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

. Let Ω be a bounded regular domain of R N , N (cid:62) 1, p ∈ (1 , + ∞ ), and s ∈ (0 , 1). We consider the eigenvalue problem


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 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 [12] 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. [15], where the following eigenvalue problem is considered.
− ∆u + a 0 (x)u = λm(x)u, in Ω, u = 0 on ∂Ω (1.2) 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, 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 [15]. Our construction of the equivalent problem is different from the one made in [15] and it is closer to the one used by Binding and Huang [3]. 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 [2,3,9,19,21]). For instance, recently [8] 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 [8] 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 • The space W s,p (Ω) is defined as the space of all u ∈ W s,p (Ω) such that u ∈ W s,p (R N ), whereũ is the extension by zero of u, outside of Ω. W s,p (Ω) is a Banach space endowed with the norm 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 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 Then there exists a positive constant C(N, p) such that 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 ofũ, the extension by 0 of any function u ∈ W s,p (Ω). (1) We will say that a function u ∈ W s,p (Ω) is a weak solution of (1.1) if 2) It should be noted that for all u ∈ W s,p (Ω), we have .
(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 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, 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 The proof of this theorem will follow from Lemma 3.4 below, based on the De Giorgi-Stampacchia iteration technique (see for instance [11,16,22], 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 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) .
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). Moreover the sequence (|w k | r p ) k converges to ((u − 1) + ) r p almost everywhere in Ω.
by the Lebesgue's dominated convergence theorem.
2. Let us first prove two claims. Claim 1. For all k ∈ N, (3.7) Claim 2. There exist D > 1 and β > 0 such that for all k ∈ N, To prove this claim, let us quote the following (trivial) inequality: By taking Besides, by taking w k+1 in the weak formulation of (3.4), we obtain from the previous inequality and therefore, using Claim 1, 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 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) and assume that ρ 1/p < σ. Choose η ∈ (ρ q−1 q , D − q q−1 ). It should be noted that Assume that (3.12) holds at order k and let us show that it holds at order k + 1. By Claim 2, Thus by passing to the limit in (3.12), we finally obtain that W k → 0.
then v ≤ 1 a.e., which gives If u − is not identically zero, we apply the same argument to −u, which is a weak solution of (1.1), to find that and estimate (3.5) follows.
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. [20], 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 and µ an eigenvalue parameter depending on the real λ.
According to [12], problem (4.1) admits a unique principal eigenvalue which we will denote µ(λ). Moreover, µ(λ) is simple and can be characterized as 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 The following proposition gives useful properties on the function λ → µ(λ). We will denote here and ϕ λ the unique positive eigenfunction of L p (Ω)-norm equal to 1 associated with µ(λ).
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 , which means that the function −µ is convex.
As a consequence of this proposition we have the following result. : is an eigenfunction associated with λ 1 (V, m) and it is sign definite. Same Proof. The proof given in [8] 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 We only give the proof of the first identity, the proof of the second one is similar.
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 := ) and using for all (a, b) ∈ R + × R + and q > 0 the trivial inequality and therefore An application of Picone's inequality to functions u T and ϕ λ0 + ε, and the fact that So when ε → 0, by the Lebesgue convergence theorem for all T > 0. Moreover, since as T → +∞ we obtain u T = u. Then by Fatou's lemma, 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 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 As a result, So when k tends to infinity we find that E V (u k ) − → λ 0 . Thus we can conclude that But ϕ λ0 is an eigenfunction associated with µ(λ 0 ) = 0, that means that 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 for all functions V ∈ Z and for all u ∈ W s,p (Ω).
Proof. This proof is a partial adaptation of Lemma 2 of [8]. 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 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 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 [10,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 Then passing to the limit we find by Fatou's lemma that 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 So we obtain The lemma follows by setting 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). 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, By using the Lebesgue dominated convergence theorem and passing to the limit we have 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 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 Let us start with the case N > ps. By the previous Sobolev embedding theorem, for some constant c > 0, we have and the estimate (5.1) follows.
If N = sp there exists some c > 0 such that for all q ≥ p, and the estimate (5.1) follows.
In the case N < sp, there exists some c > 0 such that (Ω) |N | 1/r , and the estimate (5.1) follows.
The following statement is a straightforward consequence of the above theorem. 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 which is a contradiction. 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 Since u k is an eigenfunction associated with λ k we have Thus passing to the limit and using that E V is weakly lower semi-continuous we obtain In particular u ≡ 0 and Assume first that α(V, m) > 0. Then (5.2) implies that Ω m(x)|u| p dx = 0. In fact 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 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 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 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. 6. 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 Then there exists a function u ∈ W s0,p (Ω) such that, up to a subsequence, Proof. First of all, by Poincaré's inequality, 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 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 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), Letting ε → 0 and using Fatou's lemma the conclusion 1 is reached. If s 0 = 1, by Lemma 3.10 of [5] 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, 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), Thus, letting n → ∞ in (6.2) and using that u n u in W 1−ε,p (R N ) we obtain Finally, letting ε → 0 and using Corollary 2 of [7] 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 Proof. Assume by contradiction that there exists a sequence s n → s 0 and a function u n ∈ W sn,p (Ω) such that 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 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), 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 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 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 0 ≤ u n k ∈ W s k ,p (Ω) be an eigenfunction associated with λ 1 (s n k ) such that then in particular, using u n k as test function in equation (1.1) for λ = λ(s n k ), we have By Lemma 6.1 there exists u ∈ W s0,p (Ω) such that, up to a subsequence, ,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 (6.6) and, on the other hand using (6.4) and (6.6), It remains to prove that Ω m(x)|u| p dx > 0 to conclude from the previous inequality that (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.
As in the previous case, let us prove that 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 Then, as u n k is an eigenfunction we have (6.10) By Lemma 6.1 there exists u ∈ W 1,p 0 (Ω) such that Thus, if Ω m(x)|u| p dx > 0, we can conclude, rescaling the previous inequality, that 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 [1] Let p ∈ (1, +∞). For all functions ξ and φ defined on R N such that ξ 0, and φ > 0, we have for all (x, y) ∈ R N × R N . Moreover, we have Proof. For sake of completeness we give the proof of this inequality. It uses the following convexity inequality due to [1]. 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 which is equivalent to say that 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.
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) Then for all t p we obtain 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 Therefore using Young's inequality we obtain 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 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.