Juan Casado-D Íaz

François Murat

Dpto De

Ecuaciones Diferenciales

Semilinear problems with right-hand sides singular at u = 0 which change sign

Keywords: Singular equations, monotone operators, existence, uniqueness, positive and nonpositive solutions. Mathematics Subject Classification: 35J25 certain λ

The present paper is devoted to study the existence of a solution u for a quasilinear second order differential equation with homogeneous Dirichlet conditions, where the right-hand side tends to infinity at u = 0. The problem has been considered by several authors after the 70's. Mainly, nonnegative right-hand sides are considered and thus only nonnegative solutions are possible. Here we consider the case where this right-hand side can change the sign and not restrictions on the growth at u = 0 are assumed. We show that a nonnegative solution in a sense introduced in the paper still exists. Moreover, this solution is stable with respect to the right-hand side and is unique if the right-hand side is nonincreasing in u. We also show that if the right-hand side goes to infinity at zero faster than 1/|u| then only nonnegative solutions are possible. In another case nonpositive solution or even solutions changing the sign are possible.

Introduction

The existence of a nonnegative solution for a singular semilinear second order partial differential problem with homogeneous Dirichlet conditions such as In [START_REF] Crandall | On a Dirichlet problem with a singular nonlinearity[END_REF], it is shown the existence and uniqueness of a nonnegative solution assuming that the limit in (1.1) is uniform in Ω and F = F (x, s) decreasing in s. Indeed, in [START_REF] Crandall | On a Dirichlet problem with a singular nonlinearity[END_REF] the equation is not written in a divergence form and the solution is a classical solution, i.e. it is in C 2 (Ω) ∩ C 0 ( Ω), which is strictly positive in Ω. The authors also study the behavior of the solution at the boundary and give some partial results about the existence of solution when F is not necessarily decreasing in s. In [START_REF] Coclite | On a singular nonlinear Dirichlet problem[END_REF] it is considered the case where A is the identity and F = F (x, s) = 1/s γ + (λs) q with γ, λ, q > 0. It is shown that for every γ > 0 and every q > 1, there exists λ ∈ (0, ∞) such that a solution exists if λ < λ and there is not solution if λ ≥ λ. The existence and uniqueness of classical solution for every λ ≥ 0 and p < 1 was before proved in [START_REF] Stuart | Existence and approximation of solutions of non-linear elliptic equations[END_REF].

-div(A(x)∇u) = F (x, u) in Ω u = 0 on ∂Ω, ( 1
Looking for solutions in a Sobolev space, the problem has been considered in [START_REF] Boccardo | Semilinear elliptic equations with singular nonlinearities[END_REF], where it is studied the case F (x, s) = f (x)/s γ , with γ > 0, f ≥ 0 not identically zero, f ∈ L r (Ω), for some r ≥ 1. It is proved the existence of a solution in H 1 loc (Ω) ∩ W 1,1 0 (Ω), which is strictly positive in Ω. The authors also study the integrability of u and ∇u depending on γ and m. In this paper the function u is a solution in the distribution sense, that is, taking test functions in D(Ω). It is proved in [START_REF] Boccardo | Some properties of solutions of some semilinear elliptic singular problems and applications to the G-convergence[END_REF] that more general test functions can be considered and that there is just one solution which can be obtained as the limit of the solutions corresponding to replace F by F n (x, s) = f n (x)/(s + 1/n) where f n ≥ 0 increases to f . Moreover it is studied the homogenization result corresponding to replace A in (1.1) by a sequence of matrix functions A n .

In [START_REF] Giachetti | A semilinear elliptic equation with a mild singularity at u = 0: existence and homogenization[END_REF], it is considered the case where F satisfies 0 ≤ F (x, s) ≤ h(x)

1 s γ + 1 , a.e. x ∈ Ω, ∀ s ≥ 0, (1.3)
with 0 < γ ≤ 1 and h in a certain space L r (Ω), r > 1. In this case, the authors provide a new definition of nonnegative solution which does not need the use of the strong maximum principle. It is shown that this solution is stable when we replace the function F by a sequence F n converging pointwise to F and satisfying (1.3) with h and γ independent on F . In particular, this shows that these are the solutions we find by approximating the singular function F by non-singular functions. It is also proved the uniqueness of solution if F is decreasing in s. The results are used to carry out the homogenization when the open set Ω is replaced by a sequence of open sets Ω ε satisfying similar conditions to those which appear in [START_REF] Cioranescu | Un terme étrange venu d'ailleurs[END_REF]. The extension of the existence, stability and uniqueness results to the "strong singular" case where F satisfies the weaker assumption 0 ≤ F (x, s) ≤ h(x) Γ(s) , a.e. x ∈ Ω, ∀ s ≥ 0, (1.4) with h as above and Γ a Lipschitz function, strictly increasing, and such that Γ(0) = 0 is carried out in [START_REF] Giachetti | Definition, existence, stability and uniqueness of the solution to a semilinear elliptic problem with a strong singularity at u = 0[END_REF] (see also [START_REF] Giachetti | On the definition of the solution to a semilinear elliptic problem with a strong singularity at u = 0. Nonlinear Analysis[END_REF]). The corresponding extension of the homogenization result for varying domains is considered in [START_REF] Giachetti | Homogenization of a Dirichlet semilinear elliptic problem with a strong singularity at u = 0 in a domain with many small holes[END_REF]. Excepting [START_REF] Crandall | On a Dirichlet problem with a singular nonlinearity[END_REF] where F (x, s) decreasing in s implies that F (x, s) can take negative values for s bigger enough, the rest of the papers we have mentioned above consider a function F : Ω × [0, ∞) → [0, ∞) in (1.1). Thus, by the maximum principle, if there exists a solution for (1.1), it must be nonnegative. However, taking F which can take nonnegative values introduces several questions such as: Is there still a positive solution? Can we find nonnegative solutions or even solutions changing the sign? To give an answer to these questions is the first motivation of the present paper. Indeed, with respect to the existence of nonpositive solutions we must refer for example to [START_REF] Dávila | Positive versus free boundary solutions to a singular elliptic equation[END_REF], [START_REF] Diaz | An elliptic equation with singular nonlinearity[END_REF], where it is considered the existence of nonnegative solutions for    -∆u = λG(x, u) -1 |u| β in Ω u = 0 on ∂Ω, (1.5) with λ > 0, G nonsingular and strictly positive (some other conditions are needed) and β ∈ (0, 1). Then, it is proved that a nonnegative solution exists if and only if λ is bigger than a F (x, t) dt, and can change the sign. The definition of solution given by the authors uses test functions which vanish at u = 0 and thus the equation is satisfied in Ω \ {u = 0}. It is proved the uniqueness for nonnegative solutions and g(x, .) decreasing.

In the present paper, more generally than (1.1) we deal with the problem

-div a(x, u, ∇u) = F (x, u) in Ω u = 0 on ∂Ω, (1.6)
with a : Ω × R × R N → R N a Carathéodory function satisfying usual assumptions in such way that the operator

v ∈ W 1,p 0 (Ω) → -div a(x, v, ∇v) ∈ W -1,p (Ω), (1.7)
is pseudo-monotone in the sense of , [START_REF] Lions | Quelques méthodes de résolution des problèmes aux limites non linéaires[END_REF]), for some p > 1 and it is such that a(x, 0, 0) = 0 (see (2.6), (2.7), (2.8) and (2.9) below for the exact assumptions on a).

In the first part of the paper we are interested in the existence of nonnegative solutions for (1.6) although F can take nonnegative values. We assume F = F (x, s) a Carathéodory function from Ω × [0, ∞) into R ∪ {∞}. It is nonnegative at s = 0 (being able to take an infinite value) and has a growth at s → +∞ at most of order p -1. Namely, there exists ν > 0 not too large such that for every δ > 0, there exists k δ ∈ L p (Ω), k δ ≥ 0, such that

F (x, s) ≤ k δ (x) + νs p-1 , a.e. x ∈ Ω, ∀ s ≥ δ.
Observe that this assumption is guaranteed if F satisfies (1.4). Since we do not assume F to be nonnegative, we also introduce a below estimate assuring that (2.15)). With these general hypotheses we show that a nonnegative solution still exists. In fact, similarly to [START_REF] Giachetti | A semilinear elliptic equation with a mild singularity at u = 0: existence and homogenization[END_REF] and [START_REF] Giachetti | Definition, existence, stability and uniqueness of the solution to a semilinear elliptic problem with a strong singularity at u = 0[END_REF] we introduce a suitable definition of solution (it is in particular a solution in the distribution sense) for which we show existence, stability with respect to the right-hand side and uniqueness for F (x, s) non increasing in s and a independent of s. The estimates we use to get the result are inspired in [START_REF] Giachetti | A semilinear elliptic equation with a mild singularity at u = 0: existence and homogenization[END_REF] and [START_REF] Giachetti | Definition, existence, stability and uniqueness of the solution to a semilinear elliptic problem with a strong singularity at u = 0[END_REF] although we use more general test functions. In particular, we observe that due to the nonlinearity of the operator given by (1.7), some duality arguments used in the choice of the test functions in [START_REF] Giachetti | A semilinear elliptic equation with a mild singularity at u = 0: existence and homogenization[END_REF] and [START_REF] Giachetti | Definition, existence, stability and uniqueness of the solution to a semilinear elliptic problem with a strong singularity at u = 0[END_REF] cannot be used in the present setting.

F -(x, v) is in W -1,p (Ω) for every v ∈ W 1,p 0 (Ω) (see
In the second part of the paper we consider the question relative to the existence of nonpositive solutions or even solutions changing the sign. Thus, the function F is now assumed to be defined in Ω × R. In Section 4 we show that if there exists δ, τ > 0 such that

F (x, s) ≥ τ |s| , ∀ s ∈ (-δ, 0), (1.8)
then every solution of (1.6) is necessarily nonnegative. That is, although, now F can take nonnegative values, the fact that it is very large near s = 0 is enough to preclude the existence of solutions taking negative values in a portion of Ω. Taking in particular F of the form

F (x, s) = f (x) |s| γ -g(x), (1.9)
with f (x) ≥ ρ a.e. in Ω for some ρ > 0, and γ ≥ 1, we get that all the solutions for (1.6) are nonnegatives. Now, the question is What happens if γ < 1? As we said above, references [START_REF] Dávila | Positive versus free boundary solutions to a singular elliptic equation[END_REF], and [START_REF] Diaz | An elliptic equation with singular nonlinearity[END_REF] show that it is possible to get nonpositive solutions. Here we give in Section 5 a different example in the simple case Ω = (0, l) ⊂ R, a(x, s, ξ) = ξ and F given by (1.9) with f and g positive constants and γ ∈ (0, 1). By a suitable change of variables, the problem is transformed into

   -w = 1 |w| γ -1 in (0, l) w(0) = w(l) = 0. (1.10)
We show that although there is a unique strictly positive solution, there also exist nonpositive solutions and even solutions changing the sign, but only if l is large enough. In fact the number of solutions increases with l and tends to infinity when l tends to infinity.

The paper is organized as follows:

• In Section 2 we show the existence of nonnegative solutions for (1.6) in the singular case and the stability and uniqueness of these solutions.

• In Section 3 we prove the results corresponding to Section 2.

• In Section 4 we show that (1.8) implies that only nonnegative solutions are possible. The proof of these results is given at the end of the Section.

• In Subsection 5.1 we state the results corresponding to the one-dimensional example. The corresponding proofs are given in Subsection 5.2.

2 Existence, stability and uniqueness of nonnegative solutions for the singular problem

The present section is devoted to the existence, in a sense which we describe below, of nonnegative solutions for problem

-div a(x, u, ∇u) = F (x, u) in Ω u = 0 on ∂Ω, (2.1)
where F = F (x, s) can take the value +∞ at s = 0. We also give a stability result with respect to F for this kind of solutions. To finish, we show a uniqueness result when F is decreasing in s and a = a(x, s, ξ) does not depend on s and is strictly monotone in ξ. The proof of these results will be given in Section 3.

We assume the following conditions on Ω, p, a = a(x, s, ξ) and F = F (x, s):

• For Ω and p we assume:

Ω is a bounded open set in R N , N ≥ 1. (2.2) p ∈ (1, ∞).
(2.3)

Moreover, we define p * ∈ (1, ∞], p ∈ (1, ∞) and (p * ) ∈ [1, ∞) by          p * = N p N -p if p < N p * is a fixed number in (1, ∞) if p = N p * = +∞ if p > N, (2.4)
p = p p -1 , (p * ) = p * p * -1 . (2.5) • For the function a : Ω × [0, ∞) × R N → R N we assume:
• The function a is a Carathéodory function, i.e.

a(., s, ξ) is measurable in Ω, ∀ (s, ξ) ∈ [0, ∞) × R N a(x, ., .) is continuous in [0, ∞) × R N , a.e. x ∈ Ω. ( 2

.6)

• There exist α, γ, a 0 such that

α > 0, γ ≥ 0, a 0 ∈ L (p * ) (Ω), a 0 ≥ 0 a(x, s, ξ) • ξ ≥ α|ξ| p -γs p -a 0 (x)s, ∀ (s, ξ) ∈ [0, ∞) × R N , a.e. x ∈ Ω. (2.7)
• There exist β, b such that

β > 0, b ∈ L p (Ω), b ≥ 0, |a(x, s, ξ)| ≤ β( ξ| + s + b(x) p-1 , ∀ (s, ξ) ∈ [0, ∞) × R N , a.e. x ∈ Ω. (2.8)
• The function a is monotone in the second variable, i.e.

a(x, s, ξ) -a(x, s, η)

• (ξ -η) ≥ 0, ∀ s ∈ [0, ∞), ∀ ξ, η ∈ R N , a.e. x ∈ Ω. (2.9)
Moreover, we define λ by

λ = lim inf R→∞ min R Ω |v| p dx : v ∈ W 1,p 0 (Ω) \ {0}, Ω a(x, v, ∇v) • ∇v + γ|v| p dx = R . (2.10)
• For the function F : Ω × [0, ∞) → R ∪ {+∞}, we assume:

• The function F is a Carathéodory function, i.e.

F (., s) is measurable in Ω, ∀ s ∈ [0, ∞), F (x, .) is continuous in [0, ∞), a.e. x ∈ Ω. (2.11) • F (x, 0) ≥ 0, a.e. x ∈ Ω. ( 2

.12)

• There exists ν ≥ 0 such that for every δ > 0, there exists k δ satisfying

k δ ∈ L (p * ) (Ω), k δ ≥ 0, F (x, s) ≤ k δ (x) + νs p-1 , ∀ s ≥ δ, a.e. x ∈ Ω. (2.13)
• The constants γ in (2.7), ν in (2.13), and λ defined by (2.10) are related by γ + ν < λ.

(2.14) In fact, taking in (2.7) s = 0, replacing ξ by tξ with t > 0, and dividing by t, we get

• If p ≤ N, ∃ν ≥ 0, k ∈ L (p * ) (Ω), k ≥ 0 F (x, s) ≥ -k(x) -νs p * , ∀ s ≥ 0, a.e. x ∈ Ω If p > N, ∀ m ∈ N, ∃ km ∈ L 1 (Ω), km ≥ 0, F (x, s) ≥ -km (x), ∀ s ∈ [0, m],
a(x, 0, tξ) • ξ ≥ αt p-1 |ξ| p , ∀ ξ ∈ R N , ∀ t > 0, a.e. x ∈ Ω.
Letting t tend to zero, this provides a(x, 0, 0)

• ξ ≥ 0, ∀ ξ ∈ R N ,
(u -δ) + ∈ W 1,p 0 (Ω), ∀ δ > 0 (2.21) ∇u ϕ ∈ L p (Ω) N , ∀ ϕ ∈ W 1,p 0 (Ω) ∩ L ∞ (Ω) (2.22) F (x, u) + ϕ p ∈ L 1 (Ω), ∀ ϕ ∈ W 1,p 0 (Ω) ∩ L ∞ (Ω), ϕ ≥ 0 (2.23)      Ω a(x, u, ∇u) • ∇ h(u)ϕ p ) dx = Ω F (x, u)h(u)ϕ p dx ∀ ϕ ∈ W 1,p 0 (Ω) ∩ L ∞ (Ω), ϕ ≥ 0, ∀ h ∈ W 1,∞ (0, ∞).
(

F (x, u)ϕ p ∈ L 1 (Ω), ∀ ϕ ∈ W 1,p 0 (Ω) ∩ L ∞ (Ω), ϕ ≥ 0.
(2.25)

Moreover, using ∇(h(u)ϕ p ) = h (u)∇u ϕ p + p h(u)ϕ p-1 ∇ϕ,
and assumptions (2.8), (2.22), we also have that a(x, ∇u)∇(h(u)ϕ) is in L 1 (Ω) for every ϕ and h in the conditions of (2.24). Thus, the two integrals which appear in (2.24) have a sense. Assumption (2.21) in combination with (2.20) gives

0 ≤ u < δ on ∂Ω, ∀ δ > 0,
which gives a sense to the boundary condition u = 0 on ∂Ω.

Following proposition shows that in (2.24) we can enlarge the set of test functions by introducing

W = w ∈ W 1,p (Ω), ∃ ϕ ∈ W 1,p 0 (Ω) ∩ L ∞ (Ω), ϕ ≥ 0 a.e. in Ω, |w| ≤ ϕ p a.e. in Ω, |∇w| ϕ p-1 χ {ϕ =0} ∈ L p (Ω) .
(2.26)

Observe that contrarily to the test functions used in (2.24), this set does not depend on the solution u. We have preferred to give the definition of solution taking test functions of the form

h(u)ϕ p with ϕ ∈ W 1,p 0 (Ω) ∩ L ∞ (Ω), ϕ ∈ W 1,p 0 (Ω), h ∈ W 1,∞ (Ω)
, instead of functions in W , in order to compare with the definition of solution given in [START_REF] Giachetti | A semilinear elliptic equation with a mild singularity at u = 0: existence and homogenization[END_REF] and [START_REF] Giachetti | Definition, existence, stability and uniqueness of the solution to a semilinear elliptic problem with a strong singularity at u = 0[END_REF].

Proposition 2.5. The space W is a vectorial subspace of W 1,p 0 (Ω). It contains the space of functions in W 1,p (Ω) ∩ L ∞ (Ω) with compact support in Ω and satisfies

wv ∈ W, ∀ w ∈ W, ∀ v ∈ W 1,p (Ω) ∩ L ∞ (Ω).
(2.27)

If a function u : Ω → R is a nonnegative solution of (2.1) in the sense established in (2.3), then it also satisfies F (x, u) = 0, a.e. in {u = 0} (2.28)

|∇u| p-1 |∇w| ∈ L 1 (Ω), ∀ w ∈ W (2.29) Ω a(x, u, ∇u) • ∇w dx = Ω F (x, u)w dx, ∀ w ∈ W.
(2.30)

Remark 2.6. Proposition 2.5 proves in particular that every nonnegative solution of (2.1) in the sense given by Definition 2.3 is a solution in the distribution sense, i.e. we have

-div a(x, u, ∇u) = F (x, u) in D (Ω).
(2.31) By (2.22), we also know that u is in W 1,p loc (Ω) and then, for every open set ω strictly contained in Ω, we have that div a(x, u, ∇u) is in W -1,p (ω) while by (2.23)

F (x, u) is in L 1 (ω). It is well known that then (2.31) implies that F (x, u)v belongs to L 1 (Ω) for every v ∈ W 1,p (Ω) with compact support and Ω a(x, u, ∇u) • ∇v dx = Ω F (x, u)v dx, ∀ v ∈ W 1,p (Ω) with compact support in Ω. (2.32)
The following theorem provides a stability result for the nonnegative solutions of (2.1) when the right-hand side varies. 6), (2.7), (2.8) and (2.9), and a sequence of functions F n : Ω × R → R ∪ {∞} such that (2.11), (2.12), (2.13) (2.14) and (2.15) hold, with constants ν, ν and functions k δ , k and km independent of n. We also assume the existence of

Theorem 2.7. We consider a bounded open set Ω ⊂ R N , a function a : Ω×[0, ∞)×R N → R N , satisfying (2.
F : Ω × R → R ∪ {∞} such that
For a.e. x ∈ Ω, we have:

t n ≥ 0, t n → t =⇒ F n (x, t n ) → F (x, t).
(2.33)

Then, if u n is a sequence of nonnegative solutions of -div a(x, u n , ∇u n ) = F n (x, u n ) in Ω u n = 0 in Ω, (2.34)
there exists a nonnegative solution u of (2.1) such that for a subsequence of n, still denoted by n, we have

u n → u in L q (Ω), ∀ q < p * (2.35) u n u in L p * (Ω) (2.36) (u n -δ) + (u -δ) + in W 1,p 0 (Ω), ∀ δ > 0.
(2.37)

Thanks to Theorem 2.7 we will deduce the existence of nonnegative solutions for (2.1) which is given in the following theorem. (2.8) and (2.9), and every function F which satisfies (2.11), (2.12), (2.13), (2.14) and (2.15), there exists a nonnegative solution of problem (2.1) in the sense of Definition 2.3.

Theorem 2.8. Let Ω be a bounded open set of R N . Then, for every function a : Ω × [0, ∞) × R N → R N which satisfies (2.6), (2.7),
We finish this section with the following comparison result for the nonnegative solutions of (2.1) when the operator a = a(x, s, ξ) does not depend on s. It proves in particular the uniqueness of nonnegative solutions when a is strictly monotone in ξ and F = F (x, s) is nondecreasing in s. In the case when a depends on s it is also possible to extend some uniqueness results for pseudomonotone problems (see e.g. [START_REF] Boccardo | Unicité de la solution de certaines équations elliptiques non linéaires[END_REF], [START_REF] Casado-Díaz | Uniqueness results for pseudomonotone problems with p > 2[END_REF], [START_REF] Chipot | Uniqueness results and monotonicity properties for strongly nonlinear elliptic variational inequalities[END_REF]), but the corresponding proofs are more complicated.

Theorem 2.9. Let Ω be a bounded open set of R N and let a : Ω × [0, ∞) × R N → R N be which does not depend on s and satisfies (2.6), (2.7), (2.8) and

(a(x, ξ 1 ) -a(x, ξ 2 )) • (ξ 1 -ξ 2 ) > 0, ∀ ξ 1 , ξ 2 ∈ R N , a.e. x ∈ Ω.
(2.38)

We consider two functions F 1 and F 2 which satisfy (2.11), (2.12), (2.13) and (2.15). We also assume ∃i ∈ {1, 2} such that F i (x, •) is nonincreasing, a.e. x ∈ Ω (2.39)

F 1 (x, ξ) ≤ F 2 (x, ξ), ∀ ξ ∈ R N a.e. x ∈ Ω. (2.40) Then, if u 1 , u 2 are nonnegative solutions of -div a(x, ∇u i ) = F i (x, u i ) in Ω u = 0 in Ω, i = 1, 2, (2.41)
we have u 1 ≤ u 2 a.e. in Ω.

(2.42)

3 Proof of the stability, existence and uniqueness results

Let us prove in this section the different results exposed in Section 2 relative to the existence and properties of the nonnegative solutions for the singular problem (2.1).

Proof of Proposition 2.5. In order to show that W is contained in

W 1,p 0 (Ω), we consider w ∈ W and ϕ ∈ W 1,p 0 (Ω) such that w ∈ W, ϕ ∈ W 1,p 0 (Ω) ∩ L ∞ (Ω), ϕ ≥ 0, |w| ≤ ϕ p a.e. in Ω, |∇w| ϕ p-1 χ {w =0} ∈ L p (Ω). (3.1)
Then, taking ϕ n ∈ W 1,p (Ω), ϕ n ≥ 0, with compact support which converges to ϕ in W 1,p 0 (Ω) and is bounded in L ∞ (Ω), it is immediate to show that the sequence

w n = w ∨ (-ϕ p n ) ∧ ϕ p n ∈ W 1,p 0 (Ω), converges to w in W 1,p (Ω). This proves that w is in W 1,p 0 (Ω). Now, we consider w ∈ W 1,p 0 (Ω)∩L ∞ (Ω) with compact support. Let us prove that w is in W . It is enough to observe that for every ϕ ∈ C 1 c (Ω) with ϕ ≥ w L ∞ (Ω)
in supp w, the functions w and ϕ satisfy (3.1).

Let us prove that W is stable by addition. For this purpose, we take w 1 , ϕ 1 and w 2 , ϕ 2 which are related as in (3.1), then taking into account that for i = 1, 2 ∇w i = 0 a.e. in

{w i = 0} ⊂ {ϕ i = 0} it is simple to check that w 1 + w 2 , ϕ 1 + ϕ 2 also satisfy (3.1).
To show (2.27), we take w ∈ W and v ∈ W 1,p (Ω) ∩ L ∞ (Ω), which we can assume not identically zero, then we observe that for ϕ satisfying (3.1), we have

|wv| ≤ ϕ v 1 p L ∞ (Ω) p , |∇(wv)| v p-1 p L ∞ (Ω) ϕ p-1 χ {ϕ =0} ≤   |∇w| ϕ p-1 v 1 p L ∞ (Ω) + |∇v|ϕ v p-1 p L ∞ (Ω)   χ {ϕ =0} ∈ L p (Ω).
Assume now u a nonnegative solution of (2.1) in the sense of Definition 2.3 and consider a function

ϕ ∈ W 1,p 0 (Ω) ∩ L ∞ (Ω), ϕ ≥ 0 a.e. in Ω. For S ε : [0, ∞) → R defined by S ε (s) = 1 - s ε ∨ 0, ∀ s ≥ 0, (3.2)
we take h = S ε in (2.24) to deduce

- 1 ε {u<ε} a(x, u, ∇u) • ∇u ϕ p dx + p Ω a(x, u, ∇u) • ∇ϕ S ε (u)ϕ p-1 dx = Ω F (x, u)S ε (u)ϕ p dx. (3.3)
In the first term of this equality we use (2.7) and (2.18) to deduce

1 ε {u<ε} a(x, u, ∇u) • ∇u ϕ p dx ≥ α ε {0<u<ε} |∇u| p ϕ p dx - {0<u<ε} (γε p-1 + a 0 )ϕ p dx
and then lim sup

ε→0 1 ε {0<u<ε} a(x, u, ∇u) • ∇u ϕ p dx ≥ α lim sup ε→0 1 ε {0<u<ε} |∇u| p ϕ p dx.
For the second term, we observe that (2.8), (2.22) and Hölder's inequality imply that the function a(x, u, ∇u) • ∇ϕ ϕ p-1 belongs to L 1 (Ω). Using then that S ε (u) converges a.e. to χ {u=0} and (2.18), and the Lebesgue dominated convergence theorem we deduce

lim ε→0 Ω a(x, u, ∇u) • ∇ϕ S ε (u)ϕ p-1 dx = 0. ( 3.4)
For the third term in (3.3), we use (2.25), which allows us to apply Lebesgue dominated convergence theorem to get

lim ε→0 Ω F (x, u)S ε (u)ϕ p dx = {u=0} F (x, 0)ϕ p dx.
Therefore, taking the limit in (3.5) when ε tends to zero we obtain

0 ≥ {u=0} F (x, 0)ϕ p dx + α lim sup ε→0 1 ε {0<u<ε} |∇u| p ϕ p dx,
which by (2.12) shows (2.28) and

lim ε→∞ 1 ε {0<u<ε} |∇u| p ϕ p dx = 0, ∀ ϕ ∈ W 1,p 0 (Ω) ∩ L ∞ (Ω), ϕ ≥ 0. (3.5)
Let us now prove (2.29) and (2.30), for w ∈ W . Decomposing w = w + + w -, where both w + and w -belong to W , we can assume w nonnegative. We take ϕ such that (3.1) holds. Using For ε ∈ (0, 1), we define

|∇u| p-1 |∇w| ≤ |∇u| p-1 ϕ p-1 |∇w| ϕ p-1 χ {ϕ =0} ,
Z ε ∈ W 1,∞ (0, ∞) by Z ε (s) = s ε -1 + ∧ 1. (3.6)
Then, for δ > 0, we use (2.24) with

ϕ = 2s ε -1 + ∧ 1 (w + δ) 1 p ∈ W 1,p 0 (Ω) ∩ L ∞ (Ω), h = Z ε Observing that h(u)ϕ p = Z ε (u)(w + δ), we get Ω a(x, u, ∇u) • ∇w Z ε (u) dx + 1 ε {ε<u<2ε} a(x, u, ∇u) • ∇u (w + δ) dx = Ω F (x, u) (w + δ)Z ε (u) dx.
Since Z ε (u) vanishes on the set {u < ε} and (2.21), (2.13) and (2.15) hold, we can pass to the limit when δ tends to zero, to get

Ω a(x, u, ∇u) • ∇w Z ε (u) dx + 1 ε {ε<s<2ε} a(x, u, ∇u) • ∇u w dx = Ω F (x, u) wZ ε (u) dx.
Thanks to (2.25), (2.8), (2.29), (3.1), (3.5) and Lebesgue dominated convergence theorem we can pass to the limit in this equality to deduce (2.30).

Proof of Theorem 2.7. The first part of the theorem is devoted to obtain some a priori estimates for u n .

Taking into account the definition of λ given by (2.10) and (2.14), we can fix in the proof R 0 , ε > 0 such that

       (ν + γ) Ω |v| p dx ≤ (1 -ε) Ω a(x, v, ∇v) • ∇v + γ|v| p ) dx ∀ v ∈ W 1,p 0 (Ω) with Ω a(x, v, ∇v) • ∇v + γ|v| p ) dx ≥ R 0 . (3.7)
For δ > 0, we take (u n -δ) + ∈ W , with δ > 0, as test function in (2.34). Taking into account that for every t > 0, there exists C t > 0 such that

x p ≤ (1 + t)(x -δ) p + C t δ p , ∀ x > δ > 0, (3.8)
and defining the Sobolev constant

C Ω by (C Ω does not depend on Ω if p < N ) v L p * (Ω) ≤ C Ω ∇v L p (Ω) , ∀ v ∈ W 1,p 0 (Ω), (3.9)
we have for every t > 0

{un>δ} a(x, u n , ∇u n ) • ∇u n dx = Ω F n (x, u n )(u n -δ) + dx ≤ Ω k δ + νu p-1 n (u n -δ) + dx ≤ 1 p t p k δ p L (p * ) (Ω) + t p C p Ω p {un>δ} |∇u n | p dx +ν(1 + t) {un>δ} (u n -δ) p dx + νC t δ p |Ω|. (3.10) If {un>δ} a(x, u n , ∇u n ) • ∇u n + γu p n dx ≤ R 0 , (3.11)
then, thanks to (2.7) we easily get

(u n -δ) + is bounded in W 1,p 0 (Ω). (3.12)
In another case, using (3.7) with v = (u n -δ) + we deduce from (3.10)

1 -(1 -ε)(1 + t) {un>δ} a(x, u n , ∇u n ) • ∇u n dx ≤ 1 p t p k δ p L (p * ) (Ω) + t p C p Ω p {un>δ} |∇u n | p dx + νC t δ p |Ω|,
which using (2.7) and taking t small enough to have

α 1 -(1 -ε)(1 + t) - t p C p Ω p > 0,
proves that (3.12) is also true when (3.11) does not hold.

Observe that decomposing u n as u n = (u n ∧ 1) + (u n -1) + , estimates (3.12) and (3.9) also imply

u n is bounded in L p * (Ω). (3.13)
We have obtained an estimate for ∇u n on the set {u n > δ}. Let us now obtain an estimate on the set {u n < δ}. For this purpose, we take (δ

-u n ) + ϕ p with ϕ ∈ W 1,p 0 (Ω) ∩ L ∞ (Ω), ϕ ≥ 0 a.e.
in Ω, as test function in (2.34). We get

- {un<δ} a(x, u n , ∇u n ) • ∇u n ϕ p dx + p Ω a(x, u n , ∇u n ) • ∇ϕ ϕ p-1 (δ -u n ) + dx = Ω F n (x, u n )(δ -u n ) + ϕ p dx. (3.14)
Taking into account (2.7), (2.8) and ∇u n = 0 a.e. in {u n = 0}, we deduce

α δ {un<δ} |∇u n | p ϕ p dx + Ω F + n (x, u n ) 1 - u n δ + ϕ p dx ≤ {un<δ} (γδ p-1 + a 0 )ϕ p dx + β {un<δ} |∇u n | + δ + b p-1 |∇ϕ|ϕ p-1 dx + {un<δ} F - n (x, u n )ϕ p dx, (3.15)
which using (2.15) easily implies the existence of C > 0 such that

1 δ {un<δ} |∇u n | p ϕ p dx + Ω F + n (x, u n ) 1 - u n δ + ϕ p dx ≤ C ϕ p L ∞ (Ω) + ∇ϕ p L p (Ω) N ∀ ϕ ∈ W 1,p 0 (Ω) ∩ L ∞ (Ω), ϕ ≥ 0 a.e. in Ω, ∀ δ ∈ (0, 1). (3.16)
From this inequality, (3.12) and (2.13) applied to F n , we deduce in particular

|∇u n |ϕ bounded in L p (Ω), F n (x, u n ) + ϕ p bounded in L 1 (Ω), ∀ ϕ ∈ W 1,p 0 (Ω) ∩ L ∞ (Ω). ( 3
.17) Taking into account (3.12), (3.13), (2.33), Rellich-Kondrachov's compactness theorem and Fatou's lemma we deduce the existence of a subsequence of u n , still denoted by u n , and a function u such that

u ∈ L p * (Ω), u ≥ 0 a.e. in Ω, (u -δ) + ∈ W 1,p 0 (Ω), ∀ δ > 0 |∇u|ϕ ∈ L p (Ω), F (x, u) + ϕ ∈ L 1 (Ω), ∀ ϕ ∈ W 1,p 0 (Ω) ∩ L ∞ (Ω), ( 3
F + n (x, u n ) 1 - u n δ + ϕ p dx = 0, (3.19)
for every ϕ ∈ W 1,p 0 (Ω) ∩ L ∞ (Ω), ϕ ≥ 0. Let us now prove that u is a nonnegative solution of (2.1). By (3.18), it just remains to show (2.24). Even more, let us prove (2.30).

Taking into account (3.17) and (2.8), we can assume the existence of σ ∈ L p loc (Ω) N , such that

a(x, u n , ∇u n ) ϕ p-1 σ ϕ p-1 in L p (Ω) N , ∀ ϕ ∈ W 1,p 0 (Ω) ∩ L ∞ (Ω), ϕ ≥ 0. (3.20)
Taking w ∈ W as test function in (2.34), we get

Ω a(x, u n , ∇u n ) • ∇w dx = Ω F n (x, u n ) w dx. (3.21)
In the left-hand side of this equality we use In the right-hand side of (3.21), we use (2.36), (2.37), (3.19) and assumptions (2.13), (2.15) applied to F n to also obtain

a(x, u n , ∇u n ) • ∇w = a(x, u n , ∇u n )ϕ p-1 • ∇w ϕ p-1 χ {ϕ =0} , with ϕ ∈ W 1,p 0 (Ω) ∩ L ∞ (Ω)
lim n→∞ Ω F n (x, u n ) w dx = Ω F (x, u) w dx.
Therefore, we have shown

Ω σ • ∇w dx = Ω F (x, u)w dx, ∀ w ∈ W. ( 3
) • ∇u n ϕ dx + Ω a(x, u n , ∇u n ) • ∇ϕ (u n ∧ m) dx = Ω F n (x, u n )(u n ∧ m) ϕ dx,
where we can pass to the limit thanks to (

) • ∇u n ϕ dx + Ω σ • ∇ϕ (u ∧ m) dx = Ω F (x, u)(u ∧ m) ϕ dx.
On the other hand, using w = (u ∧ m)ϕ in (3.23), we have

{u<m} σ • ∇u ϕ dx + Ω σ • ∇ϕ (u ∧ m) dx = Ω F (x, u)(u ∧ m) ϕ dx.
Thus, we have proved

lim n→∞ {un<m} a(x, u n , ∇u n ) • ∇u n ϕ dx = {u<m} σ • ∇u ϕ dx, ∀ m > 0.
This equality allows us to use the Minty rule (see e.g. [START_REF] Leray | Quelques résultats de Visik sur les problèmes elliptiques non linéaires par les méthodes de Minty-Browder[END_REF], [START_REF] Lions | Quelques méthodes de résolution des problèmes aux limites non linéaires[END_REF]) to deduce (3.24).

Proof of Theorem 2.8. For every n ∈ N, we define

F n : Ω × [0, ∞) → R by F n (x, s) = [F (x, s) ∨ (-n)] ∧ n, ∀ s ∈ R, a.e. x ∈ Ω.
Then, we extend F n to Ω × R and a to Ω × R × R N by taking

F n (x, s) = F n (x, 0), a(x, s, ξ) = a(x, 0, ξ), ∀ (s, ξ) ∈ (-∞, 0) × R N , a.e. x ∈ Ω.
Taking into account that F n is a Carathéodory function and |F n (x, s)| ≤ n a simple application of the Schauder fixed point theorem and the theory of monotone operators provides a solution

u n of -div a(x, u n , ∇u n ) = F n (x, u n ) in Ω u n ∈ W 1,p 0 (Ω).
(3.25)

Multiplying (3.25) by -u - n , we have {un<0} a(x, 0, ∇u n ) • ∇u n dx = {un<0} F n (x, 0)u n dx,
which using (2.7) and (2.12) proves u n ≥ 0, i.e. u n is a nonnegative solution of (3.25). Applying Theorem 2.7 to this sequence we deduce the existence of a subsequence of u n which converges to a nonnegative solution of (2.1).

Proof of Theorem 2.9 For δ, k > 0, we consider the function

z = u 1 -u 2 -δ + ∧ k 2 , which satisfies ∇z = 2 u 1 -u 2 -δ ∇(u 1 -u 2 )χ {u 2 +δ<u 1 <k+u 2 +δ}
and therefore

|∇z| ≤ 2 (u 1 -δ) + ∧ k |∇u 1 | + |∇u 2 | a.
(a(x, ∇u 1 ) -a(x, ∇u 2 )) • ∇(u 1 -u 2 ) u 1 -u 2 -δ 2p-1 dx = Ω F 1 (x, u 1 ) -F 2 (x, u 2 ) u 1 -u 2 -δ + ∧ k 2p dx. (3.26)
Let us first assume F 1 (x, .) non decreasing, then using F 1 (x, u 2 ) ≤ F 2 (x, u 2 ) a.e. in Ω, we have 2p

{u 2 +δ<u 1 <k+u 2 +δ} (a(x, ∇u 1 ) -a(x, ∇u 2 )) • ∇(u 1 -u 2 ) u 1 -u 2 -δ 2p-1 dx ≤ Ω F 1 (x, u 1 ) -F 1 (x, u 2 ) u 1 -u 2 -δ + ∧ k 2p dx.
Here, we observe that

u 1 ≤ u 2 =⇒ u 1 -u 2 -δ + ∧ k = 0,
while by F 1 non decreassing we have

u 1 ≥ u 2 =⇒ F 1 (x, u 1 ) -F 1 (x, u 2 ) ≤ 0,
Therefore, the right-hand side of (3.26) is nonpositive which, taking into account (2.38), proves

∇u 1 = ∇u 2 a.e. in {u 2 + δ < u 1 < k + u 2 + δ}, ∀ δ > 0, ∀ k > 0,
and then ∇u 1 = ∇u 2 a.e. in {u 2 < u 1 }, or equivalently, ∇(u 1 -u 2 ) + = 0 a.e. in Ω. This proves that (u 1 -u 2 ) + is constant in every connected component of Ω, which combined with (u i -δ) + ∈ W 1,p 0 (Ω) for every δ > 0, i = 1, 2, proves u 1 ≤ u 2 a.e. in Ω.

Assume now F 2 (x, .) non decreasing. Using F 1 (x, u 1 ) ≤ F 2 (x, u 1 ) in (3.26) we get 2p

{u 2 +δ<u 1 <k+u 2 +δ} (a(x, ∇u 1 ) -a(x, ∇u 2 )) • ∇(u 1 -u 2 ) u 1 -δ -u 2 2p-1 dx ≤ Ω F 2 (x, u 1 ) -F 2 (x, u 2 ) u 1 -u 2 -δ + ∧ k 2p dx.
This allows to repeat the above reasoning to deduce again u 1 ≤ u 2 a.e. in Ω.

4 Nonexistence of a solution taking negative values when F is too large near s = 0.

In Section 2, we have proved the existence of a nonnegative solution for problem (2.1) when the function F = F (x, s) on the right-hand side can take the value plus infinity at s = 0. In the present section we prove that if F (x, s) is bigger than τ /|s|, τ > 0, when s is close to zero, then every solution of the semilinear problem is necessarily nonnegative. This result is a consequence of the following lemma.

: Ω × R × R N → R N be a function which satisfies a(., s, ξ) is measurable in Ω, ∀ (s, ξ) ∈ R × R N a(x, ., .) is continuous in R × R N , a.e. x ∈ Ω. ( 4

.1)

• There exist α, γ, a 0 such that

α > 0, γ ≥ 0, a 0 ∈ L (p * ) (Ω), a 0 ≥ 0 a(x, s, ξ) • ξ ≥ α|ξ| p -γ|s| p -a 0 (x)|s|, ∀ (s, ξ) ∈ R × R N , a.e. x ∈ Ω. ( 4

.2)

• There exist β, b such that

β > 0, b ∈ L p (Ω), b ≥ 0, |a(x, s, ξ)| ≤ β( ξ| + |s| + b(x) p-1 , ∀ (s, ξ) ∈ R × R N , a.e. x ∈ Ω. (4.3)
Let u and H be two measurable functions in Ω such that

(u + δ) -∈ W 1,p 0 (Ω), ∀ δ > 0 (4.4) H ∈ L 1 ({u < -δ}), ∀ δ > 0 (4.5) ∃ τ, δ 0 > 0 such that H ≥ τ |u| a.e. in {-δ 0 < u < 0} (4.6)      Ω a(x, u, ∇u) • ∇v dx = Ω Hv dx ∀ v ∈ W 1,p 0 (Ω) ∩ L ∞ (Ω), such that ∃ δ > 0 with v = 0 in {u > -δ}. (4.7)
Then, we have u ≥ 0 a.e. in Ω.

(4.8)

The previous Lemma implies that any solution of problem (2.1) cannot take negative values when F (x, s) ≥ τ /|s|, τ > 0, for s small. More exactly, one has: Theorem 4.2. Let Ω be a bounded open set of R N and p > 1. We consider a function a : Ω×R×R N → R N which satisfies (4.1), (4.2) and (4.3) and a function F : Ω×R → R∪{∞}, which satisfies

F (•, s) is measurable in Ω, ∀ s ∈ R, F (x, •) is continuous in R \ {0}, a.e. x ∈ Ω (4.9) ∃ δ 0 , τ > 0, with F (x, s) ≥ τ |s| , ∀s ∈ (-δ 0 , 0), (4.10
) Then, any measurable function u : Ω → R which is a solution of

and              if p ≤ N, ∃ν δ ≥ 0, k δ ∈ L 1 (Ω), k δ ≥ 0, ∀ δ > 0 |F (x, s)| ≤ k δ (x) + ν δ |s| p * , ∀ s with |s| > δ, a.e. x ∈ Ω if p > N, ∃k m,δ ∈ L 1 (Ω), k m,δ ≥ 0, ∀ m ∈ N, ∀ δ > 0 |F (x, s)| ≤ k m,δ (
-div a(x, u, ∇u) = F (x, u) in Ω \ {u = 0} u ≥ 0 on ∂Ω, (4.12)
in the following sense: Remark 4.4. Definition 2.3 in Section 2 was concerned with nonnegative solutions. Thus, the functions a = a(x, s, ξ) and F = F (x, s) were only defined for s nonnegative. It is clear that Definition 2.3 could be extended in a natural way to the case where F is defined for s ∈ R and where the solution can take negative values. In this new setting, any such solution to (2.1) would have to be nonnegative in view of Theorem 4.2 when the function F satisfies assumption (4.10), reinforcing in this case the uniqueness result of Theorem 2.9.

         (u + δ) -∈ W 1,p 0 (Ω), ∀ δ > 0 Ω a(x, u, ∇u) • ∇v dx = Ω F (x, u)v dx ∀ v ∈ W 1,p 0 (Ω) ∩ L ∞ (Ω), such that ∃ δ > 0 with v = 0 in {|u| < δ}, (4.13
(u -δ) + ∈ W 1,p 0 (Ω), ∀ δ > 0, H ∈ L 1 ({u > δ}), ∀ δ > 0, ∃ τ, δ 0 > 0 such that H ≤ - τ |u| a.e. in {0 < u < δ}      Ω a(x, u, ∇u) • ∇v = Ω Hv dx ∀ v ∈ W 1,p 0 (Ω) ∩ L ∞ (Ω), such that ∃ δ > 0 with v = 0 in {u < δ}.
Then, instead of (4.8) we have that u ≤ 0 a.e. in Ω. The proof of this result just follows by applying Lemma 4.1 to the functions ã, ũ and H defined by

ã(x, s, ξ) = -a(x, -s, -ξ), ũ = -u, H = -H.
Using this result one can also modify Theorem 4.2 to prove that the solution u of a nonlinear problem with a singular right-hand side term F (x, u) such that

F (x, u) ≤ - τ u , if 0 < u < δ, for some δ > 0,
which is nonpositive on the boundary is necessarily nonpositive in the whole of Ω.

The results stated in Lemma 4.1 and Theorem 4.2 are based on the fact that the sign of u on the boundary is known. When no boundary conditions are given it is still possible to show Lemma 4.6 below. It proves that the solution of a nonlinear problem with a singular term, which is sufficiently large for u close to zero, cannot changes the sign. We assume there exist u ∈ W 1,p loc (Ω) and H ∈ L 1 loc (Ω) such that there exist τ, δ 0 > 0 satisfying one of the following conditions:

             |H| ≥ τ |u| a.e. in {0 < u < δ 0 } Ω a(x, u, ∇u) • ∇v dx = Ω Hv dx ∀ v ∈ W 1,p (Ω) ∩ L ∞ (Ω), spt(v) ⊂ Ω compact, ∃ δ > 0 with v = 0 in {v < δ}. (4.15) or              |H| ≥ τ |u| a.e. in {-δ 0 < u < 0} Ω a(x, u, ∇u) • ∇v dx = Ω Hv dx ∀ v ∈ W 1,p (Ω) ∩ L ∞ (Ω), spt(v) ⊂ Ω compact, ∃ δ > 0 with v = 0 in {v > -δ}. (4.16)
Then, one of the following assertions hold:

u ≥ 0 a.e. in Ω or u ≤ 0 a.e. in Ω. (4.17)

Proof of Lemma 4.1. For ε < δ 0 /2, we take

v ε := 1 + u ε - ∧ 1 ∈ W 1,p 0 (Ω) ∩ L ∞ (Ω), ∀ ε > 0. (4.18)
as test function in (4.7). We have

- 1 ε {-2ε<u<-ε} a(x, u, ∇u) • ∇u dx = Ω Hv ε dx = {u≤-δ 0 } H dx + {-δ 0 <u}
Hv ε dx, and thus -

{u≤-δ 0 } H dx = 1 ε {-2ε<u<-ε} a(x, u, ∇u) • ∇u dx + {-δ 0 <u} Hv ε dx. (4.19)
By (4.6), we know that H ≥ 0 a.e. in {-δ 0 < u < 0}. This allows us to use the monotone convergence theorem in the last term, which combined with (4.2), proves lim sup

ε→0 α ε {-2ε<u<-ε} |∇u| p dx + {-δ 0 <u<0} H dx ≤ - {u≤-δ 0 } H dx,
and thus, taking into account (4.5), we get

H ∈ L 1 ({u < 0}), lim sup ε→0 1 ε {-2ε<u<-ε} |∇u| p dx ≤ - 1 α {u<0} H dx. (4.20)
Now, we observe that thanks to (4.6), for 0 < 2ε < δ 0 , we have

1 2ε x ∈ Ω : -2ε < u < -ε ≤ {-2ε<u<-ε} 1 |u| dx ≤ 1 τ {-2ε<u<-ε} H dx,
and then, by the first assertion in (4.20)

lim ε→0 1 ε x ∈ Ω : -2ε < u < -ε = 0. (4.21)
Let us now take Ψ ∈ C ∞ c (R N ) N . Since v ε defined by (4.18) belongs to W 1,p 0 (Ω), we have

Ω v ε div Ψ dx = - Ω ∇v ε • Ψ dx = 1 ε {-2ε<u<-ε} ∇ u • Ψ dx. (4.22)
Taking into account (4.20) and (4.21), we have on the first hand lim sup

ε→0 1 ε {-2ε<u<-ε} |∇u||Ψ| dx ≤ lim ε→0 1 ε {-2ε<u<-ε} |∇u| p dx 1 p 1 ε x ∈ ω : -2ε < u < -ε 1 p Ψ L ∞ (Ω) N = 0.
On the other hand, using the definition (4.18) of v ε and Lebesgue's dominated convergence theorem, we have

lim ε→0 Ω v ε div Ψ dx = {u<0} div Ψ dx.
Thus, (4.22) provides

0 = Ω χ {u<0} div Ψ dx, ∀ Ψ ∈ C ∞ c (R N ) N =⇒ ∇χ {u<0} = 0 in D (R N ). (4.23)
This proves that χ {u<0} is a constant function in R N , but {x ∈ Ω : u(x) < 0} ⊂ Ω implies that χ {u<0} vanishes outside Ω. Therefore χ {u<0} = 0 a.e. in R N .

Proof of Theorem 4.2. It is easily deduced Lemma 4.1 with H(x) = F (x, u(x)). Namely, we observe that (4.13) implies (4.4) and (4.7), while Sobolev's embedding theorem and (4.11) provide (4.5). Finally, assumption (4.6) follows from (4.10).

Proof of Lemma 4.6. To fix ideas we assume in the following that (4.15) holds. The other case is completely similar. The proof follows the ideas of Lemma 4.1.

We take ϕ ∈ W 1,p (Ω) ∩ L ∞ (Ω). Then, for ε > 0, we define

v ε := u ε -1 + ∧ 1 . Taking v = v ε ϕ in (4.15), we get 1 ε {ε<u<2ε} a(x, u, ∇u) • ∇u ϕ dx + Ω a(x, u, ∇u) • ∇ ϕ v ε dx = Ω Hv ε ϕ dx.
Then, using Lebesgue dominated convergence theorem we can pass to the limit when ε tends to zero to deduce From (4.24) and (4.25) we then deduce similarly to (4.23) (observe that now the support ofΨ is contained in Ω).

{u>0} div Ψ dx = 0, ∀ Ψ ∈ C ∞ c (Ω) N .
This shows that the distributional gradient of χ {u>0} is zero and so that χ {u>0} is constant a.e. in Ω, i.e. u ≥ 0 a.e. in Ω or u ≤ 0 a.e. in Ω.

5 A one-dimensional example of a singular equation with many solutions which change sign.

In Section 2 we have proved the existence of a nonnegative solution for the semilinear problem (2.1) where F = F (x, s) is nonnegative at s = 0 and can take the value +∞. Later, in Section 4, we have shown that if F (x, s) is bigger than τ /|s|, τ > 0, when s is negative and close to zero, then every solution of (2.1) is nonnegative. If contrarily, the right-hand side F tends to +∞ when s tends to 0 with a speed of order 1/|s| γ , 0 < γ < 1, then some examples in [START_REF] Dávila | Positive versus free boundary solutions to a singular elliptic equation[END_REF] and [START_REF] Diaz | An elliptic equation with singular nonlinearity[END_REF] show that a nonpositive solution can also exist. This proves that condition (4.10) in Theorem 4.2 is optimal in the sense that it cannot be relaxed by

F (x, s) ≥ τ |s| γ a.e. in {δ 0 < s < 0}, with δ 0 , τ > 0, γ < 1.
In the present section we explore the simple example in dimension N = 1 given by

     -u = f |u| γ -g in (0, L) u(0) = u(L) = 0, (5.1)
with L, f, γ positive constants and g ∈ R. We will describe all the possible solutions.

Theorems 2.8 and 2.9 applied to (5.1) show the existence and the uniqueness of a nonnegative solution for (5.1), for every f, γ > 0 and g ∈ R. For γ ≥ 1, Theorem 4.2 shows that every solution of the problem is nonnegative. So, the solution given by Theorem 2.8 is the unique solution to this problem. Also, if γ < 1 and g ≤ 0, the right-hand side in problem (5.1) is nonnegative and then, the classical weak maximum principle implies that every solution has also to be nonnegative. Thus, we will assume in what follows f, g > 0, 0 < γ < 1.

We first observe that a change of scale in the variables x and u allows us to reduce the number of parameters. Namely, for r, t > 0 to be chosen, we define the new unknown function w by w(x) = ru(tx), x ∈ (0, L/t).

(

The differential equation in (5.1) is then transformed into

-w = f r γ+1 t 2 |w| γ -grt 2 in (0, L/t). Taking r = g f 1 γ , t = f 1 2γ g γ+1 2γ , l = L t , (5.3) problem
(5.1) reduces to    -w = 1 |w| γ -1 in (0, l) w(0) = w(l) = 0.
(5.4)

The main results of this section are Theorem 5.3 and 5.7 below. They show in particular the existence of solutions to problem (5.1) which take negative values and even change its sign when l is big enough. Moreover the number of such solutions increases as l increases.

Statements of the results concerning problem 5.4.

In the introduction to this section we were speaking about solutions of problem (5.4) and (5.1), but we did not define what we understand by a solution of these singular The definition that we will use is the following one: Definition 5.1. Let γ be such that 0 < γ < 1.

A measurable function w : (0, l) → R is a solution of (5.4) if it satisfies

w ∈ L 1 (0, l) (5.5)
(|w| -δ) + ∈ W 1,1 0 (0, l), ∀ δ > 0 (5.6) 1 |w| γ ∈ L 1 loc (0, l) (5.7)
-w = 1 |w| γ -1 in D (0, l).

(

This definition could seem to be weaker than the one given by Definition 2.3 for nonnegative solutions, but this is not the case because of the following Proposition. Proposition 5.2. Let w be a solution of (5.4) in the sense of Definition 5.1. Then one has

w ∈ W 2,1 (0, l) ∩ H 1 0 (0, l), (5.9)
and then in particular

1 |w| γ ∈ L 1 (0, l), w ∈ C 1 ([0, l]).
(5.10)

Moreover, defining c ≥ 0 by c = |w (0)| 2 , (5.11
)

one has |w | 2 = b(w) + c in [0, l], (5.12)
where b : R → R is the function defined by

b(s) = - 2 1 -γ s |s| γ + 2s, ∀ s ∈ R, (5.13)
where b(0) is understood as zero. Finally, when c = 0 one has

w ∈ W 2,q (0, l), 1 |w| γ ∈ L q (0, l), ∀ q < 1 γ .
(5.14)

Remark 5.5. As already said in the introduction of this section, problems (5.1) and (5.4) are equivalent through the change of (independent and dependent) variables (5.2). In the case γ = 1/2, since (5.3) implies that

g = f l L 2 3 , setting M = f T 0 L 2 3
, Theorem 5.3 implies that problem (5.1) has a unique solution (which is positive) when 0 < g < M , and a unique solution when g ≥ M . Moreover at every value g = M k 2 3 , with k ∈ N, a bundle of 4 branches of solutions appears, so that problem (5.1) has exactly 1 + 4k solutions for M k

2 3 < g < M (k + 1) 2 3 , k ≥ 0 and 4k -2 solutions for g = M k 2 3 , k ≥ 1.
Theorem 5.3 refers to the case γ = 1/2. In the case γ ∈ (0, 1/2) ∪ (1/2, γ) we do not know the exact number of solutions of problem (5.4) and the structure of such solutions. However we can still show some related results. For this purpose consider the function b defined by (5.13), whose graph is given in figure 1. The following proposition gives some properties of this function b we will need later to state and prove the results corresponding to the existence of solutions for problem (5.4).

Proposition 5.6. For γ ∈ (0, 1), the function b defined by (5.13) satisfies

             b is strictly increasing in (-∞, -1) and (1, ∞), b is strictly decreasing in (-1, 1) lim s→-∞ b(s) = -∞, b(-1) = 2γ 1 -γ , b(1) = - 2γ 1 -γ , lim s→∞ b(s) = ∞ b vanishes at s = - 1 (1 -γ) 1 γ , s = 0, s = 1 (1 -γ) 1 γ
.

(5.17)

• For 0 ≤ c < 2γ/(1 -γ), the equation b(s) + c = 0 has three solutions z 1 (c), z 2 (c) and z 3 (c) with z 1 (c) < -1, 0 ≤ z 2 (c) < 1, 1 < z 3 (c).
• For c = 2γ/(1 -γ) the equation b(s) + c = 0 has two solutions z 1 (2γ/(1 -γ)) < -1 and z 2 (2γ/(1 -γ)) = 1.

• For c > 2γ/(1 -γ), equation b(s) + c = 0 has a unique solution z 1 (c) and z 1 (c) < 1.

We define T

-: [0, ∞) → R and T + : [0, 2γ/(1 -γ)) → R by T -(c) = 2 0 z 1 (c) dt b(t) + c , ∀ c ∈ [0, ∞).
(5.18)

T + (c) = 2 z 2 (c) 0 dt b(t) + c , ∀ c ∈ 0, 2γ 1 -γ . (5.19)
Then, we have

T -∈ C 0 ([0, ∞)), T * := min c≥0 T -(c) > 0, lim c→∞ T -(c) = ∞ (5.20) T + ∈ C 0 0, 2γ 1 -γ , T + is strictly increasing, T + (0) = 0, lim c→ 2γ 1-γ T + (c) = ∞. (5.21)
Using the function T -defined by 5.18 we can now state our main result about the existence of solution for problem (5.8) for γ ∈ (0, 1). Theorem 5.7. Let γ be in (0, 1). Define T * by (5.20) and T 0 by

T 0 = T -(0).
(5.22)

Then, we have: If T -is strictly increasing in [0, ∞), then all the results in the statement of 5.3 still hold true.

If T -is not necessarily strictly increasing, we have:

1. For every l > 0, there exists 1 solution of problem (5.4) which is positive in (0, l). This solution is the unique nonnegative solution of (5.4).

2. For every l < kT * , k ≥ 1, there is not any solution of problem (5.4) which is negative on at least k nonempty open subintervals of (0, l) and vanishes on the boundary of these intervals.

3. For every l ≥ T * , there exists at least 1 solution of problem (5.4) which is negative in (0, l).

4.

For every l = kT 0 , k ≥ 1, there exists 1 solution of problem (5.4) which is negative on k-1 j=0 (jT 0 , (j + 1)T 0 ) but which vanishes at t = jT 0 , j ∈ {1, ..., k -1}. 5. For every l > kT 0 , k ≥ 1, there exist at least 4 solutions of problem (5.4) which are negative on exactly k disjoint open subintervals of (0, l), they vanish on the boundary of these subintervals and are positive on the rest of (0, l), which is composed by k -1, k or k + 1 disjoint open subintervals.

In particular, for γ ∈ (0, l), problem (5.4) has at least 1 + 4k solutions if kT 0 < l < (k + 1)T 0 , ∀ k ≥ 0, (5.23) 4k -2 solutions if l = kT 0 , ∀ k ≥ 1.

(5.24)

Remark 5.8. By Theorem 5.7, the number of solutions of (5.4) agrees with (5.15) or (5.16), when the function T -is strictly increasing. In the case where γ = 1/2 this follows from the next proposition which provides an explicit expression for T -. In the general case we do not know when this is true or not. A numerical computation provides figure 5 showing the graph for T -for several values on γ. It seems to indicate the existence of γ ≤ 1/2, close to 1/2, such that T -is not strictly increasing for γ ∈ (0, γ), while it is strictly increasing for γ ≥ γ.

Proposition 5.9. Assume γ = 1 2 . Then, the functions z 1 and z 2 defined by Proposition 5.6 are given by

z 1 (c) = -2 + c 2 + √ 4 + 2c , ∀ c ≥ 0, z 2 (c) = 2 - c 2 - √ 4 -2c, ∀ c ∈ [0, 2]. (5.25)
The functions T -and T + defined by (5.18) are given by

T -(c) = 2 √ 2   π -2 arctan -z 1 (c) -2 -z 1 (c) -z 1 (c) + -z 1 (c) -2 -z 1 (c)   , ∀ c ≥ 0, (5.26)
T + (c) = 2 √ 2   2 arctanh z 2 (c) 2 z 2 (c) -z 2 (c) -2 z 2 (c) -z 2 (c)   , ∀ s ∈ [0, 2).
(5.27)

The function T + is a strictly increasing function in [0, ∞).

5.2 Proof of the results corresponding to the one-dimensional example.

The results exposed above are essentially a consequence of Theorem 5.10 below. First of stating and proving this result, let us show Propositions 5.2 and 5.6.

Proof of Proposition 5.2. We start by observing that if w is a solution of (5.4) in the sense of Definition 5.1, then, (5.7) and (5.8) imply that w is in W 2,1 loc (0, l), and therefore w is in 1 (0, l). Multiplying equation (5.8) by w , we get

- |w | 2 2 = w |w| γ -w in (0, l),
and thus, there exists a constant c ∈ R such that

|w | 2 = - 2 1 -γ w |w| γ + 2w + c in (0, l).
(5.28)

Since w ∈ L 1 (0, l), we deduce from this equality that |w | 2 belongs to L 1 (0, l) and therefore w ∈ H 1 (0, l). Then, (5.6) implies that w beongs to H 1 0 (0, l). In particular, w belongs to C 0 ([0, l]) and then (5.28) shows that |w | is in C 0 ([0, l]).

Integrating equation (5.8) in (δ, l -δ), with δ > 0, we also have

l-δ δ dx |w| γ = -w (l -δ) + w (δ) + l -2δ. Since |w | in C 0 ([0, l]) implies w in L ∞ (0, l), the monotone convergence theorem implies that 1 |w| γ ∈ L 1 (0, l), (5.29)
and then (5.8) proves that w is in W 2,1 (0, l). In particular, this means that w is in C 0 ([0, l]) and by (5.28) that (5.12) holds. If c = 0, then, using that w ∈ C 1 ([0, l]) and that |w (s)| = c 2 for every s ∈ [0, l] such that w(s) = 0, we deduce that 1/|w| γ belongs to L q (0, l) for every q < 1/γ which combined with (5.4) shows (5.14).

Proof of Proposition 5.6. Statement (5.17

T + (c) = 2 lim c→ 2γ 1-γ z 2 (c) 0 dt b(t) + c = ∞.
(5.30)

In order to prove that T + is strictly increasing, we recall that Theorems 2.8 and 2.9 show that problem (5.1) has a unique nonnegative solution for every l ≥ 0. Taking into account Theorem 5.10 below (see (5.41)) this solution is necessarily obtained as the restriction to [0, l] of a periodic function of period T -(c) + T + (c), with l = T + (c) for some c ∈ [0, 2γ/(1 -γ)). Moreover, by (5.42), it is given by w

(x) =      G -1 (x) if x ∈ 0, l 2
G -1 (l -x) if x ∈ l 2 , l ,
with T -(c) and T (c) defined by (5.18), (5.35), i.e. Continuing with this process and taking into account that by assumption w(0) = 0 we conclude that w is periodic of period T (c) and its is given by formula (5.38) if w (0) ≤ 0 or by (5.42) if w (0) > 0. The properties of w stated in (5.40) easily follow from (5.38). The fact that l must satisfy (5.37) or (5.41) just follows from w(l) = 0. In the first case, defining G by (5.36), and using that the unique point bigger than zero, where b + c vanishes is 1, we deduce that for every δ ∈ (0, l) such that in the interval (0, δ), the function w is increasing and less than 1, we have However the function constructed in this way is positive in (0, ∞) in contradiction with w(l) = 0. This proves that only the second possibility in (5.53) can hold true. Using then that w (0) ≤ 0 and vanish on 0, l we conclude that there exists r ∈ (0, l) such that w (r) = 0. By (5.28) and z 1 the unique negative zero of b + c, we must have w(r) = z 1 . Now, we can repeat the reasoning in the case c < 2γ/(1 -γ) when we assumed (5.48). This shows that (an extension of) w satisfies (5.51) where now, thanks to (5.55), we have

           T -(c) = 2 0 z 1 (c) dt b(t) + c = -2G(z 1 (c)) T (c) = 2 z 2 (c) z 1 (c) dt b(t) + c = 2 G(z 2 (c)) -G(z 1 (c)) . (5.52) Observe that G(s) < 0 if s ∈ [z 1 (c), 0), G(s) > if s ∈ (0, z 2 (c)], implies w < 0 in r - T -(c) 2 , r + T -(c) 2 , w > 0 in r - T (c) 2 , r - T -(c) 2 r + T -(c) 2 , r + T (c) 2
G(w(x)) = x in (0, δ), (5.54
T = 2 1 z 1 (c) dt b(t) + c = ∞.
Then, (5.48) provides the expression of w in the whole of R. Since w(l) = 0, we get by symmetry that r = l/2 and we easily conclude (5.43), (5.44), (5.45) and (5.46).

Case 3: c > 2γ/(1 -γ).

Now, the unique zero of the function b + c is z 1 (c) and by (5.28), we have that w([0, l]) is contained in [1, ∞). Taking into account w(0) = w(l) = 0, we deduce as above the existence of r ∈ (0, l) such that w (r) = 0, w(r) = z 1 < -1 and then that (5.51) holds with the difference that now, there is not any point z 2 bigger than z 1 such that b(z 2 ) + c = 0. Therefore T must be defined as

T = 2 ∞ z 1 (c) dt b(t) + c = ∞.
Statement (5.51) then provides an expression of (an extension of) w in the whole of R. By w(0) = w(l) = 0 we conclude again by symmetry that (5.43), (5.44), (5.45) and (5.46) hold where now the limit of w at infinity is +∞ and not 1.

Let us now prove Theorem 5.7. Observe that Theorem 5.3 follows from this result and Proposition 5.9. Thus, the proof of Theorem 5.3 will not explicitly given.

Proof of Theorem 5.7. By Theorems 2.7 and 2.9, we know that problem (5.4) has a unique positive solution for γ ∈ (0, 1). If l < T * , then we have T -(c) > l for every c ≥ 0, but since w(0) = w(l) = 0, w < 0 in (0, l), we get that Theorem 5.10 (see 5.39), (5.43)) implies l = T -(c). Therefore, it cannot exist a negative solution of (5.4) in (0, l) for l < T * . Analogously, for l < kT * it cannot exist a solution of (5.4) which is negative in k nonempty open subintervals of (0, l) and vanishes on the boundary.

Assume l ≥ T * . Since T -is continuous and tends to infinity as infinity, we have that for every l ≥ T * , there exists c ≥ 0 such that T -(c) = l. Defining then w by (5.38) if c < 2γ/(1-γ) or (5.44) if c ≥ 2γ/(1 -γ) we deduce the existence of a negative solution of (5.4). If T -is strictly increasing then T * = T 0 and equation T -(c) = l has a unique solution, and so there is a unique negative solution for problem (5.4).

Assume l > T 0 . Joined to the negative solution in (0, l) found above, this proves the existence of at least 4 solutions of (5.4) which are negative in a nonempty interval of (0, l) and positive in zero, one or two nonempty intervals of (0, l).

In the case l > kT 0 , k ≥ 2 a similar reasoning provides 4 solutions which are negative in k nonempty intervals of (0, l) and positive in k -1, k or k + 1 intervals.

For l = kT 0 , k ≥ 2, equation (5.38) with c = 0 provides a solution which is nonpositive in (0, l) but vanishes on jT 0 for j = 1, . . . , k -1. Observe that by Theorem 5.10 and T + (c) > 0 for every c > 0, we get that this is is the unique solution satisfying this property.

If the function T -is strictly increasing, then the function mT -+ nT + is strictly increasing for every m, n ∈ N, with m + n ≥ 1 and then it is injective. Therefore the solutions described above are the only ones which are negative in k nonempty intervals of (0, l).

Proof of Proposition 5.9. We recall that z 1 (c) < z 2 (c) are the smallest solutions of the equation b(s) + c = 0, with b given by (5.13). For γ = 1/2 this equation reduces to we can transform this integral into a rational integral. Namely, denoting

r 1 = η 2 -2η η , (5.57)
we have

T -(c) = 4 √ 2 ∞ r 1 r 2 η -η + 2 (r 2 + 1) 2 dr = 8 √ 2(η -1) ∞ r 1 r 2 (r 2 + 1) 2 dr + 4 √ 2(2 -η) ∞ r 1 dr r 2 + 1 ,
where a primitive of r 2 /(1 + r 2 ) is given by r 2 (r 2 + 1) 2 dr = -

. 1 )F

 1 is a classical problem which has been first considered by several authors. Here Ω is a bounded open set of R N , A ∈ L ∞ (Ω) N ×N is uniformly elliptic and the right-hand side F : Ω×(0, ∞) → R satisfies lim s→0 (x, s) = +∞, a.e. x ∈ Ω. (1.2)

  associated to w following Definition (2.26) of W . Thanks to (3.20) we can then pass to the limit when n tends to infinity, to get lim n→∞ Ω a(x, u n , ∇u n ) • ∇w dx = Ω σ • ∇w dx. (3.22)

Lemma 4 . 1 .

 41 Let Ω be a bounded open set of R N , p > 1, and a

  x), ∀ s with δ < |s| < m, a.e. x ∈ Ω.

Remark 4 . 3 .

 43 Problem (4.12) is non standard since the equation takes place in the set Ω \ {u = 0} which depends on the solution u. In general this set is not an open set since u has not reason to be continuous.The fact that the equation takes place in Ω \ {u = 0} is reflected in the (mathematically correct) formulation (4.13) where the test functions v have to vanish on the set {u = 0}.

Remark 4 . 5 .

 45 If in Lemma 4.1 we replace assumptions (4.4), (4.5), (4.6) and (4.7) by

Lemma 4 . 6 .

 46 Assume Ω ⊂ R N , open, a : Ω × R × R N → R satisfying (4.1), (4.2) and (4.3).

  u, ∇u) • ∇u ϕ dx = {0<u} Hϕ -a(x, u, ∇u) • ∇ϕ dx, which combined with (4.2) shows lim sup ε→0 1 ε {ε<u<2ε} |∇u| p ϕ dx < +∞. (4.24)On the other hand, for ε < δ 0 /2, we have

  ) and the results about the number and position of the zeros of the function b follow immediately by studying the sign of the derivative of b. The continuity of the functions T -and T + defined by (5.18) and (5.19) is simple to check. Moreover, the definition of T + and z 2 (0) = 0 implies T + (0) = 0. Since for c = 2γ/(1 -γ), we have z 2 (c) = 1 and b(1) + c = b (1) = 0, we get lim

. Equation 5 .

 5 51 provides the function w in the interval [α, β] = [r -T (c)/2, r + T (c)/2]. Recalling (5.48) we can now extend the definition of w to [r -T (c), r + T (c)].

Case 2 :

 2 c = 2γ/(1 -γ).Using(5.28) and then that b + c is nonnegative on the range of w, we deduce by Proposition 5.6 that w([0, l]) is contained in [z 1 , +∞). By (5.28) and w(0) = 0, we get that in a neighborhood of zero, the function w is the solution of one of the two following Cauchy's problems w = b(w)

  ) but for c = 2γ/(1 -γ), we have b(1) = b (1) = 0, and thus 54) would provide the expression of G in the whole interval [0, l].

  Proposition 5.6, the function T -+ T + is continuous in [0, 2γ/(1 -γ)) and satisfiesmin c∈[0,2γ/(1-γ)) (T -(c) + T + (c)) ≥ T -(0) + T + (0) = T 0 , lim c→ 2γ 1-γ (T -(c) + T + (c)) = +∞, therefore, there exists c ∈ (0, 2γ/(1 -γ)) such that T -(c) + T + (c) = l. Equation (5.38) then provides a solution of problem (5.4) which is negative in (0, T -(c)) and positive in (T -(c), l), with 0 < T -(c) < l, while (5.42) provides a solution which is positive in (0, T + (c)) and negative in (T + (c), l). On the other hand, using also thatT -+ 2T + is continuous in [0, 2γ/(1 -γ)) and min c∈[0,2γ/(1-γ)) (T -(c) + 2T + (c)) ≥ T -(0) + 2T + (0) = T 0 , lim c→ 2γ 1-γ (T -(c) + 2T + (c)) = +∞,we can also find another number c ∈ (0, 2γ/(1 -γ)) such that T -(c) + 2T + (c) = l. Equation (5.38) then provides a solution of problem (5.4) which is negative in (0, T -(c)) ∪ (T (c), T (c) + T -(c)) and positive in (T -(c), T (c)), with 0 < T -(c) < T (c)+T -(c) < l.

4 √ 2 -

 42 -s + 2s + c = 0 if s < 0, -4 √ s + 2s + c = 0 if s > 0,whose resolution provides (5.25).Let us now compute T -, i.e. the integral in (5.18), which written as function ofη = -z 1 (c) ⇐⇒ c = -4η + 2η 2 , 2η 2 -4η , or, using the change of variables t = -p 2 ⇐⇒ p = √ p 2 -2(η -p) .Using a second change of variablesη 2 -p 2 -2(η -p) = r(η -p) =⇒ p = r 2 η -η + 2 r 2 + 1=⇒ dp = 4r(η -1) (r 2 + 1) 2 dr.

  57), we then getT -(c) = 2 √ 2π -4 √ 2 arctan η 2 -2η η + 2 2η 2 -4η.(5.58) By(5.56), this provides expression (5.26) for T -.Let us now show that T -is strictly increasing in c. By (5.58) and η = -z 1 (c) strictly increasing in c, we just need to show that the functionφ(η) = -2 √ 2 arctan η 2 -2η η + 2η 2 -4η is strictly increasing in -z 1 (0), ∞ = [2, ∞). This just follows from φ (η) = (2η -1)(η -2) (η -1) 2η 2 -4η ≥ 0, ∀ η ∈ (2, ∞).

Figure 1 :

 1 Figure 1: Graph of the function b.

Figure 2 :

 2 Figure 2: Branches of solutions for γ = 1/2.

Figure 3 :

 3 Figure 3: The cases γ = 1 2 , c = 0, c = 1.8, c = 1.9999.

Figure 4 :

 4 Figure 4: The cases γ = 1 2 , c = 2, c = 2.1.

Figure 5 :

 5 Figure 5: The graph of T -for γ = 0.2, γ = 0.4, γ = 0.5, γ = 0.7

  a.e. x ∈ Ω.

							(2.15)
	Remark 2.1. The minimum in (2.10) is well defined, thanks to (2.7).
	If a does not depend on s and satisfies the homogeneity condition
	a(x, tξ) = t p-1 a(x, ξ), ∀ ξ ∈ R N , ∀ t ≥ 0, a.e. x ∈ Ω,
	then λ agrees with the first eigenvalue of the operator v ∈ W 1,p 0 (Ω) → -div a(x, ∇v), defined by
			λ = min v∈W 1,p 0 (Ω) v =0	Ω a(x, ∇v) • ∇v dx Ω |v| p dx	.
	Moreover, if λ p denotes the first eigenvalue of the p-Laplacian operator, then, we have
				αλ p ≤ λ.		(2.16)
	This inequality can be proved as follows: We take a sequence R n tending to infinity and a
	sequence u n ∈ H 1 0 (Ω) such that
	R n =	Ω	a(x, u n , ∇u n ) • ∇u n + γ|u n | p dx, lim n→∞	R n Ω |u n | p dx	= λ.	(2.17)
	In particular u n L p (Ω) tends to infinity while the first equality, (2.7) and the definition of λ p
	give
			λ ≥ lim sup n→∞	Ω (α|∇u n | p -a 0 |u n |)dx Ω |u n | p dx	≥ αλ p .
	From (2.16) we deduce that a sufficient condition to have (2.14) is to assume
				γ + ν < αλ p .
	Remark 2.2. Condition (2.7) on a implies
			a(x, 0, 0) = 0, a.e. x ∈ Ω.		(2.18)

  .[START_REF] Lions | Quelques méthodes de résolution des problèmes aux limites non linéaires[END_REF] and such that (2.35),(2.36) and (2.37) hold.Returning now to (3.15) and using (2.15) applied to F n , (2.33), (2.12) and the Rellich-Kondrachov compactness theorem we deduce

	lim δ→0	lim sup n→∞	1 δ {un<δ} |∇u

n | p ϕ p dx + Ω

  belongs to the space W defined by (2.26). Taking z p as test function in the difference of the equations satisfied by u 1 and u 2 , we get 2p {u 2 +δ<u 1 <k+u 2 +δ}

e. in Ω.

By u 1 , u 2 nonnegative solutions of (2.41) and properties properties (2.21) and (2.22) of the nonnegative solutions, we get that z is a nonnegative function of W 1,p 0 (Ω)∩L ∞ (Ω) and therefore z p

Acknowledgements. This work has been partially supported by the projects MTM2017-83583-P of the "Ministerio de Economía, industria y competitividad" of Spain.

In the case where γ = 1/2, we are able to describe all the solutions of problem (5.4) in the sense of Definition 5.1. This is the most simple case of the main result of this section Theorem 5.3. Let γ be such that γ = 1 2 .

Define T 0 by

Then, the set of the solutions of problem (5.4) in the sense of Definition is the union of the following branches:

1. For every l > 0, there exists 1 solution of problem (5.4) which is positive in (0, l). This solution is the unique nonnegative solution of (5.4).

2. For l = T 0 , there exists 1 solution of problem (5.4) which is negative in (0, l). This solution is the unique negative solution of (5.4).

3. For every l > T 0 , there exist 4 solutions of problem (5.4) which are as follows: the first one is negative in (0, l), and is the unique solution (strictly) negative on (0, l); the 3 other ones change sign at every zero and are negative on 1 nonempty subinterval of (0, l) and positive on 1 or 2 nonempty subintervals of (0, l).

4. For l = 2T 0 , there exists 1 solution of problem (5.4) which is negative in (0, T 0 ) ∪ (T 0 , 2T 0 ) but which vanishes at T 0 .

5. For every l > 2T 0 , there exist 4 solutions to problem (5.4) which change sign at every zero and are negative on 2 nonempty subintervals of (0, l) and positive on 1, 2 or 3 nonempty subintervals of (0, l).

6. Simlilarly, for every k ∈ N, with k ≥ 3, for every l = kT 0 , there exists 1 solution of problem (5.4) which is negative on k-1 j=0 (jT 0 , (j + 1)T 0 ) but which vanishes at t = jT 0 , j ∈ {1, ..., k -1}.

7. Similarly, for every k ∈ N, with k ≥ 3, for every l > kT 0 , there exist 4 solutions to problem (5.4) which change sign at every zero and are negative on k nonempty subintervals of (0, l) and positive on k -1, k or k + 1 nonempty subintervals of (0, l).

Therefore when γ = 1/2, problem (5.4) has exactly

(5.16)

Remark 5.4. Let us denote by B j k , with k ∈ N, k ≥ 1, and j = 1, 2, 3, 4, the 4 functions which to any l > k associate the 4 solutions of problem (5.4) in (0, l), which are negative on k subintervals of (0, l), described in the latest point of Theorem 5.3, or more exactly their extensions by zero to (0, ∞) of these 4 solutions. Since these 4 solutions are uniquely defined by formulas (5.38) and (5.42) below, it is not difficult to prove that the functions B j k are continuous on (kT 0 , +∞) with value in L 1 (0, ∞) or even W 1,p 0 (0, ∞) for any p < ∞. This proves that in the case where γ = 1/2, the set of all the solutions of (5.4) coincide with a set made of a countable number of bundles of 4 continuous branches originating at every point l = kT 0 , with k ≥ 1 to which one has to add the branch of the nonnegative solution originating for l = 0. A representation of these branches with the type of solutions corresponding to each branch is given in figure 2.

with G defined by (5.36). This provides the implicit definition of w

The uniqueness of w implies then that for every l ∈ [0, there exists a unique c ∈ [0, 2γ/(1γ)), which satisfies T + (c) = l. Combined with T + (0) = 0, (5.30) and T continuous, this shows that T + is strictly increasing.

In order to study the values of T -, we consider the function R defined by

which is obtained by writing the function T -in the variable z = -z 1 (c). Then, recalling definition (5.13) of b and that z 1 (c) is the unique negative root of b + c, we get that R is given by

Then, using

we deduce that ψ(r) ≤ 0 for every r ∈ [0, z], which implies

and then, for every z > 1, we have

Taking into account (5.32), the function z → z γ+1 /(z γ -1) increasing for z > (1 + γ)

Returning to the variable c = z -1 1 (-z), we have then proved

(5.33) Theorem 5.10. Assume that w is a solution of (5.4) and define c by (5.11).

• If 0 ≤ c < 2γ/(1 -γ), then w is the restriction to [0, l] of a function in W 2,1 loc (R) (W 2,q loc (R), for every q < 1/γ if c > 0), still denoted by w, which is a solution of with T -, T + defined by (5.18), (5.19). Defining G by

(5.36)

we have that one of the two following conditions hold: a)

and for every j ∈ Z, we have

(5.38)

) with z satisfying (5.38), (5.39) and (5.40).

(5.42)

(5.43)

we have that w is the restriction to [0, l] of a function in W 2,q loc (R) for every q < 1/γ solution of (5.34) defined by

(5.44)

Moreover, w satisfies w > 0 in R \ (0, l), w < 0 in (0, l) (5.45)

Reciprocally, for every c ∈ R and l given by (5.37) or (5.41) if c < 2γ/(1 -γ) and (5.43) if c ≥ 2γ/(1 -γ), the above expressions provide a solution of (5.4).

For γ = 1/2, the graph of the different types of solutions given by Theorem 5.10 is represented in figures 3 and 4.

Proof of Theorem 5.10. We distinguish the different cases depending on the value of c:

By (5.12) and Proposition 5.6 we know that w([0, l]) is contained either in [z 1 (c), z 2 (c)] or [z 3 (c), ∞), but since z 3 (c) > 0 and w(0) = w(l) = 0, the last possibility cannot hold true. Thus, we have w

From w(0) = w(l) = 0, there exists r ∈ (0, l) such that w (r) = 0, which combined with (5.12) implies w(r) = z 1 (c) or w(r) = z 2 (c). To fix ideas we assume


Spiros Kotopoulis

Anthony Delalande

Chantal Pichon

Michiel Postema

Real-time sonoporation through HeLa cells

Keywords: Sonoporation, Ultrasound, Microbubbles, HeLa Cancer cells, Drug delivery, Low Mechanical Index PACS: 3.25.Ywm, 43.25.Nm, 43.30.Lz, 43.35.Ty

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

INTRODUCTION

In medical-diagnostics, guidelines state an MI<0.3 can be considered safe for pregnant women and neonatals, but yet diagnostic imaging machines allow the use of MI up to 1.9. Sonoporation is the transient permeabilisation and resealing of a cell membrane with the help of ultrasound and/or an ultrasound contrast agent, allowing for the transmembrane delivery and cellular uptake of macromolecules between 10 kDa and 3 MDa [START_REF] Postema | Ultrasound-directed drug delivery[END_REF]. Previous studies on non-invasive, ultrasound-induced therapeutics used acoustic amplitudes corresponding to mechanical indices between 0.2 and 7.0 [START_REF] Deng | Ultrasound-induced cell membrane porosity[END_REF][START_REF] Pua | Ultrasound-mediated drug delivery[END_REF]. Many studies have demonstrated increased drug and gene uptake of sites under sonication [START_REF] Kondo | Treatment of acute myocardial infarction by hepatocyte growth factor gene transfer: the first demonstration of myocardial transfer of a "functional" gene using ultrasonic microbubble destruction[END_REF][START_REF] Kudo | Sonoporation by single-shot pulsed ultrasound with microbubbles adjacent to cells[END_REF][START_REF] Lindner | Delivery of drugs with ultrasound[END_REF]. These studies presumed that a physical membrane disruption mechanism, i.e. sonoporation, caused the increased uptake, as opposed to naturally occurring active uptake processes, such as endocytosis, which are controlled by the system biology. Here we examin the effect of low MI ultrasound (MI=0.2) in combination with lipid shelled, gas filled microbubbles on HeLa cancer cells. Figure 1 shows six possible mechanisms of sonoporation.

METHODS

An overview of the experimental setup is shown in Figure 2. We have described our experimental setup extensively in Delalande et al. [START_REF] Delalande | Sonoporation at a low mechanical index[END_REF]. In short, a signal consisting of 40 cycles with a centre frequency of 6.6 MHz and a pulse repetition frequency of 10 kHz, was generated by an AFG3102, dual channel arbitrary function generator (Tektronix, Inc., Beaverton, OR), amplified by a 2100L, 50-dB RF amplifier (Electronics & Innovation Ltd., Rochester, NY) and fed to a custom-built 6.6-MHz ultrasound transducer. The peak-negative acoustic pressure corresponds to an MI of 0.2. The transducer was placed in a custom-built, 260×160×150 (mm) 3 Perspex sonication chamber, in which an OptiCell cell culture chamber (Nunc GmbH & Co. KG, Langenselbold, Germany) was placed. One side of the cell culture chamber contained a monolayer of HeLa cells. A customised BXFM-F microscope unit with an LCAch N 20×/0.40 NA PhC (Olym-pusDeutschland GmbH, Hamburg, Germany) and a LUMPlanFL 60×/0.90 NA waterimmersion objective (Olympus) was placed on top of the sonication chamber with the objective lens immersed in the water. The colour charge coupled device (CCD) of a PHOTRON FastCam MC-2.1 high-speed camera (VKT Video Kommunikation GmbH, Pfullingen, Germany) was connected to the microscope.

RESULTS AND DISSCUSSION

Lipid-shelledmicrobubbles were forced into cells using pulsed ultrasound at MI=0.2 at transmit frequencies of 1.0 MHz and 6.6 MHz. This phenomenon typically takes 2 s from the moment a bubble contacts the cell membrane, to complete dissolution of the gas inside the cell. Most bubbleâ Ȃ Şcell penetration occurred within 8 s from the start of sonication. These results were easily reproducible, independent of the setup geometry. We are the first to observe the translation of entire microbubbles into cells. Figure 3 shows a 3D reconstruction of the results seen. Targeted drug delivery down to the cellular level, with the use of encapsulated bubbles will allow the use of high-toxicity drugs to be injected into the body, but only delivered to a specific area. Thus, leaving healthy tissue unaffected. Our sonoporation observations could be attributed to the long pulse lengths used. Although cells themselves are acoustically active, this acoustic activity is probably negligible to that of microbubbles in high concentrations. Therefore, we expect bubble-cell interaction to be more likely in very low bubble concentrations.

CONCLUSION

It has been demonstrated that lipid-shelled microbubbles can be forced to enter cells at a low MI. Hence, if a therapeutic load is added to the bubble, ultrasound guided delivery could be facilitated at diagnostic settings. In addition, these results may have implications for the safety regulations on the use of ultrasound contrast agents for diagnostic imaging.

Figure 1 .

 1 Figure 1. Possbile mechanisms of sonoporation.

Figure 2 .

 2 Figure 2. Confocal fluorescence configuration used to simultaniously observe and sonicate HeLa cells in the pressance of fluorescence coated microbubbles.

Figure 3 .

 3 Figure 3. 3D reconstruction of microbubble translation through a cell membrane. Once the microbubble has translated through the membrane, the pore re-seals and the microbubble is seen to disolve.

NONLINEAR ACOUSTICS State-of-the-Art and Perspectives AIP Conf. Proc. 1474, 271-274 (2012); doi: 10.1063/1.4749348 © 2012 American Institute of Physics 978-0-7354-1081-7/$30.00

ACKNOWLEDGMENTS

This work has been supported by DFG Emmy Noether Programme (grant no. 38355133) and Engineering and Physical Sciences Research Council (EPSRC) (grant no. EP/F037025/1).


Mariia Nesterova
email: mariia.nesterova@ifsttar.fr

Marcel Nowak
email: marcel.nowak@tum.de

Franziska Schmidt
email: franziska.schmidt@ifsttar.fr

Oliver Fischer
email: oliver.fischer@tum.de

Reliability of a bridge with an orthotropic deck exposed to extreme traffic events

Keywords: Millau viaduct, orthotropic deck, extreme load effects, reliability, traffic actions

Introduction

The European road infrastructure includes a significant number of bridges with orthotropic decks, which are exposed to heavy traffic loads and unfavorable environmental influence. In order to estimate if a structure operates safely, such bridges need to be assessed on a regular basis to verify the original design life. This can be done by reliability analysis (e.g. reliability index at the end of the operating life) of the most critical detail of a structure.

Various methods for reliability analysis exist nowadays, each having advantages and drawbacks. A correct choice of the most appropriate reliability method depends on the studied structure, input data, initial conditions, and computational sources [START_REF] Morio | A survey of rare event simulation methods for static input-output models[END_REF]. In bridge engineering, the most frequently used are First and Second Order Reliability Methods (FORM and SORM) described, for instance, by D. Straub [START_REF] Klüppelberg | Risk -A Multidisciplinary Introduction[END_REF].

The main interest of the current study is in comparison of predicted reliability levels at the end of design life based on a limit state function (LSF) defined with various approaches of the extreme values theory (EVT) [START_REF] Coles | An introduction to statistical modeling of extreme values[END_REF]. In a recent study [START_REF] Ghasemi | Target reliability for bridges with consideration of ultimate limit state[END_REF] of various LSF, one of the conclusions shows that not only fatigue but also extreme values and serviceability have to be considered in the reliability analysis of bridges.

Such as for bridges with steel orthotropic decks, critical details of a deck have a shorter life than other structural elements of a bridge [START_REF] Lukacevic | Assessment of reliable fatigue life of orthotropic steel deck[END_REF]. The current study is focused on details of the orthotropic deck of the Millau viaduct obtained using the extreme values approaches: Peaks Over Threshold (POT), Block Maximum (BM) and level crossing counting (LCC). The stresses are derived from the FE model of a part of the deck and based on 180 days of monitored by bridge Weigh-in-Motion (BWIM) traffic data.

Methodology

To assess the reliability of a detail, first of all, the LSF G(x) is needed. It is based on one of several approaches of the EVT and is expressed as:

G(x) = R -L(x) > 0, (1)
where L(x) represents the probabilistic model of a load effect -equivalent stress in the critical welding between a rib and the deck plate -and R = F u is the probabilistic model of the material resistance. R is represented here by the ultimate strength of steel, described by a log-normally distributed random variable with coefficient of variance (CoV) CoV = 0.05.

For each described case, the reliability index β is found for the corresponding probability of failure P f , Eq. ( 2) with the probability density function f x (x) of the load effect x and the cumulative distribution function (CDF) of the standardized Normal distribution Φ -1 (3).

β = -Φ -1 (P f ) = -Φ -1 G(x)≤0 f x (x)dx (2)
Φ(x) = 1 √ 2π x -∞ e -x 2 /2 dx (3)

POT approach

The POT approach has been recently proved to be a good solution for predictions of extreme traffic actions [START_REF] Zhou | A mixture peaks over threshold approach for predicting extreme bridge traffic load effects[END_REF]. As a time-series process, "peak" values of load effect (LE), that lay above a certain threshold, are fitted to the generalized Pareto distribution (GPD). It was recently concluded [START_REF] Schendel | Confidence intervals for return levels for the peaks-over-threshold approach[END_REF] that the efficiency of this method is high enough under the assumption of normal distribution of return level estimators.

Let X be the sequence of n random variables X i representing values of LE with the distribution function F . Let Y be the sequence of m random variables Y j representing exceedance of a threshold u and defined by Y = X -u, for every threshold excess X j ∈ X so that X j > u. A few assumptions for the application of the EVT include identical distribution and statistical independence of random variables X i .

According to the general principle of the POT approach [START_REF] Pickands | Statistical inference using extreme order statistics[END_REF], the CDF of threshold exceedances tends to the upper tail of GPD (4), with shape and scale (σ > 0 and ξ). The approach is valid only if

Y j = X j -u ≥ 0, X j ≥ u for ξ ≥ 0 and u ≤ Y j ≤ u -σ/ξ for ξ < 0. G(Y j ; ξ; σ; u) = (1 -[1 + ξ( Yj σ )] -1/ξ , ξ = 0 1 -exp(- Yj σ ), ξ = 0 (4)
For a long period, estimates can be based on the CDF of extreme values from a shorter period [START_REF] Crespo-Minguillon | A comprehensive traffic load model for bridge safety checking[END_REF]. In other words, monitored values of a LE for a limited period can be used for extrapolation of the LE in time. The value of the p-observation return level L r (p) for the probability P [X ≤ X i |X i > u] with the probability of exceedance (PE) ζ u = P {X i > u}, is a quantile that exceeds once every p observations. The procedure for selecting an optimized threshold is described earlier by author [START_REF] Nesterova | Probabilistic analysis of the effect of the combination of traffic and wind actions on a cable-stayed bridge[END_REF]. Therefore, for a fixed period p, the LSF (1) based on the GPD takes the following form:

R > L r (u, σ, ξ, ζ u ) = u + σ ξ [(pζ u ) -ξ -1], ξ = 0 u + σlog(pζ u ), ξ = 0 (5)

BM method

The BM method is based on a limiting theorem stating that the generalized Extreme Values distribution (GEVD) is an approximation for describing sample maxima for large sample sizes. This holds for the condition of the BM being (approximately) independent observations with identical distributions [START_REF] Coles | An introduction to statistical modeling of extreme values[END_REF]. A BM series is generated by separating the time history for load effects X into intervals of a chosen size and determining the maxima in each interval. Different modifications of BM method are available to handle inhomogeneities in the data-set, aiming at approximately obtaining idealized conditions of independent and identically distributed observations in a remaining subset for further model fit.

One such approach [START_REF] Caprani | Characteristic traffic load effects from a mixture of loading events on short to medium span bridges[END_REF] proposes a composite distribution model that describes bridge load effects from a mixture of different loading event types, representing different number of trucks on the bridge contributing to a maximum value of the LE. Such a model accounts for the actual physical nature of the bridge loading process, and hence allows for a higher accuracy in extreme value estimates, as also shown in a recent study [START_REF] Nowak | Extended extrapolation methods for robust estimates of extreme traffic load effects on bridges[END_REF]. Based on separate GEVD fits G j (z) with model parameters [µ, σ, ξ] to the corresponding BM series for each of the different loading event types, the distribution of load effects G C (z) resulting from the mixture of these loading event types asymptotically approaches the following composite distribution:

G C (z) = N j=1 G j (z) = exp    - N j=1 1 + ξ j z -µ j σ j -1/ξj    (6)
Due to the small influence area of the considered load effect it was observed that the main contribution for extreme values of LE results from a single vehicle. Hence, there is no need for further classification for this study. The LSF (1) based on the BM takes the following form (with

T ref = 5d and R t to be inserted in [d]): R > L r (µ, σ, ξ) = µ -σ ξ 1 -(T ref /R t ) -ξ , ξ = 0 µ -σlog(T ref /R t ), ξ = 0 (7)

Rice formula

The LCC method, where Rices formula is fitted to the upper tail of an out-crossing rate histogram (ORH) is based on the assumption of a stationary Gaussian process describing the time variations of load effects on bridges [START_REF] Cremona | Optimal extrapolation of traffic load effects[END_REF]. The number of times is counted at which positive values are crossed upwardly in a LE time history.

By normalizing the resulting level crossing histogram with respect to the time history length, the ORH is obtained, representing for each level the mean rate of its crossing during a reference period T ref .

The Rices formula with its model parameters

[a 0 = ln(ν 0 ) -m 2 /2σ 2 , a 1 = m/σ 2 , a 2 = -1/2σ 2 ]
describes the mean rate ν(x) of up-crossing for a certain level during T ref , and is fitted to the significant tail regions to the ORH. The proper choice of the starting point for the fit with Rices formula is crucial. It should be as low as possible, to ensure sufficient representativeness for statistical extrapolation, but not too low to still provide a reasonable approximation of the significant tail region. In this study, optimal starting points are identified with evaluating the goodness of fit by means of a modified Kolmogorov test [START_REF] Cremona | Optimal extrapolation of traffic load effects[END_REF]. The LSF (1) based on the LCC takes the following form: Millau viaduct is a cable-stayed bridge that consists of 8 spans with a total length of 2460 m. The stiff orthotropic steel deck is suspended by 11 steel cables at each span. Each span of 342 m is composed of four or six meters long elements, each of them acting almost as a simply supported beam. The cross-section of the deck, Figure 1 [START_REF] Nesterova | Generalized Pareto Distribution for reliability of bridges exposed to fatigue[END_REF], shows that the viaduct has two lanes in each direction: (i) slow lane, where traffic is mainly composed of heavy vehicles, (ii) fast lane, that is used by lorries only to overtake.

R > L r (R t , a 0 , a 1 , a 2 ) = - a 1 2a 2 + - a 0 a 2 + a 1 2a 2 2 - ln(R t /T ref ) a 2 (8
The data on traffic actions were provided from BWIM system that was located in the middle of the first span of the viaduct. It includes axle weights [kN] and spacing [m], vehicle speed [m/s], axles configuration, pavement temperature. Recordings were made between October 2016 and June 2017 with a total 180 days of recorded traffic.

Effects of traffic actions

For the current study, the critical detail is chosen to be the welded connection between a longitudinal stiffener and the plate of the orthotropic deck that is located right under a truck wheel assuming the vehicle is passing in the middle of the slow lane.

The values of stresses that occurred in this connection were computed with a detailed finite element model (FEM) of the part of the deck. The model takes into account the self-weight of the bridge deck with its asphalt layer, the type of passing vehicle and amplitudes of axle loads from each axle, as well it considers the influence of vehicles queues and the wind [START_REF] Nesterova | Probabilistic analysis of the effect of the combination of traffic and wind actions on a cable-stayed bridge[END_REF].

The summary of random variables for each LSF based on described approaches is given in Table 1.

Comparison between various statistical approaches

In order to perform the reliability analysis, limit state functions are defined for every method described in Section 2, and then used in the software UQLab to obtain reliability indexes with SORM method, based on values of load effects described in Section 3. Table 2 gives results for three time references: one year, 50 and 120 years. First three columns show comparison of three extreme values approaches, the reliability index based on design load model is shown in the fifth column, and the last column reflects minimum values of β according to European standards [START_REF]Eurocode-1.3: Basis of structural design[END_REF].

Discussion of results

This work has been carried out on the monitored traffic data from the Millau viaduct. Three extreme value approaches have been performed in order to asses the reliability of the connection between a longitudinal stiffener and the steel plate of the orthotropic deck. The values of reliability indexes are found for the year of monitoring, the 50 years reference period and at the end of the promised operational life of the viaduct.

The results show that during the operational life, the reliability of the most critical detail of the deck is higher than the minimum value given in standards. POT approach and Rice formula show similar results that are also close to results obtained with the design load model (LM1) of standards. The reliability obtained with BM method is much higher, which could be due to the choice of a block size and high variance of distribution parameters, and should be further investigated. The choice of an appropriate method depends on available tools and on the reference data-set. For robust results, reliability levels from several methods should be compared.

) 3 .Figure 1 .

 31 Figure 1. Cross section of the deck of the Millau Viaduct

Table 1 .

 1 Random Variables for reliability modeling

	Case	Random variable	Distribution	Mean	CoV
	Resistance	steel strength	Fu	Log-Normal	400 MPa	0.05
		threshold	u	Normal	102	0.05
	POT	shape scale	ξ σ	Normal Log-Normal	-0.3 25.35	0.12 0.06
		PE	ζu	Bi-nominal	8.2 × 10 -4	0.045
		location	µ	Normal	100.07	0.059
	BM	scale	σ	Normal	23.57	0.198
		shape	ξ	Normal	-1.36 × 10 -2	17.503
		parameter	a 0	Normal	8.23	0.028
	LCC	parameter	a 1	Normal	-1.03 × 10 -2	0.798
		parameter	a 2	Normal	-8.06 × 10 -4	0.087

Table 2 .

 2 Reliability index β for different studied cases

	β, reference	POT	BM	LCC	EN, LM1 [16]	β, EN [15],
	2017	9.8	19.7	9.6	-	4.7
	50 years	7.6	12.5	8.0	8.0	3.8
	120 years	7.4	11.4	7.7	-	-

Acknowledgments

This project has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 676139. The grant is gratefully acknowledged.

The BWIM measurements for the Millau viaduct were obtained with the help from CESTEL, Slovenia and EIFFAGE, France, which is highly appreciated.