Pascal Bégout
email: pascal.begout@math.cnrs.fr

Jesús Ildefonso Díaz

Existence of weak solutions to some stationary Schr ödinger equations with singular nonlinearity

Keywords: 35Q55, (35A01, 35A02, 35B45, 35B65 Schrödinger equation, different boundary conditions, unbounded domains, non local terms, data in weighted spaces, existence, uniqueness, smoothness

We prove some existence (and sometimes also uniqueness) of solutions to some stationary equations associated to the complex Schrödinger operator under the presence of a singular nonlinear term. Among other new facts, with respect some previous results in the literature for such type of nonlinear potential terms, we include the case in which the spatial domain is possibly unbounded (something which is connected with some previous localization results by the authors), the presence of possible non-local terms at the equation, the case of boundary conditions different to the Dirichlet ones and, finally, the proof of the existence of solutions when the right-hand side term of the equation is beyond the usual L 2 -space. Contents 1 Introduction 2 2 Main results 3 3 Notations 8 4 A priori estimates 9 5 Proofs of the main results 16 † The research of J.I.

Introduction

This paper is concerned by existence of solutions for two kinds of equations related to the complex Schrödinger operator,

-∆u + a|u| -(1-m) u + bu = F, in L 2 (Ω), (1.1)
-∆u + a|u| -(1-m) u + bu + cV 2 u = F, in L 2 (Ω), (1.2)
with homogeneous Dirichlet boundary condition In Bégout and Díaz [START_REF] Bégout | Localizing estimates of the support of solutions of some nonlinear Schrödinger equations -The stationary case[END_REF], the authors study the spatial localization property compactness of the support of solutions of equation (1.1) (see Theorems 3.1, 3.5, 3.6, 4.1, 4.4 and 5.2). Existence, uniqueness and a priori bound are also established with the homogeneous Dirichlet boundary condition, F ∈ L p (Ω)

u |Γ = 0, (1.3
(2 < p < ∞) and (a, b) ∈ C 2 satisfying assumptions (2.7) below. In this paper, we give such existence and a priori bound results but for the weaker assumption F ∈ L 2 (Ω) (Theorems 2.8 and 2.9) and also for some different hypotheses on (a, b) ∈ C 2 (Theorems 2.1 and 2.3). Additionally, we consider homogeneous Neumann boundary condition (Theorems 2.8 and 2.9).

In Bégout and Díaz [START_REF] Bégout | Self-similar solutions with compactly supported profile of some nonlinear Schrödinger equations[END_REF], spatial localization property for the partial differential equation (1.2) associated to self-similar solutions of the nonlinear Schrödinger equation

iu t + ∆u = a|u| -(1-m) u + f (t, x),
is studied.

In this paper, we prove existence of solutions with homogeneous Dirichlet or Neumann boundary conditions (Theorems 2.4) and establish a priori bounds (Theorem 2.6), for both equations (1.1) and

(1.2) with any of both boundary conditions (1.3) or (1.4). We also show uniqueness (Theorem 2.10)

and regularity results (Theorem 2.12), under suitable additional conditions. We send the reader to the long introduction of Bégout and Díaz [START_REF] Bégout | Self-similar solutions with compactly supported profile of some nonlinear Schrödinger equations[END_REF] for many comments on the frameworks in which the equation arises (Quantum Mechanics, Nonlinear Optics and Hydrodynamics) and their connections with some other papers in the literature.

This paper is organized as follows. In the next section, we give results about existence, uniqueness, regularity and a priori bounds for equations (1.1) and (1.2), with boundary conditions (1.3) or (1.4), and notations are given in Section 3. Section 4, is devoted to the establishment of a priori bounds for the different truncated nonlinearities of equations studied in this paper. In Section 5, we prove the results given in Section 2. In Bégout and Díaz [START_REF] Bégout | Localizing estimates of the support of solutions of some nonlinear Schrödinger equations -The stationary case[END_REF], localization property is studied for equation (1.1).

The results we give require, sometimes, the same assumptions on (a, b) ∈ C 2 as in Bégout and Díaz [START_REF] Bégout | Localizing estimates of the support of solutions of some nonlinear Schrödinger equations -The stationary case[END_REF] but with a change of notation. See Comments 2.7 below for the motivation of this change. In Section 6

we will show the existence of solutions to equation (1.2) for data in a weighted subspace. Finally, in the last section, we state the principal results obtained in this paper and give some applications.

Existence of solutions for equation (1.2) is used in Bégout and Díaz [START_REF] Bégout | Self-similar solutions with compactly supported profile of some nonlinear Schrödinger equations[END_REF] while existence of solutions for equation (1.1) is used in Bégout and Díaz [START_REF] Bégout | A sharper energy method for the localization of the support to some stationary Schrödinger equations with a singular nonlinearity[END_REF].

Main results

Here, we state the main results of this paper. Then there exists at least a solution u ∈ H 1 0 (Ω) of (1.1). In addition, Symmetry Property 2.2 below holds. Symmetry Property 2.2. If furthermore, for any R ∈ SO N (R), RΩ = Ω and if F is spherically symmetric then we may construct a solution which is additionally spherically symmetric. For N = 1, this means that if F is an even (respectively, an odd) function then u is also an even (respectively, an odd) function.

Theorem 2.3 (A priori bound).

Let Ω an open subset of R N be such that |Ω| < ∞ and assume

0 < m < 1, (a, b) ∈ C 2 and F ∈ L 2 (Ω). If Re(b) < 0 then assume further that Im(b) = 0 or 3 -1 C 2 P < Re(b)
, where C P is the constant in (4.1) below. Let u ∈ H 1 0 (Ω) be any solution to (1.1). Then we have the following estimate.

u H 1 0 (Ω) C, where C = C( F L 2 (Ω) , |Ω|, |a|, |b|, N, m). Theorem 2.4 (Existence). Let Ω ⊆ R N be an open subset and assume V ∈ L ∞ (Ω; R), 0 < m < 1, (a, b, c) ∈ C 3 is such that Im(a)
(Ω) ∩ L m+1 (Ω) to (1.2).
2) If we assume furthermore that Ω is bounded with a C 1 boundary then the conclusion 1) still holds true with u ∈ H 1 (Ω) and the boundary condition

(1.4) instead of u ∈ H 1 0 (Ω).
If, in addition, V is spherically symmetric then Symmetry Property 2.2 holds.

Remark 2.5. Here are some comments about boundary condition.

1) If u ∈ C(Ω) and Ω has not a C 0,1 boundary, the condition u |Γ = 0 does not make sense (in the sense of the trace) and, in this case, has to be understood as u ∈ H 1 0 (Ω).

2) Assume that Ω is bounded and has a C 1,1 boundary. Let u ∈ H 1 (Ω) be any solution to

(1.2) with the boundary condition (1.4). Then u ∈ H 2 (Ω) and boundary condition ∂u ∂ν |Γ = 0 makes sense in the sense of the trace γ ∇u.ν = 0. If, in addition, u ∈ C 1 (Ω) then obviously for any x ∈ Γ, ∂u ∂ν (x) = 0. Indeed, since u ∈ H 1 (Ω), ∆u ∈ L 2 (Ω) and (1.2) makes sense almost everywhere in Ω, we have γ ∂u ∂ν ∈ H -1 2 (Γ) and by Green's formula, [START_REF] Lions | Problèmes aux limites non homogènes[END_REF] and [START_REF] Bégout | Localizing estimates of the support of solutions of some nonlinear Schrödinger equations -The stationary case[END_REF][START_REF] Brezis | Blow up for u t -∆u = g(u) revisited[END_REF][START_REF] Bégout | A sharper energy method for the localization of the support to some stationary Schrödinger equations with a singular nonlinearity[END_REF][START_REF] Drábek | Nonlinear elliptic equations. Singular and Degenerate case[END_REF] in Grisvard [START_REF] Grisvard | Elliptic problems in nonsmooth domains[END_REF], p.62).

Re Ω ∇u(x).∇v(x)dx -γ ∂u ∂ν , γ(v) H -1 2 (Γ),H 1 2 (Γ) + Re Ω f u(x) v(x)dx = Re Ω F (x)v(x)dx, (2.1) for any v ∈ H 1 (Ω), where f (u) = a|u| -(1-m) u + bu + cV 2 u (see Lemma 4.
This implies that γ ∂u ∂ν , γ(v)

H -1 2 (Γ),H 1 2 (Γ) = 0, (2.2) for any v ∈ H 1 (Ω). Let w ∈ H 1 2 (Γ). Let v ∈ H 1 (Ω) be such that γ(v) = w (Theorem 1.5.1.3,
p.38, in Grisvard [START_REF] Grisvard | Elliptic problems in nonsmooth domains[END_REF]). We then deduce from (2.2) that,

∀w ∈ H 1 2 (Γ), γ ∂u ∂ν , w H -1 2 (Γ),H 1 2 (Γ)
= 0, and so γ ∂u ∂ν = 0. But also u ∈ L 2 (Ω) and ∆u ∈ L 2 (Ω). It follows that u ∈ H 2 (Ω) (Proposition 2.5.2.3, p.131, in Grisvard [START_REF] Grisvard | Elliptic problems in nonsmooth domains[END_REF]). Hence the result.

Theorem 2.6 (A priori bound). Let Ω ⊆ R N be an open subset, let V ∈ L ∞ (Ω; R), let 0 < m < 1, let (
u 2 H 1 (Ω) + u m+1 L m+1 (Ω) M V 4 L ∞ (Ω) + 1 F 2 L 2 (Ω) ,
where M = M (|a|, |b|, |c|).

with possibly b = 0, and we worked with

-i∆u + a|u| -(1-m) u + bu = F .
Nevertheless, to maintain a closer notation to many applied works in the literature (see, e.g., the introduction of Bégout and Díaz [START_REF] Bégout | Self-similar solutions with compactly supported profile of some nonlinear Schrödinger equations[END_REF]), we do not work any more with this equation but with, 1) For any F ∈ L 2 (Ω), there exists at least a solution u ∈ H 1 0 (Ω) ∩ L m+1 (Ω) to

-∆u + a|u| -(1-m) u + bu = F,
-∆u + a|u| -(1-m) u + bu = F, in L 2 (Ω) + L m+1 m (Ω). (2.8)
2) If we assume furthermore that Ω is bounded with a C 1 boundary then the conclusion 1) still holds true with u ∈ H 1 (Ω) and the boundary condition

(1.4) instead of u ∈ H 1 0 (Ω).
In addition, Symmetry Property 2.2 holds.

Theorem 2.9 (A priori bound). Let Ω ⊆ R N be an open subset of R N , let 0 < m < 1 and let

(a, b) ∈ A 2 satisfies (2.7). Let F ∈ L 2 (Ω) and let u ∈ H 1 (Ω) ∩ L m+1 (Ω)
u 2 H 1 (Ω) + u m+1 L m+1 (Ω) M F 2 L 2 (Ω) ,
where M = M (|a|, |b|).

Theorem 2.10 (Uniqueness). Let Ω ⊆ R N be an open subset, let V ∈ L ∞ loc (Ω; R), let 0 < m < 1 and let (a, b, c) ∈ C 3 satisfies one of the three following conditions.

1) a = 0, Re(a) 0, Re(ab) 0 and Re(ac) 0.

2) b = 0, Re(b) 0, a = kb, for some k 0 and Re(bc) 0.

3) c = 0, Re(c) 0, a = kc, for some k > 0 and Re(bc) 0.

Let F ∈ L 1 loc (Ω). If there exist two solutions u 1 , u 2 ∈ H 1 (Ω) ∩ L m+1 (Ω) of (1.
2) with the same boundary condition (1.3)

or (1.4) 1 such that V u 1 , V u 2 ∈ L 2 (Ω) then u 1 = u 2 .
Remark 2.11. Here are some comments about Theorems 2.1, 2.4, 2.8 and 2.10.

1) Assume F is spherically symmetric. Since we do not know, in general, if we have uniqueness of the solution, we are not able to show that any solution is radially symmetric.

2) In Theorem 5.2 in Bégout and Díaz [START_REF] Bégout | Localizing estimates of the support of solutions of some nonlinear Schrödinger equations -The stationary case[END_REF], uniqueness for equation 3) In 2) of the above theorem, if we want to make an analogy with 1), assumption a = kb, for some k 0 has to be replaced with Re(ab) 0 and Im(ab) = 0. But, Re(ab) 0 and Im(ab) = 0 ⇐⇒ ∃k 0/a = kb .

-i∆u + a|u| -(1-m) u + bu = F ,
In the same way, Re(ac) > and Im(ac) = 0 ⇐⇒ ∃k > 0/a = kc . 4) In the case of real solutions (with F ≡ 0 and (a, b, c) ∈ R × R × {0}), it is well-known that if b < 0 then it may appear multiplicity of solutions (once m ∈ (0, 1) and a > 0). For more details, see Theorem 1 in Díaz and Hernández [START_REF] Díaz | On a numerable set of branches bifurcating from the infinity of nodal solutions for a singular semilinear equation[END_REF].

Theorem 2.12 (Regularity). Let Ω ⊆ R N be an open subset, let V ∈ L r loc (Ω; C), for any 1 < r < ∞, let 0 < m < 1, let a ∈ C, let F ∈ L 1 loc (Ω), let 1 < q < ∞
and let u ∈ L q loc (Ω) be any local solution to

-∆u + a|u| -(1-m) u + V u = F, in D ′ (Ω).
(2.9)

Let q p < ∞ and let α ∈ (0, m].

1) If F ∈ L p loc (Ω) then u ∈ W 2,p loc (Ω). If (F, V ) ∈ C 0,α loc (Ω) × C 0,α loc (Ω) then u ∈ C 2,α loc (Ω).
2) Assume further that Ω is bounded with a C 1,1 boundary,

F ∈ L p (Ω), V ∈ L r (Ω; C), for any 1 < r < ∞, u ∈ L q (Ω) and γ(u) = 0. Then u ∈ W 2,p (Ω) ∩ W 1,p 0 (Ω). If (F, V ) ∈ C 0,α (Ω) × C 0,α (Ω) then u ∈ C 2,α (Ω) ∩ C 0 (Ω). 3) Assume further that Ω is bounded with a C 1,1 boundary, F ∈ L p (Ω), V ∈ L r (Ω; C), for any 1 < r < ∞, u ∈ L q (Ω) and γ ∂u ∂ν = 0. Then u ∈ W 2,p (Ω). If (F, V ) ∈ C 0,α (Ω) × C 0,α (Ω) then u ∈ C 2,α (Ω) and for any x ∈ Γ, ∂u ∂ν (x) = 0. Remark 2.13. Assume Ω is bounded and has a C 1,1 boundary. Let V ∈ 1<r<∞ L r (Ω; C), 0 < m < 1, (a, b) ∈ C 2 , 1 < q p < ∞, F ∈ L p (Ω)
and let u ∈ L q (Ω) be any solution to (2.9). Let

T : u -→ γ(u), γ ∂u ∂ν be the trace function defined on D(Ω). By density of D(Ω) in D q (∆) def = u ∈ L q (Ω); ∆u ∈ L q (Ω)
, T has a linear and continuous extension from (Hörmander [12], Theorem 2 p.503; Lions and Magenes [START_REF] Lions | Problèmes aux limites non homogènes[END_REF], Lemma 2.2 and Theorem 2.1 p.147; Lions and Magenes [START_REF] Lions | Problemi ai limiti non omogenei[END_REF], Propositions 9.1, 9.2 and Theorem 9.1 p.82; Grisvard [START_REF] Grisvard | Elliptic problems in nonsmooth domains[END_REF], p.54). Since u ∈ L q (Ω), it follows from equation (2.9) and Hölder's inequality that u ∈ D q (∆), so that "γ(u) = 0"

D q (∆) into W -1 q ,q (Γ) × W -1-1 q ,q (Γ)
and "γ ∂u ∂ν = 0" make sense.

The main difficulty to apply Theorem 2.12 is to show that such a solution of (2.9) verifies some boundary condition. In the following result, we give a sufficient condition.

Proposition 2.14 (Regularity).

Let Ω be a bounded open subset of R N with a C 1,1 boundary, let

V ∈ L N (Ω; C) (V ∈ L 2+ε (Ω; C), for some ε > 0, if N = 2 and V ∈ L 2 (Ω; C) if N = 1), let 0 < m < 1, let a ∈ C and let F ∈ L 2 (Ω).
1) Let u ∈ H 1 0 (Ω) be any solution to (2.9). Then u ∈ H 2 (Ω) and γ(u) = 0.

2) Let u ∈ H 1 (Ω) be any solution to (2.9) and (1.4). Then u ∈ H 2 (Ω) and γ ∂u ∂ν = 0.

Remark 2.15. Any solution given by Theorems 2.1, 2.4 or 2.8 belongs to H 2 loc (Ω) (Theorem 2.12).

Notations

We indicate here some of the notations used throughout this paper which have not been defined yet in the introduction (Section 1). We write i 2 = -1. We denote by z the conjugate of the complex number z, Re(z) its real part and Im(z) its imaginary part. For 1 p ∞, p ′ is the conjugate of p defined by 1 p + 1 p ′ = 1. The symbol Ω always indicates a nonempty open subset of R N (bounded or not); its closure is denoted by Ω and its boundary by Γ. For A ∈ {Ω; Ω}, the space

C(A) = C 0 (A)
is the set of continuous functions from A to C and C k (A) (k ∈ N) is the space of functions lying in C(A) and having all derivatives of order lesser or equal than k belonging to C(A). For 0 < α 1

and k ∈ N 0 def = N ∪ {0}, C k,α loc (Ω) = u ∈ C k (Ω); ∀ω ⋐ Ω, |β|=k H α ω (D β u) < +∞ , where H α ω (u) = sup (x,y)∈ω 2 x =y |u(x)-u(y)| |x-y| α . The notation ω ⋐ Ω means that ω is a bounded open subset of R N and ω ⊂ Ω. In the same way, C k,α (Ω) = u ∈ C k (Ω); |β|=k H α Ω (D β u) < +∞ .
H m (Ω) = W m,2 (Ω) and H m 0 (Ω) = W m,2 0
(Ω). For a Banach space E, its topological dual is denoted by

E ⋆ and . , . E ⋆ ,E ∈ R is the E ⋆ -E duality product. In particular, for any T ∈ L p ′ (Ω) and ϕ ∈ L p (Ω) with 1 p < ∞, T, ϕ L p ′ (Ω),L p (Ω) = Re Ω T (x)ϕ(x)dx. We write, W -m,p ′ (Ω) = (W m,p 0 (Ω)) ⋆ (p < ∞)
and

H -m (Ω) = (H m 0 (Ω)) ⋆ .
Unless if specified, any function belonging in a functional space W m,p (Ω), C k (Ω), etc is supposed to be a complex-valued function W m,p (Ω; C), C k (Ω; C), etc . We denote by SO N (R) the special orthogonal group of R N . Finally, we denote by C auxiliary positive constants, and sometimes, for positive parameters a 1 , . . . , a n , write C(a 1 , . . . , a n ) to indicate that the constant C continuously depends only on a 1 , . . . , a n (this convention also holds for constants which are not denoted by "C").

A priori estimates

The proofs of the existence theorems relies on a priori bounds, in order to truncate the nonlinearity and pass to the limit. These bounds are formally obtained by multiplying the equation by u and iu, integrate by parts and by making some linear combinations with the obtained results. Now, we recall the well-known Poincaré's inequality.

If |Ω| < ∞ then, ∀u ∈ H 1 0 (Ω), u L 2 (Ω) C P ∇u L 2 (Ω) . (4.1)
where C P = C P (|Ω|, N ). We will frequently use Hölder's inequality in the following form. If |Ω| < ∞ and 0 m 1 then L 2 (Ω) ֒→ L m+1 (Ω) and

∀u ∈ L 2 (Ω), u m+1 L m+1 (Ω) |Ω| 1-m 2 u m+1 L 2 (Ω) . (4.2)
Finally, we recall the well-known Young's inequality. For any real x 0, y 0 and µ > 0, one has

xy µ 2 2 x 2 + 1 2µ 2 y 2 . (4.3) Lemma 4.1. Let Ω an open subset of R N be such that |Ω| < ∞, let ω an open subset of R N be such that ω ⊆ Ω, let 0 m 1, let (a, b) ∈ C 2 , let α, β 0 and let F ∈ L 2 (Ω). Let u ∈ H 1 0 (Ω) satisfies ∇u 2 L 2 (Ω) + Re(a) u m+1 L m+1 (ω) + α u L 1 (ω c ) + Re(b) u 2 L 2 (ω) + β u L 1 (ω c ) Ω |F u|dx, (4.4) Im(a) u m+1 L m+1 (ω) + α u L 1 (ω c ) + Im(b) u 2 L 2 (ω) + β u L 1 (ω c ) Ω |F u|dx. (4.5)
Here, ω c = Ω \ ω. Assume that one of the three following assertions holds.

1) Re(b) 0. If Re(a) < 0 and |ω| < |Ω| then assume further that α u L 1 (ω c ) u m+1 L m+1 (ω c ) . 2) Re(b) < 0 and Im(b) = 0. If |ω| < |Ω| then assume further that α u L 1 (ω c ) u m+1 L m+1 (ω c ) , F ∈ L ∞ (Ω) and -α|Im(a)| + β 2 |Im(b)| > F L ∞ (Ω) . 3) -C -2 P < Re(b) < 0, where C P is the constant in (4.1), α u L 1 (ω c ) u m+1 L m+1 (ω c ) and β u L 1 (ω c ) u 2 L 2 (ω c ) .
Then we have the following estimate.

u H 1 0 (Ω) C, (4.6)
where

C = C( F L 2 (Ω) , |Ω|, |a|, |b|, N, m). Remark 4.2. Obviously, if |ω| = |Ω| then α u L 1 (ω c ) u m+1 L m+1 (ω c ) and β u L 1 (ω c ) u 2 L 2 (ω c ) .
Proof of Lemma 4.1. By Poincaré's inequality (4.1), it is sufficient to establish

∇u L 2 (Ω) C( F L 2 (Ω) , |Ω|, |a|, |b|, N, m). (4.7)
Moreover, it follows from (4.3) and (4.1),

Ω |F u|dx C 2 P 2 F 2 L 2 (Ω) + 1 2 ∇u 2 L 2 (Ω) . (4.8)
Finally, it follows from (4.2) and (4.1) that if α u L 1 (ω c ) u m+1 L m+1 (ω c ) then one has,

u m+1 L m+1 (ω) + α u L 1 (ω c ) u m+1 L m+1 (Ω) C m+1 P |Ω| 1-m 2 ∇u m+1 L 2 (Ω) . (4.9)
We divide the proof in 3 steps.

Step 1. Proof of (4.7) with Assumption 1).

Assume hypothesis 1) holds true. If Re(a) 0 then (4.7) follows from (4.4) and (4.8), while if Re(a) < 0 we then deduce from (4.4), (4.8) and (4.9) that,

∇u 1-m L 2 (Ω) -2|Re(a)|C m+1 P |Ω| 1-m 2 ∇u m+1 L 2 (Ω) C 2 P F 2 L 2 (Ω) .
Hence (4.7).

Step 2. Proof of (4.7) with Assumption 2).

As for Step 1, it follows from (4.5), (4.2), (4.3) and Hölder's inequality that

|Im(b)| u 2 L 2 (ω) + β u L 1 (ω c ) |Im(a)||Ω| 1-m 2 u m+1 L 2 (ω) + α|Im(a)| u L 1 (ω c ) + 1 2|Im(b)| F 2 L 2 (ω) + |Im(b)| 2 u 2 L 2 (ω) + F L ∞ (ω c ) u L 1 (ω c ) .
Recalling that when |ω| < |Ω|, -α|Im(a

)| + β 2 |Im(b)| > F L ∞ (Ω) , the above estimate yields |Im(b)| u 1-m L 2 (ω) -2|Im(a)||Ω| 1-m 2 u m+1 L 2 (ω) + β|Im(b)| u L 1 (ω c ) 1 |Im(b)| F 2 L 2 (ω) . (4.10) If |Im(b)| u 1-m L 2 (ω) -2|Im(a)||Ω| 1-m 2 1 then u L 2 (ω) C( F L 2 (Ω) , |Ω|, |a|, |b|, m) not. = C 0 , (4.11)
and it follows from (4.5), (4.2), (4.11) and Hölder's inequality that,

β|Im(b)| -α|Im(a)| u L 1 (ω c ) C(C 0 ) + F L ∞ (ω c ) u L 1 (ω c ) C(C 0 ) + β 2 |Im(b)| -α|Im(a)| u L 1 (ω c ) ,
so that,

β u L 1 (ω c ) C( F L 2 (Ω) , |Ω|, |a|, |b|, m) not. = C 1 . (4.12) But if |Im(b)| u 1-m L 2 (ω) -2|Im(a)||Ω| 1-m 2
> 1 then (4.11) and (4.12) come from (4.10).

Finally, by (4.4), (4.8), (4.9), (4.11) and (4.12), one obtains

∇u 2 L 2 (Ω) |Re(a)|C m+1 P |Ω| 1-m 2 ∇u m+1 L 2 (Ω) + C(C 0 , C 1 ) + C 2 P 2 F 2 L 2 (Ω) + 1 2 ∇u 2 L 2 (Ω) . It follows that ∇u 1-m L 2 (Ω) -C ∇u m+1 L 2 (Ω) C + C 2 P F 2 L 2 (Ω)
, from which we easily deduce (4.7).

Step 3. Proof of (4.7) with Assumption 3).

Let µ > 0. By Assumption 3), (4.1), (4.3) and (4.9)

∇u 2 L 2 (Ω) C ∇u m+1 L 2 (Ω) + |Re(b)|C 2 P + C 2 P 2µ 2 ∇u 2 L 2 (Ω) + µ 2 2 F 2 L 2 (Ω) ,
where C = C(|Ω|, |a|, N, m). We then deduce,

1 -|Re(b)|C 2 P - C 2 P 2µ 2 ∇u 1-m L 2 (Ω) -C ∇u m+1 L 2 (Ω) µ 2 2 F 2 L 2 (Ω) . Since |Re(b)| < C -2 P , there exists µ 0 > 0 such that C 2 def. = 1 -|Re(b)|C 2 P - C 2 P 2µ 2 0 > 0. For such a µ 0 , C 2 ∇u 1-m L 2 (Ω) -C ∇u m+1 L 2 (Ω) µ 2 0 2 F 2 L 2 (Ω)
, from which (4.7) follows.

Corollary 4.3. Let (Ω n ) n∈N a sequence of open subsets of R N be such that sup n∈N |Ω n | < ∞, let 0 < m < 1, let (a, b) ∈ C 2 and let (F n ) n∈N ⊂ L ∞ (Ω n ) be such that sup n∈N F n L 2 (Ωn) < ∞. If Re(b) < 0 then assume further that Im(b) = 0 or -1 C 2 P < Re(b), where C P is the constant in (4.1). Let (u n ℓ ) (n,ℓ)∈N 2 ⊂ H 1 0 (Ω n ) be a sequence satisfying ∀n ∈ N, ∀ℓ ∈ N, -∆u n ℓ + f ℓ u n ℓ = F n , in L 2 (Ω n ), (4.13)
where for any ℓ ∈ N,

∀u ∈ L 2 (Ω n ), f ℓ (u) =      a|u| -(1-m) u + bu, if |u| ℓ, aℓ m u |u| + bℓ u |u| , if |u| > ℓ. (4.14)
Then there exists a diagonal extraction u n ϕ(n) n∈N of (u n ℓ ) (n,ℓ)∈N 2 such that the following estimate holds.

∀n ∈ N, u n ϕ(n) H 1 0 (Ωn) C,
where

C = C sup n∈N F n L 2 (Ωn) , sup n∈N |Ω n |, |a|, |b|, N, m .
Proof. Choosing u n ℓ and iu n ℓ as test functions, we get

∇u n ℓ 2 L 2 (Ωn) + Re(a) u n ℓ m+1 L m+1 ({|u n ℓ | ℓ}) + ℓ m u n ℓ L 1 ({|u n ℓ |>ℓ}) + Re(b) u n ℓ 2 L 2 ({|u n ℓ | ℓ}) + ℓ u n ℓ L 1 ({|u n ℓ |>ℓ}) = Re Ωn F n u n ℓ dx, Im(a) u n ℓ m+1 L m+1 ({|u n ℓ | ℓ}) + ℓ m u n ℓ L 1 ({|u n ℓ |>ℓ}) + Im(b) u n ℓ 2 L 2 ({|u n ℓ | ℓ}) + ℓ u n ℓ L 1 ({|u n ℓ |>ℓ}) = Im Ωn F n u n ℓ dx,
for any (n, ℓ) ∈ N 2 . We first note that,

∀(n, ℓ) ∈ N 2 ,    ℓ m u n ℓ L 1 ({|u n ℓ |>ℓ}) u n ℓ m+1 L m+1 ({|u n ℓ |>ℓ}) , ℓ u n ℓ L 1 ({|u n ℓ |>ℓ}) u n ℓ 2 L 2 ({|u n ℓ |>ℓ}) , (4.15)
For each n ∈ N, we choose ϕ(n) ∈ N large enough to have ϕ(n) 1-m > 2

F n L ∞ (Ωn) < -ϕ(n) m |Im(a)| + ϕ(n) 2 |Im(b)|. (4.16)
With help of (4.15) and (4.16), we may apply Lemma 4.1 to Let α, β, R 0, let F ∈ L 2 (Ω) and let

u n ϕ(n) , for each n ∈ N, with ω = x ∈ Ω n ; u n ϕ(n) (x) ϕ(n) , α = ϕ(n) m and β = ϕ(n).
A =      max 1, 1+|b|+R 2 |c| |Im(b)| , |Re(a)| |Im(a)| , if Re(a) 0, max 1, 1+|b|+R 2 |c| |Im(b)| , if Re(a) > 0. If |ω| < |Ω| then assume further that F ∈ L ∞ (Ω) and β 2A F L ∞ (Ω) + 1. Let u ∈ H 1 (Ω) satisfies ∇u 2 L 2 (Ω) + Re(a) u m+1 L m+1 (ω) + α u L 1 (ω c ) -(|b| + R 2 |c|) u 2 L 2 (ω) + β u L 1 (ω c ) Ω |F u|dx, (4.17) |Im(a)| u m+1 L m+1 (ω) + α u L 1 (ω c ) + |Im(b)| u 2 L 2 (ω) + β u L 1 (ω c ) Ω |F u|dx. (4.18)
Then there exists a positive constant M = M (|a|, |b|, |c|) such that,

∇u 2 L 2 (Ω) + u 2 L 2 (ω) + u m+1 L m+1 (ω) + u L 1 (ω c ) M (R 4 + 1) F 2 L 2 (Ω) . (4.19)
Proof. Let A be as in the lemma. We multiply (4.18) by A and sum the result to (4.17). This yields,

∇u 2 L 2 (Ω) + A 0 u m+1 L m+1 (ω) + α u L 1 (ω c ) + u 2 L 2 (ω) + β u L 1 (ω c ) 2A Ω |F u|dx,
where A 0 = A|Im(a)| + Re(a). Applying Hölder's inequality and (4.3), we get

∇u 2 L 2 (Ω) + u 2 L 2 (ω) + A 0 u m+1 L m+1 (ω) + β u L 1 (ω c ) 2A F L ∞ (Ω) u L 1 (ω c ) + 2A 2 F 2 L 2 (Ω) + 1 2 u 2 L 2 (ω) ,
from which we deduce the result if |ω| = |Ω|. Now, suppose |ω| < |Ω|. The above estimate leads to,

∇u 2 L 2 (Ω) + u 2 L 2 (ω) + A 0 u m+1 L m+1 (ω) + β -2A F L ∞ (Ω) u L 1 (ω c ) 4A 2 F 2 L 2 (Ω) ,
from which we prove the lemma since β -2A F L ∞ (Ω) 1.

Let δ 0. Let (F n ) n∈N ⊂ L ∞ (Ω)∩L 2 (Ω) be bounded in L 2 (Ω) and let (u n ℓ ) (n,ℓ)∈N 2 ⊂ H 1 (Ω)∩L m+1 (Ω) be a sequence satisfying ∀n ∈ N, ∀ℓ ∈ N, -∆u n ℓ + δu n ℓ + f ℓ u n ℓ = F n , in L 2 (Ω), (4.24)
with boundary condition (1.3) or (1.4), where for any ℓ ∈ N,

∀u ∈ L 2 (Ω), f ℓ (u) =      a|u| -(1-m) u + (b -δ)u + cV 2 u, if |u| ℓ, aℓ m u |u| + (b -δ)ℓ u |u| + cV 2 ℓ u |u| , if |u| > ℓ. (4.25)
For (1.4), Ω is assumed to have a C 1 boundary. Then there exist

M = M V L ∞ (Ω) , |a|, |b|, |c| and a diagonal extraction u n ϕ(n) n∈N of (u n ℓ ) (n,ℓ)∈N 2 for which, ∇u n ϕ(n) 2 L 2 (Ω) + u n ϕ(n) 2 L 2 u n ϕ(n) ϕ(n) + u n ϕ(n) m+1 L m+1 u n ϕ(n) ϕ(n) + u n ϕ(n) L 1 u n ϕ(n) >ϕ(n) M sup n∈N F n 2 L 2 (Ω) ,
for any n ∈ N. The same is true if we replace the conditions on (a, b, c) by (a, b, c) ∈ A × A × {0} satisfies (2.7) and δ δ ⋆ , where δ ⋆ is given by Lemma 4.5. In this case, M = M (|a|, |b|).

Proof. Choosing u n ℓ and iu n ℓ as test functions, we obtain

∇u n ℓ 2 L 2 (Ω) + Re(a) u n ℓ m+1 L m+1 ({|u n ℓ | ℓ}) + ℓ m u n ℓ L 1 ({|u n ℓ |>ℓ}) + Re(b) -V 2 L ∞ (Ω) |Re(c)| u n ℓ 2 L 2 ({|u n ℓ | ℓ}) + ℓ u n ℓ L 1 ({|u n ℓ |>ℓ}) Re Ω F n u n ℓ dx, (4.26) Im(a) u n ℓ m+1 L m+1 ({|u n ℓ | ℓ}) + ℓ m u n ℓ L 1 ({|u n ℓ |>ℓ}) + Im(b) u n ℓ 2 L 2 ({|u n ℓ | ℓ}) + ℓ u n ℓ L 1 ({|u n ℓ |>ℓ}) + Im(c) V u 2 L 2 ({|u n ℓ | ℓ})) + ℓ V 2 u L 1 ({|u n ℓ |>ℓ})) = Im Ω F n u n ℓ dx, (4.27) for any (n, ℓ) ∈ N 2 . If (a, b, c) ∈ A × A × {0} satisfies (2.7
), then we obtain

∇u n ℓ 2 L 2 (Ω) + δ u n ℓ 2 L 2 (Ω) + Re(a) u n ℓ m+1 L m+1 ({|u n ℓ | ℓ}) + ℓ m u n ℓ L 1 ({|u n ℓ |>ℓ}) + Re(b) -δ u n ℓ 2 L 2 ({|u n ℓ | ℓ}) + ℓ u n ℓ L 1 ({|u n ℓ |>ℓ}) = Re Ω F n u n ℓ dx, (4.28) Im(a) u n ℓ m+1 L m+1 ({|u n ℓ | ℓ}) + ℓ m u n ℓ L 1 ({|u n ℓ |>ℓ}) + Im(b) u n ℓ 2 L 2 ({|u n ℓ | ℓ}) + ℓ u n ℓ L 1 ({|u n ℓ |>ℓ}) = Im Ω F n u n ℓ dx, (4.29)
for any (n, ℓ) ∈ N 2 . For this last case, it follows from Lemma 4.5, Hölder's inequality and (4.3) that

∇u n ℓ 2 L 2 (Ω) + L 2 u n ℓ 2 L 2 ({|u n ℓ | ℓ}) + L u n ℓ m+1 L m+1 ({|u n ℓ | ℓ}) + Lℓ -M F n L ∞ (Ω) u n ℓ L 1 ({|u n ℓ |>ℓ}) M 2 2L F n 2 L 2 (Ω) .
Then the result follows by choosing for each n ∈ N, ϕ(n) ∈ N large enough to have Lϕ(n) -

M F n L ∞ (Ω)
1. Now we turn out to the case (4.26)-(4.27). Let M and A be given by Lemma 4.

4 with R = V L ∞ (Ω) . For each n ∈ N, let ϕ(n) ∈ N be large enough to have ϕ(n) 2A F n L ∞ (Ω) + 1, if |ω| < |Ω| and ϕ(n) = n, if |ω| = |Ω|.
For each n ∈ N, with help of (4.26) and ( 4.27), we may apply

Lemma 4.4 to u n ϕ(n) with ω = x ∈ Ω; u n ϕ(n) (x) ϕ(n) , α = ϕ(n) m , β = ϕ(n) and R = V L ∞ (Ω) .
Hence the result.

Proofs of the main results

Proof of Theorem 2.12. Property 1) follows from Proposition 4.5 in Bégout and Díaz [START_REF] Bégout | Localizing estimates of the support of solutions of some nonlinear Schrödinger equations -The stationary case[END_REF] while Property 2) comes from Remark 4.7 in Bégout and Díaz [START_REF] Bégout | Localizing estimates of the support of solutions of some nonlinear Schrödinger equations -The stationary case[END_REF]. It remains to establish Property 3).

Assume first that F ∈ L p (Ω) and V ∈ 1<r<∞ L r (Ω). It follows from the equation that for any ε ∈ (0, q -1), ∆u ∈ L q-ε (Ω). We now recall an elliptic regularity result. If for some 1 < s < ∞, u ∈ L s (Ω) satisfies ∆u ∈ L s (Ω) and γ(∇u.ν) = 0 then u ∈ W 2,s (Ω) (Proposition 2.5.2.3, p.131, in Grisvard [START_REF] Grisvard | Elliptic problems in nonsmooth domains[END_REF]).

Since for any ε ∈ (0, q -1), u, ∆u ∈ L q-ε (Ω) and γ(∇u.ν) = 0 (by assumption), by following the bootstrap method of the proof p.52 of Property 1) of Proposition 4.5 in Bégout and Díaz [START_REF] Bégout | Localizing estimates of the support of solutions of some nonlinear Schrödinger equations -The stationary case[END_REF], we obtain the result. Indeed, therein, it is sufficient to apply the global regularity result in Grisvard [START_REF] Grisvard | Elliptic problems in nonsmooth domains[END_REF] (Proposition 2.5.2.3, p.131) in place of the local regularity result in Cazenave [START_REF] Cazenave | An introduction to semilinear elliptic equations[END_REF] (Proposition 4.1.2, p.101-102). Now, you turn out to the Hölder regularity. Assume F ∈ C 0,α (Ω) and V ∈ C 0,α (Ω). By global smoothness property in W 2,p proved above, we know that u ∈ W 2,N +1 (Ω) and γ(∇u.ν) = 0 in L N +1 (Γ). It follows from the Sobolev's embedding, W 2,N +1 (Ω) ֒→ C 1, 1 N +1 (Ω) ֒→ C 0,1 (Ω), that for any x ∈ Γ, ∂u ∂ν (x) = 0 and u ∈ C 0,1 (Ω). A straightforward calculation yields,

∀(x, y) ∈ Ω 2 , |u(x)| -(1-m) u(x) -|u(y)| -(1-m) u(y) 5|u(x) -u(y)| m 5|x -y| m . Setting, g = F -(a|u| -(1-m) u + (b -1)u + cV u), we deduce that g ∈ C 0,α (Ω). Let v ∈ C 2,α (Ω) be the unique solution to    -∆v + v = g, in Ω, ∂v ∂ν = 0, on Γ,
(see, for instance, Theorem 3.2 p.137 in Ladyzhenskaya and Ural'tseva [START_REF] Ladyzhenskaya | Linear and quasilinear elliptic equations[END_REF]). It follows that u and v are two H 1 -solutions of the above equations and since uniqueness holds in

H 1 (Ω) (Lax-Milgram's
Theorem), we deduce that u = v. Hence u ∈ C2,α (Ω). This concludes the proof 2 .

Proof of Proposition 2.14. We first establish Property 1). Since Ω has C 0,1 boundary and u ∈ H 1 0 (Ω), it follows that γ(u) = 0. Moreover, Sobolev's embedding and equation (2.9) imply that ∆u ∈ L 2 (Ω). We then obtain that u ∈ H 2 (Ω) (Grisvard [11], Corollary 2.5.2.2, p.131). Hence Property 1). We turn out to Property 2). It follows from equation (2.9) that ∆u ∈ L 2 (Ω), so that (2.9) makes sense a.e. in Ω. Then Property 2) comes from the arguments of 2) of Remark 2.5.

Lemma 5.1. Let O ⊂ R N be a bounded open subset, let V ∈ L ∞ (Ω; C), let 0 < m < 1, let (a, b, c) ∈ C 3 and let F ∈ L 2 (O). Let δ ∈ [0, 1]. Then for any ℓ ∈ N, there exist a solution u 1 ℓ ∈ H 1 0 (O) to -∆u ℓ + δu ℓ + f ℓ (u ℓ ) = F, in L 2 (O), (5.1)
with boundary condition (1.3) and a solution u 2 ℓ ∈ H 1 (O) to (5.1) with boundary condition (1.4) (in this case, O is assumed to have a C 1 boundary and δ > 0), where

∀u ∈ L 2 (Ω), f ℓ (u) =      a|u| -(1-m) u + (b -δ)u + cV 2 u, if |u| ℓ, aℓ m u |u| + (b -δ)ℓ u |u| + cV 2 ℓ u |u| , if |u| > ℓ. (5.2)
If, in addition, V is spherically symmetric then Symmetry Property 2.2 holds.

Proof. We proceed with the proof in two steps. Let H = H 1 0 (O), in the homogeneous Dirichlet case, and

H = H 1 (O), in the homogeneous Neumann case. Let δ ∈ [0, 1] with additionally δ > 0 and Γ of class C 1 if H = H 1 (O) .
Step 1 below being obvious, we omit the proof.

Step 1. ∀G ∈ L 2 (O), ∃!u ∈ H s.t. -∆u + δu = G. Moreover, ∃α > 0 s.t. ∀G ∈ L 2 (O), (-∆ + δI) -1 G H 1 (O) α G L 2 (O)
. Finally, Symmetry Property 2.2 holds.

Step 2. Conclusion.

For each ℓ ∈ N, we define

g ℓ = -f ℓ + F ∈ C L 2 (O); L 2 (O)
. With help of the continuous and compact embedding i : H ֒→ L 2 (O) and Step 1, we may define a continuous and compact sequence of mappings (T ℓ ) ℓ∈N of H as follows. For any ℓ ∈ N, set

T ℓ : H i ֒→ L 2 (O) g ℓ -→ L 2 (O) (-∆+δI) -1 -------→ H u -→ i(u) = u -→ g ℓ (u) -→ (-∆ + δu) -1 (g ℓ )(u) Set ρ = 2α(|a| + |b| + |c| + 1) V 2 L ∞ (Ω) + 2 ℓ|O| 1 2 + F L 2 (O) . Let u ∈ H. It follows that, T ℓ (u) H 1 (O) = (-∆ + δI) -1 (g ℓ )(u) H 1 (O) α g ℓ (u) L 2 (O) ρ.
Existence comes from the Schauder's fixed point Theorem applied to T ℓ . The Symmetry Property 2.2 is obtained by working in H rad in place of H and in H even and H odd for N = 1 .

Proof of Theorem 2.1.

Let for any u ∈ L 2 (Ω), f (u) = a|u| -(1-m) u + bu. Set Ω n = Ω ∩ B(0, n). Let (G n ) n∈N ⊂ D(Ω) be such that G n L 2 (Ω) ----→ n→∞ F. Let u n ℓ (n,ℓ)∈N 2 ⊂ H 1 0 (Ω n )
a sequence of solutions of (5.1) be given by Lemma 5.1 with O = Ω n , c = δ = 0 and F n = G n|Ωn . We define u n ℓ ∈ H 1 0 (Ω) by extending u n by 0 in Ω ∩ Ω c n . We also denote by f ℓ the extension by 0 of

f ℓ in Ω ∩ Ω c n . By Corollary 4.3, there exists a diagonal extraction u n ϕ(n) n∈N of u n ℓ (n,ℓ)∈N 2 which is bounded in H 1 0 (Ω)
. By reflexivity of H 1 0 (Ω), Rellich-Kondrachov's Theorem and converse of the dominated convergence theorem, there exist u ∈ H 1 0 (Ω) and g ∈ L 2 loc (Ω; R) such that, up to a subsequence that we still denote by u n ϕ(n) n∈N C(g m + g) ∈ L 2 loc (Ω), a.e. in Ω.

From the dominated convergence Theorem,

f ϕ(n) u n ϕ(n) L 2 loc (Ω) -----→ n→∞ f (u). Let ϕ ∈ D(Ω).
Let n ⋆ ∈ N be large enough to have supp ϕ ⊂ Ω n⋆ . We have by (5.1),

∀n > n ⋆ , -i∆u n ϕ(n) + f ϕ(n) u n ϕ(n) -F n , ϕ |Ωn D ′ (Ωn),D(Ωn) = 0.
The above convergencies lead to,

-∆u + f (u) -F, ϕ D ′ (Ω),D(Ω) = -u, ∆ϕ D ′ (Ω),D(Ω) + f (u) -F, ϕ D ′ (Ω),D(Ω) = lim n→∞ -u n ϕ(n) , ∆ϕ D ′ (Ω),D(Ω) + lim n→∞ f ϕ(n) u n ϕ(n) -G n , ϕ D ′ (Ω),D(Ω) = lim n→∞ -∆u n ϕ(n) + f ϕ(n) u n ϕ(n) -F n , ϕ |Ωn D ′ (Ωn),D(Ωn) = 0.
By density, we then obtain that u ∈ H 1 0 (Ω) is a solution to -∆u + f (u) = F, in L 2 (Ω). Finally, if F is spherically symmetric then u (obtained as a limit of solutions given by Lemma 5.1) is also spherically symmetric. For N = 1, this includes the case where F is an even function.

Proof of Theorems 2.3 and 2.9. Choosing u and iu as test functions, we obtain

∇u 2 L 2 (Ω) + Re(a) u m+1 L m+1 (Ω) + Re(b) u 2 L 2 (Ω) = Re Ω F udx, Im(a) u m+1 L m+1 (Ω) + Im(b) u 2 L 2 (Ω) = Im Ω F udx.
Theorem 2.3 follows immediately from Lemma 4.1 applied with ω = Ω, while Theorem 2.9 is a consequence of Lemma 4.5 applied with δ = 0 and (4.3). This ends the proof.

Proof of Theorem 2.6. Choosing u and iu as test functions, we obtain

∇u 2 L 2 (Ω) + Re(a) u m+1 L m+1 (Ω) + Re(b) -|Re(c)| V 2 L ∞ (Ω) u 2 L 2 (Ω) Ω |F u|dx, |Im(a)| u m+1 L m+1 (Ω) + |Im(b)| u 2 L 2 (Ω) + |Im(c)| V u 2 L 2 (Ω) Ω |F u|dx.
The theorem follows Lemma 4.4 applied with ω = Ω, R = V L ∞ (Ω) and α = β = 0.

Proof of Theorems 2.4 and 2.8. We first assume that Ω is bounded. Let H = H 1 0 (Ω), in the homogeneous Dirichlet case, and H = H ℓ (n,ℓ)∈N 2 which is bounded in W 1,1 (Ω) ∩ Ḣ1 (Ω). Let 1 < p < 2 be such that W 1,1 (Ω) ֒→ L p (Ω). Then u n ϕ(n) n∈N is bounded in W 1,p (Ω) and there exist u ∈ W 1,p (Ω) ∩ Ḣ1 (Ω) and g ∈ L p (Ω; R) such that, up to a subsequence that we still denote by

u n ϕ(n) n∈N , u n ϕ(n) L p (Ω) ----→ n→∞ u, ∇u n ϕ(n) ⇀ ∇u in L 2 w (Ω) N , as n -→ ∞, u n ϕ(n) a.e. in Ω -----→ n→∞ u, u n ϕ(n) g, a.e. in Ω and u n ϕ(n) 1 u n ϕ(n) ϕ(n) n∈N is bounded in L 2 (Ω),
where the last estimate comes from Corollary 4.6. By these three last estimates and Fatou's Lemma,

u ∈ L 2 (Ω), f ϕ(n) u n ϕ(n) a.e. in Ω -----→ n→∞ f (u) -δu and f ϕ(n) u n ϕ(n) C(g m + g) ∈ L p (Ω), a.e. in Ω. It follows that u ∈ H 1 (Ω). From the dominated convergence Theorem, f ϕ(n) u n ϕ(n) L p (Ω) ----→ n→∞ f (u) -δu.
Consider the Dirichlet boundary condition. We recall a Gagliardo-Nirenberg's inequality.

∀w ∈ H 1 0 (Ω), w N +2 L 2 (Ω) C w 2 L 1 (Ω) ∇w N L 2 (Ω) ,
where C = C(N ). In particular, C does not depend on Ω. Since

u n ϕ(n) n∈N ⊂ H 1 0 (Ω) is bounded in W 1,1 (Ω) ∩ Ḣ1 (Ω),
+ δu n ϕ(n) + f ϕ(n) u n ϕ(n) , v L p (Ω),L p ′ (Ω) -F n , v L 2 (Ω),L 2 (Ω) = 0. (5.3)
Above convergencies lead to allow us to pass in the limit in (5.3) and by density of D(Ω) in H 1 0 (Ω) and density of H m0 (Ω) in H 1 (Ω) (see, for instance, Corollary 9.8, p.277, in Brezis [START_REF] Brezis | Functional analysis, Sobolev spaces and partial differential equations[END_REF]), it follows that

∀v ∈ H, ∇u, ∇v L 2 (Ω),L 2 (Ω) + f (u), v L 2 (Ω),L 2 (Ω) = F, v L 2 (Ω),L 2 (Ω) .
This finishes the proof of the existence for Ω bounded. Approximating Ω by an exhaustive sequence of bounded sets (Ω ∩ B(0, n)) n∈N , the case Ω unbounded can be treated in the same way as in the proof of Theorem 2.1. The symmetry property also follows as in the proof of Theorem 2.1.

Proof of Theorem 2.10. Let u 1 , u 2 ∈ H 1 (Ω) ∩ L m+1 (Ω) be two solutions of (1.2) such that V u 1 , V u 2 ∈ L 2 (Ω). We set u = u 1 -u 2 , f (v) = |v| -(1-m) v and g(v) = af (v) + bv + cV 2 v. From
Lemma 9.1 in Bégout and Díaz [START_REF] Bégout | Localizing estimates of the support of solutions of some nonlinear Schrödinger equations -The stationary case[END_REF], there exists a positive constant C such that,

C ω |u 1 (x) -u 2 (x)| 2 (|u 1 (x)| + |u 2 (x)|) 1-m dx f (u 1 ) -f (u 2 ), u 1 -u 2 L m+1 m (Ω),L m+1 (Ω) , (5.4)
where ω = x ∈ Ω; |u 1 (x)|+ |u 2 (x)| > 0 . We have that u satisfies -∆u + g(u 1 )-g(u 2 ) = 0. Choosing v = au as a test function, we get

Re(a) ∇u 2 L 2 + |a| 2 f (u 1 ) -f (u 2 ), u 1 -u 2 L m+1 m ,L m+1 + Re(ab) u 2 L 2 + Re (ac) V u 2 L 2 = 0.
It follows from the above estimate and (5.4) that,

Re(a) ∇u 2 L 2 + C|a| 2 ω |u 1 (x) -u 2 (x)| 2 (|u 1 (x)| + |u 2 (x)|) 1-m dx + Re(ab) u 2 L 2 + Re (ac) V u 2 L 2 0,
which yields Property 1). Properties 2) and 3) follow in the same way.

Remark 5.2. It is not hard to adapt the above proof to find other criteria of uniqueness.

6 On the existence of solutions of the Dirichlet problem for data beyond L 2 (Ω)

In this section we shall indicate how some of the precedent results of this paper can be extended to some data F which are not in L 2 (Ω) but in the more general Hilbert space L 2 (Ω; δ α ), where δ(x) = dist(x, Γ) and α ∈ (0, 1).

In order to justify the associated notion of solution, we start by assuming that a function u solves equation

-∆u + f (u) = F, in Ω, ( 6.1)
with the Dirichlet boundary condition (1.3), u |Γ = 0, and we multiply (formally) by v(x)δ(x), with v ∈ H 1 0 (Ω; δ α ) the weighted Sobolev space associated to the weight δ α (x) , we integrate by parts (by Green's formula) and we take the real part. Then we get,

Re Ω ∇u.∇v δ α dx + Re Ω v ∇u.∇δ α dx + Re Ω f (u) v δ α dx = Re Ω F v δ α dx. (6.2)
To give a meaning to the condition (6.2), we must assume that

F ∈ L 2 (Ω; δ α ), (6.3)
where The justification of the second term in (6.2) is far to be trivial and requires the use of a version of the following Hardy type inequality,

F 2 L 2 (Ω;δ α ) = Ω |F (x)| 2 δ α (x)
Ω |v(x)| 2 δ -(2-α) (x)dx C Ω |∇v(x)| 2 δ α (x)dx, (6.5)
which holds for some constant C independent of v, for any v ∈ H 1 0 (Ω; δ α ) once we assume that Ω is a bounded open subset of R N with Lipschitz boundary (6.6) (see, e.g., Kufner [13] and also Drábek, Kufner and Nicolosi [START_REF] Drábek | Nonlinear elliptic equations. Singular and Degenerate case[END_REF], Kufner and Opic [START_REF] Kufner | Hardy-type inequalities[END_REF], Kufner and Sänding [START_REF] Kufner | Some applications of weighted Sobolev spaces[END_REF] and Nečas [START_REF] Nečas | Sur une méthode pour résoudre les équations aux dérivées partielles du type elliptique, voisine de la variationnelle[END_REF]). Notice that under (6.6), we know that δ ∈ W 1,∞ (Ω) and so

Ω v ∇u.∇δ α dx = Ω δ α 2 ∇u . v δ α 2 ∇δ α dx α ∇δ L ∞ (Ω) ∇u L 2 (Ω;δ α ) v L 2 (Ω;δ -(2-α) ) < ∞,
by Cauchy-Schwarz's inequality and (6.5).

Definition 6.1. Assumed (6.3), (6.6) and α ∈ (0, 1), we say that u ∈ H 1 0 (Ω; δ α ) is a solution of (6.1) and (1.3) in H 1 0 (Ω; δ α ) if (6.4) holds and the integral condition (6.2) holds for any v ∈ H 1 0 (Ω; δ α ).

eigenfunction ϕ 1 of the Laplace operator,

-∆ϕ 1 = λ 1 ϕ 1 , in Ω, ϕ 1|Γ = 0, on Γ. It is well-known that ϕ 1 ∈ W 2,∞ (Ω) ∩ W 1,∞ 0
(Ω) and that C 1 δ(x) ϕ 1 (x) C 2 δ(x), for any x ∈ Ω, for some positive constants C 1 and C 2 , independent of x. Now, with this new weighted function, it is easy to see that the second term in (6.2) does not play any important role since, for instance, when taking v = u as test function, we get that

Re

Ω u ∇u.∇ϕ α 1 dx = 1 2 Ω ∇|u| 2 .∇ϕ α 1 dx = - 1 2 Ω |u| 2 ∆ϕ α 1 dx = αλ 1 2 Ω |u| 2 ϕ α 1 dx + α(1 -α) 2 Ω |u| 2 ϕ -(2-α) 1 |∇ϕ 1 | 2 dx 0.

Conclusions

In this section, we summarize the results obtained in Section 2 and give some applications.

The next result comes from Theorems 2.1, 2.3 and 2.10. In the above theorem, the complex numbers a and b are seen as vectors -→ a and -→ b of R 2 . Consequently, -→ a .

-→ b denotes the scalar product between these vectors of R 2 . ). Knowing that for such (a, b) equation (7.1) admits solutions, it would be interesting if, whether or not, solutions with compact support exist, as in Bégout and Díaz [START_REF] Bégout | Localizing estimates of the support of solutions of some nonlinear Schrödinger equations -The stationary case[END_REF].

The novelty of

By Theorems 2.4, 2.6 and 2.10, we get the following result. Since, now, we are able to show that equation (7.2) admits solutions, we can study the propagation support phenomena. Indeed, we can show that, under some suitable conditions, there exists a selfsimilar solution u to iu t + ∆u = a|u| -(1-m) u + f (t, x), in R N , such that for any t > 0, supp u(t) is compact (see Bégout and Díaz [START_REF] Bégout | Self-similar solutions with compactly supported profile of some nonlinear Schrödinger equations[END_REF]). Now, we turn out to equation (7.1) by extending some results found in Bégout and Díaz [START_REF] Bégout | Localizing estimates of the support of solutions of some nonlinear Schrödinger equations -The stationary case[END_REF]. These results are due to Theorems 2.8, 2.9 and 2.10. For any F ∈ L 2 (Ω), there exists at least a solution u ∈ H 1 (Ω) ∩ L m+1 (Ω) to

-∆u + a|u| -(1-m) u + bu = F, in L 2 (Ω) + L m+1 m (Ω), (7.3)
with boundary condition (1.3) or (1.4) 1 (in this last case, Ω is assumed bounded). Furthermore, • If Ω = R N and if F ∈ L 2 (R N ) has compact support then equation (7.3) admits solutions and any solution is compactly supported.

u 2 H 1 (Ω) + u m+1 L m+1 (Ω) M (|a|, |b|) F 2 L 2 (Ω) .
• If F L 2 (Ω) is small enough and if F has compact support then equation (7.3) admits solutions with the homogeneous Dirichlet boundary condition and any solution is compactly supported in Ω.

• If F L 2 (Ω) is small enough, if -→ a .

-→ b > 0 and if F has compact support then equation (7.3) admits a unique solution with the homogeneous Neumann boundary condition and, in fact, this solution is compactly supported in Ω.

For more details, see Bégout and Díaz [START_REF] Bégout | A sharper energy method for the localization of the support to some stationary Schrödinger equations with a singular nonlinearity[END_REF]. Finally, in Section 6 we extended our techniques of proofs to the case in which the datum F is very singular near the boundary of Ω but still is in some weighted Lebesgue space (see Theorem 6.5).

2 j

 2 a subset of R N with boundary Γ, 0 < m < 1, (a, b, c) ∈ C 3 and V ∈ L ∞ (Ω; R) is a real potential. Here and in what follows, when Γ is of class C 1 , ν denotes the outward unit normal vector to Γ. Moreover, ∆ = is the Laplacian in Ω.

Theorem 2 . 1 ( 1 C 2 P<

 2112 Existence). Let Ω an open subset of R N be such that |Ω| < ∞ and assume 0 < m < 1, (a, b) ∈ C 2 and F ∈ L 2 (Ω). If Re(b) < 0 then assume further that Im(b) = 0 or -Re(b), where C P is the Poincaré's constant in (4.1) below.

1 )

 1 0, Im(b) < 0 and Im(c) 0. If Re(a) 0 then assume further that Im(a) < 0. Then we have the following result. For any F ∈ L 2 (Ω), there exists at least a solution u ∈ H 1 0

  a, b, c) ∈ C 3 be such that Im(a) 0, Im(b) < 0 and Im(c) 0. If Re(a) 0 then assume further that Im(a) < 0. Let F ∈ L 2 (Ω) and let u ∈ H 1 (Ω) be any solution to (1.2) with boundary condition (1.3) or (1.4) 1 . Then we have the following estimate.

Comments 2 . 7 .

 27 In the context of the paper of Bégout and Díaz [1], we can establish an existence result with the homogeneous Neumann boundary condition (instead of the homogeneous Dirichlet condition) and F ∈ L 2 (Ω) instead of F ∈ L m+1 m (Ω) . In Bégout and Díaz [1], we introduced the set, A = C \ z ∈ C; Re(z) = 0 and Im(z) 0 , and assumed that ( a, b) ∈ C 2 satisfies, ( a, b) ∈ A × A and              Re( a)Re( b) 0, or Re( a)Re( b) < 0 and Im( b) > Re( b) Re( a) Im( a),

1

  for which we additionally assume that Ω has a C 1 boundary. and b = 0. This means that we chose, a = ia, b = ib and F = iF. Then assumptions on (a, b) are changed by the fact that for z = iz, Re(z) = Re(-i z) = Im( z), (2.4) Im(z) = Im(-i z) = -Re( z). (2.5) It follows that the set A and (2.3) become, A = C \ z ∈ C; Re(z) 0 and Im(z) = 0 , (2.6) (a, b) ∈ A × A and            Im(a)Im(b) 0, or Im(a)Im(b) < 0 and Re(b) > Im(b) Im(a) Re(a).

  b) ∈ A × A satisfies (2.3) ⇐⇒ (a, b) ∈ A × A satisfies (2.7) . Assumptions (2.7) are made to prove the existence and the localization property of solutions to equation (1.1). Now, we give some results about equation (1.1) when (a, b) ∈ A × A satisfies (2.7).Theorem 2.8 (Existence). Let Ω ⊆ R N be an open subset of R N , let 0 < m < 1 and let (a, b) ∈ A 2 satisfies (2.7).

  holds if a = 0, Im( a) 0 and Re( a b) 0. By (2.4)-(2.5), those assumptions are equivalent to 1) of Theorem 2.10 above for equation (1.1) (of course, c = 0). It follows that Theorem 2.10 above extends Theorem 5.2 of Bégout and Díaz[START_REF] Bégout | Localizing estimates of the support of solutions of some nonlinear Schrödinger equations -The stationary case[END_REF].

  The space C 0 (Ω) consists of functions belonging to C(Ω) and vanishing at the boundary Γ, D(Ω) is the space of C ∞ functions with compact support and D(Ω) is the restriction to Ω of functions lying in D(R N ). The trace function defined on D(Ω) is denoted by γ. For 1 p ∞ and m ∈ N, the usual Lebesgue and Sobolev spaces are respectively denoted by L p (Ω) and W m,p (Ω), W m,p 0 (Ω) is the closure of D(Ω) under the W m,p -norm,

Fn

  L ∞ (Ωn ) +|Im(a)| |Im(b)| , when Im(b) = 0 and ϕ(n) = n, when Im(b) = 0. Thus for any n ∈ N, as soon as Im(b) = 0, one has

Lemma 4 . 4 .

 44 Let Ω ⊆ R N be an open subset, let ω an open subset of R N be such that ω ⊆ Ω, let m 0 and let (a, b, c) ∈ C 3 be such that Im(b) = 0. If Re(a) 0 then assume further that Im(a) = 0.

Lemma 4 . 5 . 1 +Case 1 .C 1 +Case 2 .C 1 +Case 4 .C 1 +Corollary 4 . 6 .

 45111214146 Let (a, b) ∈ A 2 satisfies (2.7). Then there exists δ ⋆ = δ ⋆ (|a|, |b|) ∈ (0, 1], L = L(|a|, |b|) and M = M (|a|, |b|) satisfying the following property. If δ ∈ [0, δ ⋆ ] and C 0 , C 1 , C 2 , C 3 , C 4 are six nonnegative real numbers satisfyingC 1 + δC 2 + Re(a)C 3 + Re(b)δ C 4 LC 3 + LC 4 M C 0 . (4.22)Proof. We split the proof in 4 cases. Let γ > 0 be small enough to be chosen later. Note that when Im(a)Im(b) 0 then estimate (4.21) can be rewritten as |Im(a)|C 3 + |Im(b)|C 4 C 0 . (4.23) Re(a) 0, Re(b) 0 and Im(a)Im(b) 0. We add (4.23) with (4.20) and obtain, Re(a) + |Im(a)| C 3 + Re(b)δ ⋆ + |Im(b)| C 4 2C 0 . Re(a) 0, Re(b) < 0 and Im(a)Im(b) 0 or Im(a)Im(b) < 0 . Then, C 1 + Re(a)Im(b) -Re(b)Im(a) + γIm(a) Im(b) C 3 + (γδ ⋆ )C 4 |Re(b)| + |Im(b)| + γ |Im(b)| C 0 . where we computed (4.20) -Re(b)-γ Im(b) (4.21). Case 3. Re(a) < 0, Re(b) 0 and Im(a)Im(b) 0. By computing (4.20) -Re(a)-γ Im(a) (4.21), we get, γC 3 + Re(b)Im(a) -Re(a)Im(b) + γIm(b) Im(a) δ ⋆ C 4 |Re(a)| + |Im(a)| + γ |Im(a)| C 0 . Re(a) < 0, Re(b) < 0 and Im(a)Im(b) 0. Note that since (a, b) ∈ A 2 then necessarily Im(a)Im(b) = 0. Thus, we can compute (4.20) + max |Re(a)|+γ |Im(a)| , |Re(b)|+γ |Im(b)| (4.23) and obtain, γC 3 + (γδ ⋆ )C 4 |Re(a)| + |Im(a)| + γ |Im(a)| + |Re(b)| + |Im(b)| + γ |Im(b)| C 0 . In both cases, we may choose γ > 0 small enough to have          Re(a)Im(b) -Re(b)Im(a) + γIm(a) Im(b) > 0, in Case 2, Re(b)Im(a) -Re(a)Im(b) + γIm(b) Im(a) > 0, in Case 3. Then we choose 0 < δ ⋆ < min 1, γ, |Im(b)| + |Re(b)| such that δ ⋆ < Re(b)Im(a) -Re(a)Im(b) + γIm(b) Im(a) , in Case 3. This ends the proof. Let Ω ⊆ R N be an open subset, let V ∈ L ∞ (Ω; R), let 0 < m < 1 and let (a, b, c) ∈ C 3 be such that Im(a) 0, Im(b) < 0 and Im(c) 0. If Re(a) 0 then assume further that Im(a) < 0.

  g, a.e. in Ω, By these two last estimates, f ϕ(n) u n ϕ(n) a.e. in Ω -----→ n→∞ f (u) and f ϕ(n) u n ϕ(n)

  1 (Ω), in the homogeneous Neumann case. Let δ ⋆ be given by Lemma 4.5 and let for anyu ∈ L 2 (Ω), f (u) = a|u| -(1-m) u + bu + cV 2 u (with c = 0 in the case of Theorem 2.8). Let (F n ) n∈N ⊂ D(Ω) be such that F n L 2 (Ω) ----→ n→∞ F. Let u n ℓ (n,ℓ)∈N 2 ⊂ H asequence of solutions of (5.1) be given by Lemma 5.1 with O = Ω, δ = 1 for Theorem 2.4, δ = δ ⋆ for Theorem 2.8 and such F n . By Corollary 4.6, there exists a diagonal extraction u n ϕ(n) n∈N of u n

  dx, and to include in the definition of solution the conditions u ∈ H 1 0 (Ω; δ α ) and f (u) ∈ L 2 (Ω; δ α ).(6.4)

Theorem 7 . 1 . 1 C 2 PFurthermore, u H 1 0

 71121 Let Ω an open subset of R N be such that |Ω| < ∞ and assume 0 < m < 1, (a, b) ∈ C 2 and F ∈ L 2 (Ω). Assume that Re(b) >or Im(b) = 0, where C P is the Poincaré's constant in (4.1).Then there exists at least a solution u ∈ H 1 0 (Ω) to-∆u + a|u| -(1-m) u + bu = F, in L 2 (Ω). (7.1) (Ω) C( F L 2 (Ω) , |Ω|, |a|, |b|, N, m). Finally, if Re(a) 0 and -→ a . -→ b 0 then the solution is unique.

  Theorem 7.1 is about the range of (a, b) : we obtain existence of solution with, for instance, (a, b) ∈ R -× (-ε, 0), with ε > 0 small enough, or (a, b) = (-1 + i, -1i). Recall that, up to today, existence was an open question when (a, b)∈ R -× R -or [a, b] ∩ R -× i{0} = ∅ (Bégoutand Díaz[START_REF] Bégout | Localizing estimates of the support of solutions of some nonlinear Schrödinger equations -The stationary case[END_REF]

Theorem 7 . 2 .

 72 Let Ω ⊆ R N be a bounded open subset, let 0 < m < 1 and let (a, b, c) ∈ C 3 be such that Im(a) < 0, Im(b) < 0 and Im(c) 0. For any F ∈ L 2 (Ω), there exists at least a solution u ∈ H 1 (Ω)to -∆u + a|u| -(1-m) u + bu + c|x| 2 u = F, in L 2 (Ω),(7.2)with boundary condition (1.3) or (1.4) 1 . Furthermore,u H 1 (Ω) C(|a|, |b|, |c|)(R 2 + 1) F L 2 (Ω) ,where B(0, R) ⊃ Ω. Finally, if Re(a) 0, -→ a .-→ b 0 and -→ a . -→ c 0 then the solution is unique.

Theorem 7 . 3 .

 73 Let Ω ⊆ R N be an open subset of R N , let 0 < m < 1 and let (a, b) ∈ A 2 satisfies (2.7).

Finally, if

 if Re(a) 0 and -→ a . -→ b 0 then the solution is unique. When |Ω| < ∞, Theorem 7.3 is an improvement of Theorem 4.1 of Bégout and Díaz [1], since we may choose F ∈ L 2 (Ω), instead of F ∈ L m+1 m (Ω) and that L m+1 m (Ω) L 2 (Ω). In addition, this existence result extends to the homogeneous Neumann boundary condition. In this context, we may show three kinds of new results, under assumptions of Theorem 7.3.

  it follow from the above Gagliardo-Nirenberg's inequality that u n (Ω). Now, we show that u ∈ H is a solution. Let m 0 ∈ N be largeenough to have H m0 (Ω) ֒→ L p ′ (Ω). Let v ∈ D(Ω), if H = H 1 0 (Ω) and let v ∈ H m0 (Ω), if H = H 1 (Ω).

	bounded in H 1 0 (Ω), so that u ∈ H 1 0 By (5.1), we have for any n ∈ N, ∇u n ϕ(n) , ∇v L 2 (Ω),L 2 (Ω)	ϕ(n) n∈N	is

More directly, we could have said that since u ∈ W 2,N+1 (Ω), γ(∇u.ν) = 0 and ∆u ∈ C 0,α (Ω) (by the estimate of the nonlinearity) then by Theorem 6.3.2.1, p.287, in Grisvard[START_REF] Grisvard | Elliptic problems in nonsmooth domains[END_REF], u ∈ C 2,α (Ω). But this theorem requires Ω to have a C 2,1 boundary.

Acknowledgements

The first author is very grateful to Professor Thierry Cazenave for helpful comments about the presentation of this paper and both authors thank the anonymous referee for stimulating remarks.

Díaz was partially supported by the project ref. MTM2011-26119 of the DGISPI (Spain) and the Research Group MOMAT (Ref. 910480) supported by UCM. He has received also support from the ITN FIRST of the Seventh Framework Program of the European Community's (grant agreement number 238702) 2010 Mathematics

Remark 6.2. Notice that H 1 0 (Ω; δ α ) ֒→ L 2 (Ω) (by the Hardy's inequality (6.5) and (6.6)). Moreover, since δ -sα ∈ L 1 (Ω), for any s ∈ (0, 1), (6.7) we know (Drábek, Kufner and Nicolosi [START_REF] Drábek | Nonlinear elliptic equations. Singular and Degenerate case[END_REF], p.30) that

. This fact is crucial when the nonlinear term f (u) involves a singular term of the form as in (1.2) but with m ∈ (-1, 0) (see Díaz, Hernández and Rakotoson [START_REF] Díaz | On very weak positive solutions to some semilinear elliptic problems with simultaneous singular nonlinear and spatial dependence terms[END_REF] for the real case). Remark 6.4. We point out that in most of the papers dealing with weighted solutions of semilinear equations, the notion of solution is not justified in this way but merely by replacing the Laplace operator by a bilinear form which becomes coercive on the space H 1 0 (Ω; δ α ). The second integral term in (6.2) is not mentioned since, formally, the multiplication of the equation is merely by v ∈ H 1 0 (Ω; δ α ) but then it is quite complicated to justify that such alternative solutions satisfy the pde equation (1.2) when they are assumed, additionally, that ∆u ∈ L 2 loc (Ω). We also mention now (although it is a completely different approach) the notion of L 1 (Ω; δ)-very weak solution developed recently for many scalars semilinear equations: see, e.g., Brezis, Cazenave, Martel and Ramiandrisoa [START_REF] Brezis | Blow up for u t -∆u = g(u) revisited[END_REF], Díaz and Rakotoson [START_REF] Díaz | On very weak solutions of semi-linear elliptic equations in the framework of weighted spaces with respect to the distance to the boundary[END_REF] and the references therein).

By using exactly the same a priori estimates, but now adapted to the space H 1 0 (Ω; δ α ), we get the following result. (a,b,c) ∈ C 3 as in Theorem 2.4 and let F ∈ L 2 (Ω; δ α ). Then we have the following result.

1) There exists at least a solution u ∈ H 1 0 (Ω; δ α ) to (1.2). Furthermore, any such solution belongs to H 2 loc (Ω).

2) If, in addition, we assume the conditions of Theorem 2.10, this solution is unique in the class of H 1 0 (Ω; δ α )-solutions.

Remark 6.6. In the proof of the a priori estimates, it is useful to replace the weighted function δ by a more smooth function having the same behavior near Γ. This is the case, for instance of the first


J L Bretelle

J Bourgoin

L Viricel

J Thomazet

Y Musante

C Brun

Ftance S Fnevnrol'ae

F Anthoni

Dacquait

Extended fuel cycle : results of the first two cyclesof ( DUO experimentation >>

Introduction

French nuclear plants primary coolant conditioning is essentially based on the boron-lithium coordinated chemistry. Today, the pH target value is 7.2 at 300oC for all the 58 French units, and the maximum authorised lithium concentration is2.2mgfr;g [l]. To decrease dose rates, primary coolant pH in EDF nuclear plants was changed from 7 .0 to 7 .2 l2l.

Since EDF 1300 MWe units began operating 18 month-extended frrel cycles, increasing fi¡el uranium oxide en¡ichment from 3.2 to 4yo had been necessary. This evolution led to : a significant increase in the boron concentration at the beginning of fuel cycles (progressively for the first two conversion cycles) ; a decrease ofthe pH at the beginning ofthe cycles, since today the maximum lithium concentration in normal operation is 2.2 mgllg. This issue does not concern the twenty 1300 MWe units operating extended cycle only. All the other French nuclear reactors can be impacted if higher boron concentrations than those used until now are imposed by ner¡/ core desrgn (lr4ixed OXide, ...).

As shown by experience feedback, operating at PH¡oo around 6.8 may make it ditricult to keep corrosion products as low as possible. Hence it may lead to : deposits in RCS, and particularly on the fi¡el cladding which may in turn lead to axial ofrset anomalies t3l-t4l , inc¡ease in dose rates.

One solution is to adjust the pH by increasing the lithium concentration at the beginning ofthe cycles, and hence to operate a modified chemistry reg¡m€. This solution is e¿rsy to implement, as soon as decidãd and without any modification on the units. A new maximum lithium concentration, consistent with materials behaviour and especiatly with fuel cladding has to be fixed.

Duo experimentation features

This experimentation deals with the specific surveillance, from the first to the third conversion cycle, of two 1300 MWe units starting implementing extended cycle operation. The reference unit and the pilot unit for this experimentation were chosen to be Golfech I and Cattenom 2 respectively. They had to have simila¡ desigt (steam generator tubes bundle in 600 TT alloy, fircl grids assemblies, no hot spots, no flux anomalies, ...).

Golfech 1 implements a classic boron-lithium coordination ÞHsoo7.Z (ñgure 1). Cattenom2 operates EDF modified chemistry at the beginning of the cycles, with maximum authorised lithium concentration of 3.5 mgkg(figure 2).

The mqjor goal of the experimentation is to assess the impact of elevated lithium concentrations at the beginning of cycles on ñlel cladding oxide behaviour, mass transport and dose rates. During each @nversion cycle, the primary coolant surveillance deals with its chemical and radiochemical features (table 1).

During shutdown transients and outages

For the th¡ee conversion glcles, the surveillance concerns primary coolant activity during shutdown, surface activity and ñ¡el oxide thickness measurements.

Scrapping of crud deposits (if any) on the firel cladding are planned at the end of the third conversion cycle (table 1).

These controls aim at assessing the performance of an extended cycles modified chemistry when elevated initial boron concentrations are used compared to a classic chemistry, which may lead to operate, at the bginning of cycles, at risþ pH for ñrel crud deposits and dose rates.

Table I : Surveyedparameters Duo experimentation partial results

Chemistry

During the first two conversion cycles on Golfech I and Cattenom 2, despite load following the boron- lithium coordinated chemistry regime matched the speciñcations (Fig. 3 and4). Data out of the normal operating areaaÍe essentially due to shutdowns, physical WATER CHEMISTRY OF NUCLEAR REACTOR SYSTEMS B testing, rapid load variations and makeups for tritium dilution in RCS. No specific pollution was observed during these conversion cycles in Gottech I and in Cattenom 2.

The initial boron concentrations at hot zero power and ñrll power are reported intable 2.

In the Cattenom 2 pilot unit, operation with a lithium concentration between 3.5 and 2.2 mgkg increased from about one week to two weeks from the first to the second conversion cycle; operation with a lithium content between 2.5 and2.Zmgllcg increased from about two to seven weeks.

In the Golfech 1 reference unit, pH was around 6.8 at 300oC at the beginning of the second conversion cycle (similar to fi¡st conversion cycle) and 2 to 3 days were needed to reach 6.9. It took overall about 24 weelc to reach a pH of '1.2 at 300"C. In Cattenom 2, pH at the beginning of the cycles was always above 6.9 at 300oC, and25 weeks were needed during the second conversion cycle to reach a pH of 7 .2 at 300"C. However, just before criticality, pH is better controlled in Cattenom than in Golfech.

(a il) : reftding $rtdwr 12) : qitiølity adp@qiñqae i3) : load redudio to O o/c awq Water and boron makeups may generate oxygen ingress in RCS which may impact the coolant-materials interactions. Table 3 summarises the makeups volumes in RCS during the first two conversion cycles in Golfech 1 and Cattenom 2. The greater makeup volumes in Golfech 1 compared to Cattenom 2 canbe explained by more load following in Golfech. Similarly, the smaller volumes used during the second conversion cycle on both units compared to the first conversion cycle is explained by less load variations. 5 MBq/t for @Co in both units ; 30 MBq/t for 51Cr in Golfech and 2}MBq/t in Cattenom. Some higher activities could sometimes be observed most of them related to significant makeups or load reductions to 0 %o po$aer.

' (z) <2 (2\ (, -(1.) h,-\ (l)' {t.i' t3).
The activities measured during shutdowns are presented in table 5. In table 4 are reported the results of corrosion products measurements performed in Golfech I and Cattenom 2 during power increase at the beginning of the conversion cycles or during power decrease before refireling shutdown. These results ag¡ee with the values typically observed in French plants. For both units, a significant increase ofsoluble nickel is observed between the beginning of the first and second conversion rycle. In Golfech 1, higher ch¡omium values can also be noted. These data should be discussed û¡rther afrer the third conversion cycle is completed in both units.

Radiochemistry

In Gotfech 1, the cobalt 58 peak activity was lower for the second conversion cycle than for the previous two cycles, whe¡eas the total released activity slightly increased from the ñrst to the second conversion rycle. In Cattenom 2, activities remained overall stable between the first and the second conversion cycle. So today, no significant effect of the conversion cycles is óserved.

Deposiß

Cobalt 58 and cobalt 60 surface activity were measured on stearn generators (SG) tt¡be bundle, hot legs (HL) and crossover legs (CL): results are reported in tables 6 and 7 for Gotlech 1 and Cattenom 2 reqpectively. Moreover, dose rates were measured around steam generators and legs (table 8). Compared with tle previous two cycles, the results ftom the second conversion cycles show that : 58Co aøivity seems stable, or slightly decreasing in both units, except on Cattenom 2 steam generators where an increase has been observed for the last cycle (figure 5). However, these va¡iations are t¡¡pical for 58Co; they are usually observed on plants and also in Cattenom 2 before the first conversion cycle ; 60Co activity has increased in Golfech 1 since the beginning of conversion. The origin of this increase is probably an abnormal input of cobalt (i.e. stellites), because this nuclide is the only concerned. An analysis of the behaviour of other 1300 MWe units will probably confirm this hlpothesis; 60Co activity seems stabilised, or slightly decreases in Cattenom 2, after an increase at the end of the first conversion cycle : specrogammamelry was useo to measure cooalr )Õ, cobalt 60 and ch¡omium 51 in RCS during the first two conversion cycles in Golfech 1 and Cattenom 2. Activities dose rates seem stable in both units. Some values presented in table 8 can be explained by fuel rod clad failures prior to conversion and dose rates impacted by fission products.

Except on Cattenom 2 steam generators, cobalt 58 surface activities tend to decrease.

Overall, at the end of the second conversion cycle, deposited activities in Golfech 1 and Cattenom 2 do not show any peculiar trend compared to usual French results. These trends should be confirmed after the third conversion cycle is completed in both units. Four characterised fuel assemblies were loaded in both the Gotfech I and the Cattenom 2 reactors. The fr¡el rods claddings are made of. Zfucaloy-4, alloy M4 or alloy M5m. Each alloy comes from the same tube batch.

The chemical composition of the alloys is reported in table 9.

The firel rods were loaded into eight assemblies so as to distribute uniformly all the rods cladding batches into each assembly. Their distribution in the assemblies is shown in figure 6. Concerning the advanced alloys, the location ofthe corresponding rods were chosen in order to be able to measure zirconia layer thickness along four al(es 90o apart. However, some of them were also loaded in peripheral location in order to run in-depth visual examinations.

Table 9 : During each cycle outâge, the device is installed on a criss-cross measurement t¿ble, allowing blade insertion into each row of rods at a chosen measurement altitude.

All analysed rod, particularly those with claddings made of advanced alloys, are therefore measured during blade insertion and extraction from all four assembly faces leading to more than 2000 measurements collected per campaign. Statistical process using the < Lagos > and <<Zircone ) sofü¡Ëre prog¡ams is used for this analysis.

The measurement uncertainty is estimated to be + 5 pm.

During the first two conversion cycles, neither in core In order to assess the impact of the reactor coolant chemistry on the fi¡el cladding corrosion, oxide thickness measurements and crud analyses were planned on characterised firel assemblies loaded in the Golfech 1 and . PwRraooltw" axial offset anomalies nor abnormal cruds on assemblies had to be reported. Hence, only oxide thickness measurements were carried out on firel rods.

Concerning Zircaloy-4, the maximum oúde thicknesses measured at Golfech I and Cattenom 2 atthe end of each cycle are plotted on figure 7. For both reactors and given the burnups reached, the results are within the scattering generally observed on this alloy in French (900 and 1300 Mrù/e) and foreign rcactors implementing a 2.2 mgllgmaximum lithium chemistry. Similarly, the results obtained on the advanced alloys on both reactors, are within the scattering observed on these alloys in French (900 and 1300 MWe) and foreign reactors implementing elevated lithium chemistry (figures 8 and 9, alloy M4 and M5d respectively). The high lithium chemistry does not seem to have an obvious impact on the corrosion behaviour of fuel cladding during the first two cycles.

Discussion

The results of the Duo experimentation surveillance program show a good behaviour of the ñrel cladding in the Cattenom 2, the pilot unit implementing EDF modiÉed chemistry vcith maldmum authorised lithium concentrations of 3.5 mÚkg at the beginning of the cycles.

There is no significant difference in both units behaviour (contamination and mass transport), other than the differences that were existing prior to the beginning of the Duo experimentation.

The results from the third conversion cycles will be important to confirm the observedüends.

No a:rial ofrset anomaly has been noticed in Golfech 1

and Cattenom 2, nor in all the twenty EDF 1300 MWe units on-going conversion cycles ; most of these 20 units already began their third conversion cycles, being thus confronted to the worst conditions at the beginning of the cycles.

This reinforces the position on which all the specialists agree, stating that core features are also an important factor which is not taken into account into the Duo experimentation [6].

From these results, it appears possible to improve pH at beginning of cycles, beneficial to a better behaviour of RCS. It can be noticed that the impact of boron concentrations up to 1800 mgiþ at beginning of cycles can be managed with EDF modified chemistry, ensuring PH¡oo > 6.9.

Studying operating veith a constant lithium concentration of 3.5 mg/kg could also be of interest in order to reach optimum PHaoo: 7.2 as soon as possible.

Non destructive steam generator tubing examinations proved that no abnormal corrosion due to elevated lithium concentrations occurred on 600fi alloy, since the beginning of extended cycle operation in Cattenom 2.

These ¡esults will have to be confirmed afrer the completion of the third conversion cycle. cycle operation with a low pH at the beginning of the cycle does not seem to generate any significant crud deposits on flrel rod cladding. Hence, no a<ial offset anòmaþ nor increase ofdose rates due to the presence of crud deposits has been observed. However, a few more cycles with elevated initial boron concentrations are needed to assess the influence of increased cycle lengfh on dose rates.

Oxide thickness measurements results show that operating with EDF modiñed chemistry, hence lithium concentrations up to 3.5 melkg at the beginning of the

  Fig. I : Classic B-Li coordinated chemistry pH3ss 7.2

  Fig. 3 : Golfech L,2ndconversion rycle: B-Li diagram

  Fig. 6 : Characterised assemblies location map The zirconia layer thickness measurements were performed using the blade device developed by FRA[,IAToÀ,ß, based on the principle of measurements by the Eddy Current technique [5].

  Fig. 5 : Cobalt 58 activity on SG tubes bundle in Cattenom 2 and other French 1300 MWe PWRs Fuel assemblies:

  Fig. 7 : Zircaloy 4 external corrosion experience

  Fig. 8 : Alloy M5ru external corrosion experience

  s core desrgn of French plants and the available results from the Duo experimentation, extended

  Fig. 9 : Alloy M4 external conosion experience

  This is what EDF has decided to test through the

  This paper deals with the results of the controls and expertise performed during the first two conversion cycles of the Duo Experimentation.

Table 3 :

 3 Golfech 1 and Cattenom 2 makeups to RCS 387 are overall below: 30 MBq/t for rsCo in Gotfech and 40 MBq/t in

	Cattenom;

Table 5 :

 5 Shutdown activities

		lvfakeip wder	Bømmakapwater
	Ccoversicn cycle	Volume t-'l	Meae. dissolved Oz (pglkg)	Volume t-tl	Meae dissolved Oz (Fg/ke)
			Golfedr 1
	1	1230ø.	37	8í|7	1700
	a	9242	49	774	1777
			Caɡnotn2
	I	9600	45	467	2200
	t	8528	51	444	1523

Table 4 :

 4 Soluble corrosion products in Golfech I and Ddectio ümit = 4 ng/kg Co ; NA =

	Cattenom 2

=

Table 6 :

 6 Surface activity in Golfech 1

		SG(cBqh) HL(GBqln) CL(cBq/br)
		sco eco 58co @co 58co @co
	T ^q cycleb€fore coversim	2.4	0.3	1.8	0.7	4.3	0.6
	lst ccoversicn cycle	2.3	0.5	1.5	0.9	3.7	0.8
	2nd cmversicn cycle	2.4	0.6	7.2	2.0	3.5	1.0

Table 7 :

 7 Surface activity in Cattenom 2

	Table8:Doserates

While any beneficial impact of a constant pH of 7 .2 at 300'C during throughout the cycle on dose rates is not proved, the easy implementation of increased lithium concentration at the beginning of the cycles seems to be enough to face issues due to French l8-months extended cycles management. However, the examination of the French 1300 MIVe beha¡iour after complete conversion to extended cycles is essential to conclude on the Duo experimentation.

Achieving very high burnups (60 GrWd/tLD or issues with waste management could lead EDF to change its stratery and examine again the possible use of enriched boron 10. According to recent studies ca¡ried out in the Robust Fuel Program, the impac't of boron on flux anomalies will also have to be taken into account when defining EDF strates¡ on using enriched boron 10.

Today, given the pH improvement obtained by operating EDF modified chemistry with increased lithium concentrations up to 3.5rnglkg at the beginning of frrel cycles, generalising this chemistry regime to all the French plants will be considered.


Gamma energy KeV (% emission)

Beta energy keV (% emission ) Co58 / 71 d 811 (100%) Co60 / 5,3 y

1 Study of various chemical species behaviour for contamination risk J.L. Bretelle, Electricité de France/GDL A. Rocher, Electricité de France/GPR M. Berger, Electricité de France/SEPTEN F. Dacquait, Commissariat a l'Energie Atomique -France R. Rosset, ESPCI -France

1-Introduction

The goal of this paper is to precise the behaviour of different radiochemical species in the primary coolant of PWR plants. Managing these pollutants must lead :

-To limit workers contamination risk to avoid incidental dissemination in the containment atmosphere when the primary circuit is opened ; -To limit Reactor Coolant Sysytem (RCS) walls "over-contamination" to decrease the dose rates during the maintenance operations. In French Plants, Co 60, iodine, silver and antimony represent the major radiochemical pollutants which require a good understanding of the different phenomena to ensure the lowest contamination risks. The stakes deal with the control and the optimization of collective and individual doses including waste treatment with low costs. These stakes represent primordial elements of nuclear acceptability.

2-Normal contamination of circuits

Contamination is one of the dose causes only. Nevertheless smearable contamination can impact radiological cleanliness of facilities. In French PWR plants :

-More than 90% of integrated doses are due to contaminated walls in contact with the primary coolant because of activated corrosion products. -80% of total dosimetry are integrated during outages.

In the absence of specific pollution, the mean contribution of out of core deposits to dose rates is as follows (results expressed as percentage equivalent dose rates in Sv/h) :

-Co 60 = 50% (the older the plant, the greater the Co 60 quantity).

-Co 58 = 30% (with a higher level in 1300 MW units) -Ag 110m + Sb 124 + Cr 51 + ... = 10% Nota : Considering that Co 60 is the major contaminant in French facilities, sensitive and accurate Co 60 measurement and control devices are needed at the entrance and way out of controlled areas as well as individual electronic dosimeters.

Usual contaminants come mostly from :

-Either materials/coolant interactions leading to soluble, particulate and colloidal products transport and corrosion products activation (Co 58, Cr 51, Mn 54 ….) [START_REF] Dacquait | Corrosion Products transport in French PWR'S during cold Shutdown[END_REF] ; -Or in core pollutions leading to fission products with a slight impact on contamination and dose rates [table 1].

Table 1 -Gamma and beta energies and emission percentage (> 50%) of major nuclides

The dose rates can increase in case of more important corrosion products transport or of incidental pollution. In the latter case the ratio of species contributing to dose rates is modified [START_REF] Nordmann | Overwiew of PWR chemistry options[END_REF].

3-Incidental contamination

The major pollutants responsible for over-contamination in EDF plants are :

-Co 60 coming from stellite degradation ; -Ag 110 m coming from control rod perforation and probably from materials like seals ; -Sb 124 coming from pumps of auxiliary system ; -Iodine due to fuel cladding degradation.

Given the importance of over-contamination with respect to dosimetry, representing 10 to 30 % of outages dosimetry in the concerned units, a specific program is being set up in France, in the context of the ALARA project. The goal is to optimise diagnosis, surveillance, prevention and remedies to fight pollutions even though priority is set on prevention.

-Co 60

In most cases, hot spots are due to particles of cobalt activated by a neutron flux (Co 60) mainly from stellite surface equipment (rich in cobalt) [START_REF] Nordmann | Overwiew of PWR chemistry options[END_REF] in the RCS (valves, pumps, internals, etc.) [START_REF] Rocher | Program to eradicate hot spots (Co 60) carried out in French PWR Units[END_REF].

The contribution of hot spots to shutdown dosimetry may appear to be marginal in French reactors (2 to 4 %) but becomes heavier (15 to 25 %) for the units affected. This excess dosimetry has to be taken into account, particularly for the most exposed workers. Approximately ten French units have been affected by this phenomenon over the last 15 years. A set of actions, aimed at eradicating hot spots in PWRs, was set up in 1997. It consists of three complementary phases : prevention, surveillance and curative actions.

-Prevention

Prevention consists in avoiding the occurrence of hot spots. It concerns all the units by performing the following actions:

limiting the use of cobalt based components in contact with the primary coolant ; removing stellite particles produced during some maintenance operations (e.g. lapping) ; filtering effluents able to carry hot spots outside the Reactor Building. The simplified diagram below illustrates the principles of the preventive filtration methods which are proposed (Fig. 1).

R C S /

P R E V E N T IV E A C T IO N N ° 1 : F IL T R A T IO N O F P O O L B O T T O M S R E A C T O R C O N T A I N M E N T F U E L P O O L C O O L IN G A N D P U R IF IC A T IO N S Y S T E M

Fig. 1 : Preventive filtration methods

The drains of pools are important routes for hot spots, before they spread through the systems. The installation of fine filters, an initial containment barrier, is proposed for the drain orifices of each pool :

-Pools located in the reactor building ; -Pools located in the fuel building ; Appropriate filters are required :

-Truncated sieves ; -50 µm mesh.

The drain lines of the primary cooling circuit represent the second main vector for the movement of hot spots. As a preventive measure, the installation of sleeves on the main migration channels enables sensitive systems to be filtered. This filtration prevents the spread of hot spots and ensures that they are eliminated.

The filtration system is installed in the reactor building or the fuel building, depending on the shutdown phases, thus enabling all the pools to be treated with the same device.

-Surveillance

Surveillance is designed to inform the site, as early as possible, of the presence of hot spots (mapping) in order to take the appropriate measures to prevent their propagation and/or to eradicate them. During unit operation, most hot spots will become fixed to the fuel. Others may fall, by gravity, to the bottom of the pool or be trapped in the special devices or the low points of the primary coolant system. The most common locations are as follows:

-Thermal sleeves of the pressuriser -Steam generator packing glands -Valves of the primary cooling system.

etc.

After connection of the Residual Heat Removal System (RHRS), some hot spots may migrate into this circuit and be deposited or fixed. The most common locations are the pumps, heat exchangers and valves of the circuit.

An underwater pool cleaner should pass through the pool out after discharging. In this case, particularly high equivalent dose rates, equal to or greater than 1 Sv per hour, measured in contact with the filters, represent the last indicator of the possible presence of hot spots, before the draining of the pools.

Since no warning signs have been identified yet, to indicate the occurrence of hot spots, it was decided to concentrate on preventive filtering, trapping hot spots as close as possible to their source to eliminate them.

-Metastable silver 110 contamination

In some PWR units, the primary and auxiliary systems are subject to metastable silver 110 (Ag 110 m) contamination, due to silver 109 neutronic activation :

109 Ag + 1 n à 110m Ag + γ
The most likely root causes for silver leading to the observed Ag 110m pollutions may be as follows :

the neutron-absorbing alloy Ag-In-Cd contained in control rods ; some silver coated seals. For French plants, the major source for Ag110m is natural silver. Moreover, for the French plants polluted with Ag 110m, the quantity of metallic silver released in the primary system could be about 1-10 grams.

3.2.1-Impact on dose rates

Ag 110 m systems contamination can be very penalising since it impacts :

dosimetry integrated during maintenance operations ; shutdown schedule ; control rods management (examination and rejection). It can represent 5 to 15 % of a shutdown total dosimetry. Systems most sensitive to silver contamination are :

the Residual Heat Removal System (RHRS) and Chemical and Volume Control System (CVCS) coolers ; the purification system downstream the CVCS demineralisers ; the reactor cavity. Ag 110m can contribute to more than 90 % of the dose rates around some parts of auxiliary systems.

-Surveillance

The volumic activity of Ag 110m in the primary coolant stays at a low level (1-10MBq/t) due to the low silver solubility in reducing medium. Activity increases higher than 10 MBq/t indicate a poor silver removal in the CVCS or an important silver source. During the shutdown, in case of silver pollution, the observed levels of activity increase dramatically (several decade) specially after the oxygenation. This can be explained by :

-Changing from reducing to oxidising medium ; -Simultaneously by temperature and pH changes. At the opposite from other radioelements, for which dissolution is maximum at the "oxygenation peak", silver dissolution shows a trend to go on after the effective oxygenation (dissolved oxygen close to 1 mg/kg). Generally, the silver peak activity appears 1 to 12 hours later after the Co 58 peak. The table 2 gives some data of silver peak for the French plants in the year 2000 during shutdown. There is no correlation between the Ag 110 activity peak and the other corrosion products. A first level of assessment can lead to consider that RCS is polluted when Ag 110m peak during shutdown is higher than 0,5 GBq/t.

Shutdown

-Silver behaviour

The observation of different and unpredictable behaviour of silver, mostly measured by gamma-spectroscopy and particularly during different cold shutdowns, even in a same plant, can be surprising when the operation schedule seems stable. Measurements demonstrated that it is possible to decrease a lot the steam generator channel head silver contamination and to increase the CVCS exchanger. Depending on chemical environment and physical properties (pH, redox potential ...) silver can be :

-In ionic form in solution, Ag cation (I) specially ; -In metallic form, Ag (0) ;

- In colloidal form, 200-600 Ă.
Studies, conducted by EDF, demonstrated the essential impact of redox potential, pH, and temperature on silver thermodynamic stability. In a large range of pH (corresponding to the nominal operating conditions or to the shutdown ones at 300°C, 80°C and 30°C) and in different locations of RCS, CVCS and RHR, Ag 0 and Ag + can be found simultaneously and are very sensitive to the redox variation and the concentration. Actually, it seems that the physical and chemical conditions of the primary coolant set silver mostly on the limits of different existence areas considering the corresponding pH-potential diagrams.

On the opposite, pH is determinant on colloids behaviour. Repulsion between them is the lowest at iso-electric point, meaning at pH for which the zeta potential is zero. Thus, optimization of its removal is difficult, the preferential deposition of Ag 110m taking place in "cold" points of auxiliary exchangers where thermal gradient is important. This also elucidates the apparent unoperating conditions of the purification. The goal will become to manage the silver so as to transport it in the form able to be removed by the purification system.

-Preventive actions .

Preventive actions to control silver root source are :

-Control rods examination ; -Primary Ag 110 m volumic activity monitoring during operation. -purification during end of cycle shutdown ; -elution of activity confined in dead legs ; -other complementary actions (cleaning system).

Control rods examination

Purification during end of cycle shutdown

Ag 110 m purification is performed on a dedicated ion exchanger operating at the maximum possible flowrate.

During shutdown, when the filters are replaced, purification must not be suspended. Filters consistent with colloids removal are needed (lower mesh with polarised membrane). Thus, silver removal is improved and downstream resins pollution avoided (poisoning by colloids). The filter replacement must lead to reduce by half the CVCS flowrate purification to minimize resins pollutions. The background and the resin features show that macroporous mixed-bed will be the most adapted for soluble silver and silver colloids. The pressure drop must be specially observed.

Until now, the feedback demonstrated that withdrawing lithium of the primary coolant with a non lithiated resin during the shutdown, giving the lower pH, improves the silver removal efficiency. Without cladding failure, the lithium can be eliminated starting from the rods drop (see &5)

Crossover legs drainage

At Tricastin, forecasting Ag 110 m activity level allows to get ready, before shutdown, for a specific operation to decrease Ag 110 m volumic activity confined in the main primary system crossover legs, prior to reactor cavity filling and refueling. Usually, crossover legs drainage is performed after core unloading. In the presence of Ag 110 m, scheduling a crossover legs rinsing prior to system drainage allows to elute the activity confined in them, and hence to lower the risks for contamination of the water during cavity filling. Ag 110 m final volumic activity of the water in the reactor cavity can be reduced by a factor of 3, leading to limited dose integrated during fuel handling operations.

Complementary actions

Other remedial actions can be undertaken depending on the types of operations to be performed during the end of cycle outage (cleaning of some systems parts, such as RHRS and CVCS coolers).

Coolers contamination occurs mostly during outages. Contamination of CVCS pumps and valves dowstream the purification system is mostly due to the lithiated CVCS demineraliser unefficiency in purifying Ag 110 m during operation.

-Antimony contamination

In Sb peak of 100 GBq/t.

-Antimony behaviour

EDF studies, elaborating pH-Potential curves at normal operating conditions (300°C) and at forced oxygenation at 80°C confirmed the behaviour observed on plants. It seems that, in nominal conditions at 300°C, metallic antimony is likely the more stable in aqueous reducing medium (fig 3). For shutdown conditions at forced oxygenation, it seems that the more stable species is at the limit of the SBO 3 area in aqueous solution. Eliminating antimony as well as possible becomes easier in this zone (fig 4).

-Impact on dose rates

The important antimony released activity has an impact on dosimetry and waste management. 124 Sb deposited activities on the out-of-core surfaces increase a lot as it is shown in table 4 Sb peak of 70 GBq/t Thus, the124 Sb contribution to the dose rates can reach 10 % in some parts of circuits. And the dosimetry for maintenance during a plant shutdown can increase to about 5 % because of the 124 Sb contamination.

-Remedial actions

To avoid antimony pollution, the incriminated pump bearings have been replaced by pump bearings without antimony. As it is indicated in the During oxygenation, antimony seems soluble (< 0.45 µm). When an antimony pollution occurs, to improve purification by the ion exchange resins, reactor coolant lithium content has to be lower than 1 ppm [START_REF] Gaudard | Improvements of Primary Coolant Shutdown Chemistry and Reactor coolant cleanup system[END_REF]. In order to obtain an acidic medium, purification can be carried out thanks to non-saturated lithium cation bed demineralizers of the Boron Recycle System. Nevertheless, it should be noted that an acidic reactor coolant seems to increase antimony deposition on the primary circuit surfaces [START_REF] Gaudard | Improvements of Primary Coolant Shutdown Chemistry and Reactor coolant cleanup system[END_REF].

As for silver behaviour, to manage an antimony pollution the operators face some difficulties linked to the different physical and chemical conditions of the RCS and the auxiliary systems. Except for removing the root cause, the objective consists in the optimization of the form of the antimony to enable the purification line to remove the pollution.

-Iodine contamination

Iodine released in the primary coolant, when a fuel cladding failure has occurred during load reduction, leads to volumic activity increases of I 131 (several GBq/t :5 GBq/t in Gravelines 1 cycle 12). After the forced oxygenation, the iodine concentration shows decreases which could be explained by the efficiency of the purification system. Actually, on 15 units examined, a decrease of about 70% of I 131 measured by gamma spectroscopy can be observed with no real correlation with the resin capacities. Then absorption becomes an assumption to examine. If studies have been conducted in radioactive accident scenario, little concerned the normal operation.That is why EDF tried to define iodine behaviour [START_REF] Gaudard | Improvements of Primary Coolant Shutdown Chemistry and Reactor coolant cleanup system[END_REF] [5].

-Iodine behaviour

Considering the physical and chemical parameters of the primary coolant at nominal power and during transients, iodine can be found principally in iodide, iodate or molecular form following the equilibrium :

dismutation 3 I 2 + 3 H 2 O 5 I -+ IO 3 -+ 6 H + amphoterization
However, the pH then ranging from 7 to 7.2 at 300°C and the redox conditions of the hydrogenated (thus reducing) RCS, strongly favor iodide formation. At low concentration, which is the RCS situation, these ions are soluble and are removed by the anionic resin of the lithiated mixed bed even at the CVCS temperature. Dismutation reaction is almost total but kinetics is low and increases with temperature. Therefore, the risk of absorption is very weak during the operation.

On the opposite, amphoterization reaction rate is high. It occurs at 80°C when RCS is oxygenated. This was verified on a lot of units. This was studied in French laboratories where the iodine absorption was examined on alloy 600, Zircaloy and stainless steel with the chemical primary coolant conditions. A large absorption was demonstrated. It depends on temperature and pH. pH interferes with dismutation equilibrium : the lower the pH, the higher the molecular iodine ratio free for absorption. The figure 5 shows the variation of the iodine absorption versus time and pH on alloy 600. Similar tests were conducted versus temperature and the experimentation showed that the lower the temperature, the higher the absorption. Whatever the materials studied, the trend stays exactly the same. Meanwhile laboratories results demonstrated that in given conditions, materials showed different behaviours concerning absorption leading to the following classification :

"Stainless steel < Zircaloy < Alloy 600"

Tests on desorption demonstrate that desorbed iodine increases with temperature and dynamic venting. The desorption is maximum on alloy 600 compared to zircaloy and stainless steel.

Considering that the steam generator tubes represent the major surface, the zone where iodine molecular absorption is greater forms also the area where iodine can potentially desorb the most.

-Impact on dose rates

Contamination measurements, performed by CEA have shown that the percentage of deposited activity due to iodine is weak and the dose rates are not a key problem for the workers. Taking into account the potential desorption of iodine when the RCS is opened, the major issue is to ensure personnel protection against contamination.

-Remedial actions

Laboratory tests conducted with fresh mixed bed resins, saturated with lithium and equilibrated with boron and close to CVCS conditions, showed that iodine removal performance was not impacted by lithium concentration in reducing medium. Taking into account this result and the behaviour of iodine, recommendations can be made to improve the operating conditions to limit its effects. The goal is to ensure that the maximum iodine can be eliminated before the forced oxygenation of the primary coolant. From the beginning of temperature decrease, it is better to purify with the maximum flowrate, avoiding introducing oxygenated water in the primary coolant (during the coolant contraction), keeping the maximum concentration of lithium. Thus, a maximum quantity of soluble iodine can be transported upstream the resins to be removed and the potential quantity of iodine able to absorb on the materials walls is reduced. Then, the operating strategy becomes :

-Optimize chemistry before the primary coolant oxygenation ; -Keep the coolant hydrogenated the longer ; -Use a mixed bed resin to favour the purification (the lithium removal leads to a pH decrease increasing the molecular iodine to be potentially absorbed, therefore not removable by ionic exchanger).

Some opposite requirements can be noticed when iodine and silver are simultaneously observed in the primary coolant. In this configuration two ways of managing both pollutions can be proposed.

The first is to focus on iodine removal before primary coolant oxygenation then to manage silver from the beginning of forced oxygenation.

The second is to keep the coolant hydrogenated from the temperature decrease and to remove lithium favourising an acidic medium. In this configuration the major part of iodine can be removed considering the local alkaline conditions of anionic resins able to shift the iodine equilibrium towards dismutation. This second one is not the more efficient to eliminate iodine in the RCS and auxiliaries. However EDF considers that as soon as primary water equivalent I 131 activity exceeds 50 MBq/t before RCS opening, the personnel protection against contamination have to be settle. The preventive strategy consists in implementing special equipments implying dynamic venting for risky works. Pressure reduction devices and iodine traps are connected to the containment building venting system.

-Conclusion

EDF plants faces radioactive pollutions (i.e. Co 60, Ag 110m, antimony and iodine) leading to significant increases of shutdown dosimetry in affected units (10 -30%) or increasing the contamination risk for personnel.

Early detection in operation as well as during shutdown requires a good knowledge of their behaviour to set up appropriate solutions. Replacing materials as soon as it will be possible should limit the source term. Limiting pollution in circuits need to implement chemistry and shutdown procedure to minimize over-contamination. To manage iodine behaviour allows to eliminate it and to perform devices to protect personnel before opening the RCS. It is obvious that implementing specific chemistry and optimizing purification features (filters, resins and flowrate) warrant limiting the effects of this type of pollutions. That's why studies are yet in progress to propose operators a chemical policy and adapted procedures for each type of radioactive pollution able to reduce the impact on dosimetry.

Fig. 2 :

 2 Fig. 2 : Ag 110 m activity in Tricastin 2 unit 3.2.5 -Remedial actions Remedial actions depend on Ag 110 m activity released during cycle operation or during shutdown. They deal with :-purification during end of cycle shutdown ; -elution of activity confined in dead legs ; -other complementary actions (cleaning system).

Fig 3 :

 3 Fig 3 : Simplified Pourbaix diagram for antimony at 300°C. The dotted line limits the RCS operating conditions

Fig 4 :

 4 Fig 4 : Simplified Pourbaix diagram for antimony at 80°C. The dotted line limits the RCS shutdown conditions at forced oxygenation.3.3.3 -Impact on dose ratesThe important antimony released activity has an impact on dosimetry and waste management.

Fig. 5 :

 5 Fig. 5 : Iodine absorbed in µg/cm 2 on passivated alloy 600 versus pH and time (Iodine initial concentration 50 mg/kg)

Table 2 :

 2 Ag 110m, 900 and 1300 MWe French Plants in 2000.

		Ag110m	Ag110m	Ag110m
	Year 2000	average	maximum	minimum
	900 MW	1.47 GBq/t	6.17 GBq/t	0.07 GBq/t
	1300 MW	0.16 GBq/t	0.48 GBq/t	0.04 GBq/t

of cycle number Ag 110 m max volumic activity during end of cycle shutdown (MBq/t)

  Eddy current and ultrasonic non destructive control rod examination is performed. Doubtful rods or rods with perforation signals are systematically replaced with either chromium or nitride coated rods. However, such examinations do not allow for detection of fast evolution kinetics flaws. It was the case in Tricastin 2, at the end of cycle 17 shutdown, as Ag 110 m volumic activity reached 16 GBq/t during primary coolant oxygenation. Such an activity called for unscheduled examination of the control rods, revealing an early perforation of 10 standard rods, which were rejected and remplaced with coated rods.Primary Ag 110 m volumic activity monitoring during operationA tight monitoring of Ag 110 m volumic activity in operation can allow to anticipate an anomaly before the shutdown transient. It consists in comparing the on-going cycle volumic activity with that of the previous cycles. Ag 110 m activity level likely to be observed as the primary coolant is oxygenated during the end of cycle shutdown can be forecasted by correlating Ag 110 m mean activity during operation to Ag 110 m maximum activity at end of cycle shutdown oxygenation (fig2). Hence, Ag 110 m maximum volumic activity, as the primary coolant is oxygenated, matches the total activity released during the outage.

	16 000						8
	14 000		Ag 110 m activity during shutdown Ag 110 m activity in operation	7
	4 000 6 000 8 000 10 000 12 000						2 3 4 5 6	Ag 100 m mean activity during	operation (MBq/t)
	2 000						1
	0	9	1 0	1 1	1 2	1 3 End 0 1 4 1 5 1 6 1 7 1 8 1 9

Table 3 :

 3 some French PWRs, Mean and maximum volumic activities encountered at the oxygenation peak

	122 Sb and Co peaks (table 3). They impacted dosimetry and waste management. 124 Sb volumic activity peaks have been observed. These peaks were higher 58 Sb and than 122 124 Sb are two multigamma emitters with radioactive half-life of 2.7 days and 60.2 days respectively. Thanks to its short radioactive half-life, 122 Sb has a low impact on health physics, but it allows to determine
	antimony sources.
	PWR UNITS (MWe)	SB 122 (GBq/t)	Sb 124 (GBq/t)
	900 first series	Mean	21	14
		Maximum	200	110
	900	Mean	25	18
		Maximum	160	140
	1300	Mean	14	11
		Maximum	70	55
	3.3.1 -Antimony sources
	Two major sources have been identified in French PWRs :
	1. Beryllium-Antimony source rod failure, but there are source rods only for the first reactor cycles ;
	2. bearing wear of pumps in the Boron Recycle System. These bearings contain graphite impregnated with
	about 10 % of antimony. A few activated antimony grams can explain such peaks of 124	122 Sb and	124 Sb. So, about 7 grams can lead to a

Table 4 :

 4 . RCS recontamination after a

	124 Sb (GBq/m²) Before H 2 O 2 After H 2 O 2
	Hot leg	0.29	1.02
	SG tubing	0.32	0.95
		124

Table 5 :

 5 table below (table 5), this allows to make the Bearing replacement during the 14 th cycle

	Cycles	13	14*	15
	124 Sb (GBq/t)	100	60	20

Proceedings of the International conference on water chemistry in nuclear reactors systems, 22-26/04/02 Avignon (France), papier 123

Sb peaks decrease.