeakers, the voice-coil is rather close to the magnetic rings so that approximations of [START_REF] Ravaud | Analytical calculation of magnetic field created by permanent magnet rings[END_REF] are sufficient to predict the T -------- real behavior of magnetic structures. Nevertheless all results are verified thanks to semi-analytical resolutions [START_REF] Rakotoarison | Using coulombian approach for modeling scalar potential and magnetic field of a permanent magnet with radial polarization[END_REF][START_REF] Ravaud | Influence of the magnetic pole volume density on the magnetic field created by ring permanent magnets whose polarization is radial[END_REF][START_REF] Furlani | A three-dimensional field solution for radially polarized cylinders[END_REF]. Thanks to the analytical resolution of (2), it is now possible to calculate quickly and without approximation the magnetic field created by these structures, with axial or radial magnetization. This tool is powerful and makes easier and faster the study of the influences of each annular magnets characteristic, especially their radius.

III. BEHAVIOR OF IRONLESS STRUCTURES

This section describes the three structures of ironless loudspeaker motor of Fig. 1a, 1b and 1c. They are made of rings with rectangular section, whose magnetization J or j can be either radial or axial. For the studies, the magnetization J is always equal to 1T. The point O represents the location of the voice-coil in its rest position.

In the considered applications, the voice-coil oscillates in the direction Oz parallel to the revolution axis of the loudspeaker. The induction has to be uniform along this direction: the more uniform the induction, the more accurate the loudspeaker; and the more intense the magnetic induction, the more efficient the loudspeaker.

The induction is calculated on the Oz axis at a distance a from the inner radius of the rings. This location corresponds to the mean value of the voice-coil thickness (the coil thickness is the radial length of the coil).

For the study, let us consider a parameter called z uni . Given a structure, this parameter represents the axial distance across which the magnetic induction seen by the voice-coil has a peak-to-peak variation smaller than 1%. Thus, across this distance, the induction is considered uniform. In classical loudspeakers, the induction variation easily exceeds 10% across a distance that corresponds to the trajectory of the voice-coil; that is why significant nonlinear effects are produced and it is important to search for structures that lead to more uniform induction.

A. Two axially magnetized annular rings Fig. 1a shows an ironless motor made of two axially magnetized concentric rings. These rings are positioned one above the other, at an axial distance e from each other. They are separated by air and their respective magnetizations are in repulsion. Thus the magnetic flux in the coil is expected to be radial. This kind of structure has been patented by the Harman society [START_REF] House | Transducer motor assembly[END_REF].

Fig. 2 shows how the created induction varies when the magnets deviate. This induction varies too much to obtain a significant z uni . When the magnets are close to each other, the induction can be intense, up to 0.6T, but then the possible linear coil displacement is very small. Thus it is quite clear that this structure should be dedicated to loudspeakers in which the voice-coil has very small displacements, so preferably to tweeters.

B. One radially magnetized ring

Let us consider the structure of Fig. 1b. It is constituted by one radially magnetized ring facing the moving coil. Previous studies [START_REF] Lemarquand | Ironless loudspeakers[END_REF] on the induction created by a rectangular magnet show that a good combination between its dimensions can always lead to a quite uniform induction in front of the magnets. In the best case, i.e. when dimensions are analytically optimized, the axial distance z uni is about 60% of the considered magnet height. The same conclusions are available for annular magnets with rectangular section.

Fig. 3 shows that, for a given length, the induction can be quite constant over an axial range if the magnet height is optimal (dot-dashed line in Fig. 3). Thus this structure can be dedicated to every loudspeaker motors. But it generally needs significant volumes of magnets and its created induction intensity is modest.

Fig. 3 also shows that small height magnets only act over short axial areas (see bold line in Fig. 3). This localized action will be useful in further structures.

C. Combination of the first two solutions

The Sony society patented the structure of Fig. 1c [3]. It is constituted by a stack of three rings of equal radial length. The central ring is radially magnetized while the external rings are axially magnetized. According to the patent, the magnetizations J and j are equal. The magnetizations of the external rings help the total magnetic flux to close; so this structure combines the effects of the first two structures (a) and (b). That is why it creates high magnetic induction levels with improved uniformity compared to the structure (a), while using relatively small volume of magnets.

Fig. 4a shows the induction calculated along the observation axis created by the whole structure (bold line) and by the separated elements. The induction variation along the Oz axis over a range corresponding to the height of the central magnet represents 19% of the central value (0.5T). Because of this significant variation, z uni is only 1.28mm, that is 14% of the total height of the structure. Note that the induction variation is 10% across the distance of 4.2mm that separates the two maxima of this induction. The structure (c), which is 11% more compact than the previous optimal structure (b), produces a more intense induction but with a larger induction variation. The induction intensity increases by 34% while the distance z uni decreases by 76% compared to the optimal structure (b).

In addition, Fig. 4a shows the contribution of each ring to the total induction and especially that the external rings provide a really high magnetic induction compared to the induction provided by the central ring (see dashed line in Fig. 4a). Fig. 4b shows the result when the external magnetizations are lowered to 0.52T. In this case z uni equals 2.1mm, and now the induction variation is only 1.8% across the 3.4mm distance that separates the two maxima of this induction. The uniformity improvement is significant but leads to an 8% decrease of the total induction.

One has to note that this structure should not be used for low frequency drivers. When the intended z uni increases (over 10mm), the two external magnets do not contribute to increase the induction intensity anymore. In this frequency range, the structure (c) would act like the structure (b) while being more expensive.

IV. LINEAR STRUCTURE

The structure of Fig. 1d goes further in the idea of using magnets whose magnetizations differ in order to give the intended flat shape to the observed induction. It is constituted by a stack of three radially magnetized rings. It combines the higher magnetization of the external rings and their small height to use their much localized action (as described in Fig. 3). Contrary to the previous structure (c), the external rings are only aimed to compensate the bell shape of the induction created by the central ring; that is why it creates less intense inductions.

Fig. 5a shows the contribution of each ring and the resulting induction (bold line). The set of dimensions is chosen in order to have the same magnet section as the one of the previous optimal structure (b) (81mm 2 ). In this case, the performance improvement compared to the structures (b) is not significant. But the structure (d) can be optimized. Indeed, the distance z uni can always correspond to the dimension H, without any influence on the induction intensity. The other dimensions and the magnetizations are calculated in order to optimize the volume of the whole structure, and the uniformity and the intensity of the magnetic field.

Thus, the geometry has been optimized in order to obtain an induction of equal intensity (0.33T) and having about the same z uni of 5.4mm. The optimal dimensions we find correspond to a more spacesaving structure whose total height As a result, the structure (d) leads to very compact devices that have a very uniform induction over a range that can be chosen. This is also applicable when searching for higher distances z uni , i.e. for low frequency loudspeakers. Fig. 6 shows the induction created by both structures (b) and (d), whose volumes are optimized in order to obtain the same z uni equal to 40mm, with the same created induction intensity. The volume of (d) is 20% lower than the one of the structure (b).

Thus, since voice-coils can easily remain in a uniform induction, the structure (d) leads to highly linear loudspeaker motors and represents a real benefit in space occupation.

V. CONCLUSION Analytical calculations of the induction created by structures totally made of permanent magnets make it possible to only concentrate in an automatic optimization of one parameter as z uni . This parameter represents the axial distance across which the induction is considered uniform; the uniformity of the induction being related to the linearity of the loudspeaker motor.

  ) makes it possible to calculate the field B  created at the point r  by a magnet of magnetization J  and volume V m bounded by the closed surface S m :

Fig. 1 .

 1 Fig. 1. The four studied ironless loudspeaker motor structures. a) two axially magnetized rings; b) one radially magnetized ring; c) one radially magnetized ring and two axially magnetized rings; d) three axially magnetized rings.

Fig. 2 .

 2 Fig. 2. Radial magnetic induction, Br (T), along the observation axis at a distance a = 0.3mm, created by the structure (a) for H = 4.5mm, L = 9mm and for various values of the distance between the magnets, from top to bottom, e = 1mm (bold line), 3mm, 5mm, 7mm and 9mm (dashed line).

Fig. 3 .

 3 Fig. 3. Radial magnetic induction, Br (T), along the observation axis at a distance a = 0.3mm, created by the structure (b) for L = 9mm and for various values of the height: H = 1mm (bold line), then from top to bottom H = 3mm, 5mm, 7mm, 9mm (dot-dashed line) and H = 18mm (dashed line).

Fig. 4 .

 4 Fig. 4. (a) Radial magnetic induction, Br (T), along the observation axis at a distance a = 0.3mm, created by the structure (c) (bold line) with its central magnet with H = 5mm, L = 8mm, J = 1T (full line) and its two external magnets with h = 2mm, l = 8mm, j = 1T (dashed line). (b) Magnetization j is lowered to 0.52T. The dotted-line remains the induction created by the patented structure where the magnetizations J and j are equal to 1T.

Fig. 5 .

 5 Fig. 5. Radial magnetic induction, Br (T), along the observation axis at a distance a = 0.3mm, created by the structure (d) (bold line) with : (a) its central magnet with H = 5mm, L = 9mm, J = 1T (full line) and its two external magnets with h = 2mm, l = 9mm, j = 1.01T (dashed line); (b) its central magnet with H = 5.2mm, L = 7mm, J = 1T (full line) and its two external magnets with h = 1mm, l = 8.5mm, j = 1.09T (dashed line).

is 7 .

 7 2mm (Fig.5b). The total section of this optimized structure (d) is 53.4mm 2 (34% decrease compared to the previous structure). As a matter of fact, one can easily understand the benefit in linearity and in volume compared to the structure (c), despite the loss of induction intensity. With about a 35% decrease of the induction intensity, the volume of the structure (d) is half the volume of the structure (c) to obtain a uniform induction over the same z uni as the structure (c).

Fig. 6 .

 6 Fig. 6. Radial magnetic induction, Br (T), along the observation axis at a distance a = 2mm, created by the structure (d) (bold line) -with its central magnet with H = 40mm, L = 28mm, J = 1.43T (full line) and its two external magnets with h = 7mm, l = 17mm, j = 1.49T (dashed line) -and by the structure (b) (dashed line) with H = 60mm and L = 46mm and J = 1.43T. The linear excursion of the voice-coil is ±20mm. So the parameter z uni should be directly related to the frequency range of a loudspeaker. Thus, the structure (a) should only be dedicated to tweeters, while structure (c) can also be dedicated to high-medium frequency loudspeakers. Structures (b) and (d) are more universal, but create less intense inductions. For very small displacements, the dimensions of the external rings of the structure (d) become negligible to compensate the bell-shaped induction created by the central ring. So the structure (b) is sufficient compared to the structure (d). But as soon as the intended z uni increases, the structure (d) represents a real benefit in linearity and in space occupation compared to the other structures.