Laure Cardoulis
email: laure.cardoulis@univ-amu.fr

APPLICATIONS OF CARLEMAN INEQUALITIES FOR A TWO-BY-TWO PARABOLIC SYSTEM IN AN UNBOUNDED GUIDE

Keywords: 2010 Mathematics Subject Classification. Primary 35K05, 35R30 inverse problems, Carleman inequalities, heat operator, system, unbounded guide

In this article we consider the inverse problem of determining some of the coefficients of a two-by-two parabolic system defined on an unbounded guide. Using an adapted Carleman estimate, we establish local stability results for at least two coefficients of this system in any finite portion of the guide. These estimates are obtained with data of the solution at a fixed time and boundary measurements for observations.

Introduction

Let ω be a bounded connex domain in R n-1 , n ≥ 2 with C 2 boundary. Denote Ω := R × ω, Q = Ω × (0, T ) and Σ = ∂Ω × (0, T ). We consider the following system

       ∂ t u -∆u + au + bv = g 1 in Q, ∂ t v -∆v + cu + dv = g 2 in Q, u = h 1 and v = h 2 on Σ, u(x, 0) = u 0 (x) and v(x, 0) = v 0 (x) in Ω, (1.1)
where a, b, c, d are bounded coefficients defined on Ω such that a, b, c, d ∈ Λ(M 0 ) := {f ∈ L ∞ (Ω), f L ∞ (Ω) ≤ M 0 } for some M 0 > 0.

Our inverse problem is to estimate at least two coefficients between a, b, c, d from the data of the solution (u, v) at T /2 and the measurement of (u, v) on a part of the boundary. We will consider (u, v) (resp. ( u, v)) a solution of (1.1) associated with (a, b, c, d, u 0 , v 0 , g 1 , g 2 , h 1 , h 2 ) (resp. ( a, b, c, d, u 0 , v 0 , g 1 , g 2 , h 1 , h 2 )) and two positive reals l, L such that l < L. Denote Ω L = (-L, L) × ω and Ω l = (-l, l) × ω.

The first result of this paper gives a Hölder result (3.3) for the coefficients b and c in the case where a = a, d = d and is the following (see Theorem 3.1)

b -b 2 L 2 (Ω l ) + c -c 2 L 2 (Ω l ) ≤ K (u -u)(., T 2 ) 2 H 2 (Ω L ) + (v -v)(., T 2 ) 2 H 2 (Ω L ) + γ L ×(0,T ) 2 k=0 (|∂ ν (∂ k t (u -ũ))| 2 + |∂ ν (∂ k t (v -ṽ))| 2 ) dσ dt κ
where K is a positive constant, κ ∈ (0, 1), γ L is a part of the boundary (see (2.2)), and assuming that the hypothesis (3.2) is satisfied. The second result (3.15) of this paper is also a Hölder stability result for the four coefficients a, b, c, d (see Theorem 3.2)

a -a 2 L 2 (Ω l ) + b -b 2 L 2 (Ω l ) + c -c 2 L 2 (Ω l ) + d -d 2 ≤ K 1 k=0 ∂ k t (u -u)(., T 2 ) 2 H 2 (Ω L ) + 1 k=0 ∂ k t (v -v)(., T 2 ) 2 H 2 (Ω L ) + γ L ×(0,T ) 2 k=0 (|∂ ν (∂ k t (u -ũ))| 2 + |∂ ν (∂ k t (v -ṽ))| 2
) dσ dt κ with stronger hypotheses (3.13) and (3.14) than those in Theorem 3.1 (see (3.2)).

The third theorem of this paper gives a Hölder stability result (3.34) (see Theorem 3.3) for the following reaction-diffusion system

       ∂ t u -∆u + au + bv + A 1 • ∇u + A 2 • ∇v = g 1 in Q, ∂ t v -∆v + cu + dv + A 3 • ∇u + A 4 • ∇v = g 2 in Q, u = h 1 and v = h 2 on Σ, u(x, 0) = u 0 (x) and v(x, 0) = v 0 (x) in Ω, (1.2)
where all the coefficients a, b, c, d, A 1 , A 2 , A 3 , A 4 are bounded (a, b, c, d ∈ Λ(M 0 ) and A 1 , A 2 , A 3 , A 4 ∈ Λ(M 0 ) n ∩ H 1 (Ω) n ). We obtain a stability result for the coefficients b and A 3 (assuming A 3 has the form A 3 = ∇g) with the same kind of observations in the right-hand side of (3.34) as we have obtained in (3.3) or (3.15). Assuming that the Assumptions (3.32) and (3.33) hold, we get the following result

b -b 2 L 2 (Ω l ) + A 3 -A 3 2 (L 2 (Ω l )) n ≤ K 1 k=0 ∂ k t (u -u)(., T 2 ) 2 H 2 (Ω L ) + 1 k=0 ∂ k t (v -v)(., T 2 ) 2 H 3 (Ω L ) + γ L ×(0,T ) 2 k=0 (|∂ ν (∂ k t (u -ũ))| 2 + |∂ ν (∂ k t (v -ṽ))| 2 ) dσ dt κ .
Of course each of these above stability results implies an uniqueness result. Up to our knowledge, there are few results concerning the simultaneous identification of more than one coefficient in each equation (see for example [START_REF] Benabdallah | Inverse problem for a parabolic system with two components by measurements of one component[END_REF] and also [START_REF] Cardoulis | Simultaneous identification of the diffusion coefficient and the potential for the Schrödinger operator with only one observation[END_REF] where the authors give a stability result for the diffusion coefficient a and the potential b of the Schrödinger operator i∂ t q +a∆q +bq).

In previous papers, stability results have been obtained for parabolic systems but, as far as we know, these papers have investigated the case of bounded domains and have given results with observations on a subdomain of their domain ( [START_REF] Benabdallah | Inverse problem for a parabolic system with two components by measurements of one component[END_REF][START_REF] Cristofol | Inverse problems for a 2x2 reaction diffusion system using a Carleman estimate with one observation[END_REF]...). Furthermore, there is no result for a twoby-two parabolic system with only one observation on a part of the boundary and without any data of the solution at a fixed time even in a bounded domain. We will use here the global Carleman estimate (2.5) for one equation given in [START_REF] Cardoulis | An inverse problem for the heat equation in an unbounded guide[END_REF] based on a classical Carleman estimate given in [START_REF] Yamamoto | Carleman estimates for parabolic equations and applications[END_REF][START_REF] Yuan | Lipschitz stability in the determination of the principal part of a parabolic equation[END_REF].

Our choice of weight functions is adapted for this unbounded domain but will give us Hölder, and not Lipschitz, estimates of the coefficients. Recall that the method using Carleman estimates for solving inverse problems has been initiated by [START_REF] Bukhgeim | Uniqueness in the Large of a Class of Multidimensional Inverse Problems[END_REF]. Our results extend to a system previous results for one equation defined on an unbounded guide (see [START_REF] Cardoulis | An inverse problem for the heat equation in an unbounded guide[END_REF] for the heat operator ∂ t u -∆u + qu and [START_REF] Cardoulis | A stability result for the diffusion coefficient of the the heat operator defined on an unbounded guide[END_REF] for the heat operator ∂ t u -∇ • (c∇u) where stability results are given either for the potential q or for the diffusion coefficient c). This Paper is organized as follows. In section 2, we specify the weight functions used for our Carleman estimate (cf (2.1), (2.3)) and due to the particular symmetric form of these weight functions with respect to x 1 and t -T /2 we recall from [START_REF] Cardoulis | An inverse problem for the heat equation in an unbounded guide[END_REF] the inequality (2.4), crucial for our final estimates (3.3), (3.15) and (3.34). Then in section 3 we state and prove our stability results, first for the coefficients b, c, after for a, b, c, d and finally for b, A 3 .

Carleman estimate

Denote Q L = Ω L × (0, T ) = (-L, L) × ω × (0, T ), x = (x 1 , • • • , x n ) ∈ R n , x = (x 2 , • • • , x n
) and define the operator Au = ∂ t u -∆u. Let l > 0, following [START_REF] Cardoulis | An inverse problem for the heat equation in an unbounded guide[END_REF] in this section, we consider some positive real L > l and choose a

∈ R n \ Ω such that if d(x) = |x -a | 2 -x 2 1 for x ∈ Ω L , then d > 0 in Ω L , |∇ d| > 0 in Ω L . (2.1)
Moreover define

Γ L = {x ∈ ∂Ω L , < x -a, ν(x) >≥ 0} and γ L = Γ L ∩ ∂Ω. (2.2)
Here < ., . > denotes the usual inner product in R n and ν(x) the outward unit normal vector to ∂Ω L at x. Notice that γ L does not contain any cross section of the guide. From [START_REF] Yamamoto | Carleman estimates for parabolic equations and applications[END_REF] we consider weight functions as follows:

for t ∈ (0, T ), if M 1 > sup 0<t<T (t -T /2) 2 = (T /2) 2 , ψ(x, t) = d(x) -t - T 2 2
+ M 1 , and φ(x, t) = e λψ(x,t) .

(2.

3)

The constant λ > 0 will be set in Proposition 2.2 and is usually used as a large parameter in Carleman inequalities. Since we will not use it, we will consider λ fixed in the article. We recall from [START_REF] Cardoulis | An inverse problem for the heat equation in an unbounded guide[END_REF] the following result.

Proposition 2.1. There exists T > 0, L > l, a ∈ R 2 \ Ω and ˜ > 0 such that (2.1) holds and, setting

O L,˜ = (Ω L × ((0, 2˜ ) ∪ (T -2˜ , T ))) ∪ (((-L, -L + 2˜ ) ∪ (L -2˜ , L)) × ω × (0, T )),
we have

d 1 < d 0 < d 2 (2.4
) where

d 0 = inf Ω l φ(., T 2 ), d 1 = sup O L,˜ φ and d 2 = sup Ω L φ(., T 2
).

We will use the following notations:

Let α = (α 1 , • • • , α n ) be a multi-index with α i ∈ N ∪ {0}. We set ∂ α x = ∂ α1 1 • • • ∂ αn n , |α| = α 1 + • • • + α n and define H 2,1 (Q L ) = {u ∈ L 2 (Q L ), ∂ α x ∂ αn+1 t u ∈ L 2 (Q L ), |α| + 2α n+1 ≤ 2}
endowed with its norm

u 2 H 2,1 (Q L ) = |α|+2αn+1≤2 ∂ α x ∂ αn+1 t u 2 L 2 (Q L ) .
We recall here a global Carleman-type estimate proved in [START_REF] Cardoulis | An inverse problem for the heat equation in an unbounded guide[END_REF], based on a classical Carleman estimate (see Yamamoto [START_REF] Yamamoto | Carleman estimates for parabolic equations and applications[END_REF]Theorem 7.3]).

Proposition 2.2. There exist a value of λ > 0 and positive constants s 0 and C = C(λ, s 0 ) such that

I(u) := Q L 1 sφ (|∂ t u| 2 + |∆u| 2 ) + sφ |∇u| 2 + s 3 φ 3 |u| 2 e 2sφ dx dt ≤ C e sφ Au 2 L 2 (Q L ) + Cs 3 e 2sd1 u 2 H 2,1 (Q L ) + Cs γ L ×(0,T ) |∂ ν u| 2 e 2sφ dσ dt, (2.5)
for all s > s 0 and all u ∈ H 2,1 (Q L ) satisfying u(., 0) = u(., T ) = 0 in Ω L , u = 0 on ∂Ω L × (0, T ).

We denote ∂ ν u = ν • ∇u.

In fact the above Proposition 2.2 is still valid for a more general function u: we can replace the condition u = 0 on ∂Ω L × (0, T ) in Proposition 2.2 by u = 0 on (∂Ω ∩ ∂Ω L ) × (0, T ). Since the method of Carleman estimates requires several time differentiations, we assume in the following that u, v (resp. u, v) belong to H = H 3 (0, T, H 3 (Ω)) satisfying the a-priori bound u H < M 2 and v H < M 2 for given M 2 > 0.

From now on, we use the notation w( T 2 ) = w(., T 2 ) for any function w.

Inverse problems

3.1. The first result. Consider here (u, v) (resp. ( u, v)) a strong solution of (1.1) associated with

(a, b, c, d, u 0 , v 0 , g 1 , g 2 , h 1 , h 2 ) (resp. (a, b, c, d, u 0 , v 0 , g 1 , g 2 , h 1 , h 2 ))
. Assume that all the coefficients a, b, c, d, b, c belong to Λ(M 0 ). From [8, Lemma 4.2], we derive the following result, also used in [START_REF] Cardoulis | An inverse problem for the heat equation in an unbounded guide[END_REF] Lemma 3.1. There exist some positive constants C, s 1 such that

Ω L e 2sφ( T 2 ) |z(T /2)| 2 dx ≤ Cs Q L e 2sφ φ 2 |z| 2 dx dt + C s Q L e 2sφ |∂ t z| 2 dx dt,
for all s ≥ s 1 and z ∈ H 1 (0, T ; L 2 (Ω L )).

For the sake of completeness, we recall its proof.

Proof. Consider η defined by (3.4) and any w ∈ H 1 (0, T ; L 2 (Ω L )). Since η( T 2 ) = 1 and η(0) = 0, we have

Ω L w(x, T /2) 2 dx = Ω L (η(T /2)w(x, T /2)) 2 dx = Ω L T /2 0 ∂ t (η 2 (t)|w(x, t)| 2 )dt dx = 2 T /2 0 Ω L η 2 (t)w(x, t)∂ t w(x, t)dx dt + 2 T /2 0 Ω L η(t)∂ t η(t)|w(x, t)| 2 dx dt.
As 0 ≤ η ≤ 1, using Young's inequality, it comes that for any s > 0,

Ω L w(x, T /2) 2 dx ≤ Cs Q L |w| 2 dx dt + C s Q L |∂ t w| 2 dx dt. (3.1)
Then we can conclude replacing w by e sφ z in (3.1).

We can state our first main result for a two-by-to linear system which extend precedent results for one equation (see [START_REF] Cardoulis | An inverse problem for the heat equation in an unbounded guide[END_REF] and [START_REF] Cardoulis | A stability result for the diffusion coefficient of the the heat operator defined on an unbounded guide[END_REF]). We do not follow here the proof of [1, Theorem 1.2] and rather use ideas from [START_REF] Cardoulis | An inverse problem for the heat equation in an unbounded guide[END_REF]. Theorem 3.1. Let l > 0. Let T > 0, L > l and a ∈ R n \ Ω satisfying the conditions of Proposition 2.1. We make the following assumption

| u(., T 2 )| ≥ R and | v(., T 2 )| ≥ R in Ω L for some R > 0. (3.2)
Then there exists a sufficiently small number δ 0 such that if δ ∈ (0, δ 0 ),

(u-u)(., T 2 ) 2 H 2 (Ω L ) + (v-v)(., T 2 ) 2 H 2 (Ω L ) + γ L ×(0,T ) 2 k=0 (|∂ ν (∂ k t (u-ũ))| 2 +|∂ ν (∂ k t (v-ṽ))| 2 )dσdt ≤ δ then the following Hölder stability estimate holds b -b 2 L 2 (Ω l ) + c -c 2 L 2 (Ω l ) ≤ Kδ κ for all δ ∈ (0, δ 0 ). (3.3)
Here, K > 0 and κ ∈ (0, 1) are two constants depending on R, r, L, l, M 0 , M 1 , M 2 , T and a.

Proof. Let χ, η be C ∞ cut-off functions defined by χ, ∇χ, ∆χ ∈ Λ(M 0 ), 0 ≤ χ ≤ 1, 0 ≤ η ≤ 1, χ(x) = 0 if x ∈ ((-∞, -L + ˜ ) ∪ (L -˜ , +∞)) × ω), χ(x) = 1 if x ∈ (-L + 2˜ , L -2˜ ) × ω, η(t) = 0 if t ∈ (0, ˜ ) ∪ (T -˜ , T ), η(t) = 1 if t ∈ ×(2˜ , T -2˜ ).
(3.4) Denote also

y = u -u, y 0 = χηy, y 1 = ∂ t y 0 , y 2 = ∂ t y 1 , z = v -v, z 0 = χηz, z 1 = ∂ t z 0 and z 2 = ∂ t z 1 .
Note that (y 0 , z 0 ) satisfies

   ∂ t y 0 -∆y 0 + ay 0 + bz 0 = ρ 1 := ( b -b)χη v + (∂ t η)χy -(∆χ)ηy -2∇χ • ∇(ηy) in Q L , ∂ t z 0 -∆z 0 + cy 0 + dz 0 = ρ 2 := ( c -c)χη u + (∂ t η)χz -(∆χ)ηz -2∇χ • ∇(ηz) in Q L , y 0 = z 0 = 0 on ∂Ω L × (0, T ). (3.5) and (y 1 , z 1 ), (y 2 , z 2 ) satisfy    ∂ t y 1 -∆y 1 + ay 1 + bz 1 = ∂ t ρ 1 in Q L , ∂ t z 1 -∆z 1 + cy 1 + dz 1 = ∂ t ρ 2 in Q L , y 1 = z 1 = 0 on ∂Ω L × (0, T ) and    ∂ t y 2 -∆y 2 + ay 2 + bz 2 = ∂ 2 t ρ 1 in Q L , ∂ t z 2 -∆z 2 + cy 2 + dz 2 = ∂ 2 t ρ 2 in Q L , y 2 = z 2 = 0 on ∂Ω L × (0, T ).
• First step: Applying (3.5) for t = T 2 , if we denote

J := Ω L e 2sφ( T 2 ) χ 2 [|c -c| 2 | u( T 2 )| 2 + |b -b| 2 | v( T 2 )| 2 ] dx then we get J ≤ Ce 2sd2 F 0 ( T 2 ) + C Ω L e 2sφ( T 2 ) (|∂ t y 0 ( T 2 )| 2 + |∂ t z 0 ( T 2 )| 2 ) dx with F 0 (T /2) = z 0 (T /2) 2 H 2 (Ω L ) + z(T /2) 2 H 1 (Ω L ) + y 0 (T /2) 2 H 2 (Ω L ) + y(T /2) 2 H 1 (Ω L ) . Note that F 0 (T /2) ≤ CF (T /2) with F (T /2) = y(T /2) 2 H 2 (Ω L ) + z(T /2) 2 H 2 (Ω L )
. Moreover, since ∂ t y 0 = y 1 , ∂ t z 0 = z 1 and 1 ≤ φ, using Lemma 3.1, we obtain

J ≤ Ce 2sd2 F (T /2) + Cs Q L e 2sφ φ 3 (|y 1 | 2 + |z 1 | 2 ) dx dt + C s Q L e 2sφ φ 3 (|y 2 | 2 + |z 2 | 2 ) dx dt. (3.6)
• Second step: Now we evaluate J with the Carleman inequalities (2.5) for y i and z i , i = 1, 2. Note that all the terms in e sφ Ay i 2

L 2 (Q L ) or e sφ Az i 2 L 2 (Q L )
with derivatives of χ or η will be bounded above by Ce 2sd1 with C a positive constant. Therefore, for s sufficiently large, there exists a positive constant C such that

I(y i ) + I(z i ) ≤ C Q L e 2sφ χ 2 [|c -c| 2 + |b -b| 2 ] dx dt + C Q L e 2sφ (|y i | 2 + |z i | 2 ) dx dt + Ce 2sd1 +Cs 3 e 2sd1 ( y i 2 H 2,1 (Q L ) + z i 2 H 2,1 (Q L ) ) + Cs γ L ×(0,T ) e 2sφ (|∂ ν y i | 2 + |∂ ν z i | 2 ) dσ dt.
Since e 2sφ ≤ e 2sφ(T /2) , we deduce that

I(y i ) + I(z i ) ≤ C Ω L e 2sφ(T /2) χ 2 [|c -c| 2 + |b -b| 2 ] dx dt + Cs 3 e 2sd1 +Cs γ L ×(0,T ) e 2sφ (|∂ ν y i | 2 + |∂ ν z i | 2 ) dσ dt. Thus s 3 Q L e 2sφ φ 3 (|y i | 2 + |z i | 2 ) dx dt ≤ C Ω L e 2sφ(T /2) χ 2 [|c -c| 2 + |b -b| 2 ] dx +Cs 3 e 2sd1 + Cs γ L ×(0,T ) e 2sφ (|∂ ν y i | 2 + |∂ ν z i | 2 ) dσ dt. (3.7)
Therefore, from (3.6) and (3.7), we get for s sufficiently large

J ≤ Ce 2sd2 F (T /2) + C s 2 s 3 e 2sd1 + Ω L e 2sφ(T /2) χ 2 [|c -c| 2 | + |b -b| 2 ] dx +s γ L ×(0,T ) e 2sφ 2 i=1 (|∂ ν y i | 2 + |∂ ν z i | 2 ) dσ dt .
So we have

J ≤ Ce 2sd2 G(T /2) + Cse 2sd1 + C s 2 Ω L e 2sφ(T /2) χ 2 [|c -c| 2 | + |b -b| 2 ] dx (3.8)
with

G(T /2) = F (T /2) + γ L ×(0,T ) 2 k=0 (|∂ ν ∂ k t y| 2 + |∂ ν ∂ k t z| 2 ) dσ dt.
• Third and last step: In this step, we come back to the coefficients b -b and c -c. First, from the hypothesis (3.2) we derive from (3.8), for s sufficiently large

Ω L e 2sφ(T /2) χ 2 (| b -b| 2 + | c -c| 2 ) dx ≤ Ce 2sd2 G(T /2) + Cse 2sd1 . (3.9)
Moreover, since e 2sd0 ≤ e 2sφ(T /2) in Ω l and χ = 1 in Ω l , we deduce from (3.9) that

e 2sd0 ( b -b 2 L 2 (Ω l ) + c -c 2 L 2 (Ω l ) ) ≤ Ce 2sd2 G(T /2) + Cse 2sd1
. This last inequality can be rewritten in the following form for s sufficiently large (s

≥ s 2 ) b -b 2 L 2 (Ω l ) + c -c 2 L 2 (Ω l ) ≤ C(e 2s(d2-d0) G(T /2) + se 2s(d1-d0) ). (3.10)
Note that if G(T /2) = 0, since (3.10) holds for any s ≥ s 2 and d 1 -d 0 < 0 we get (3.3). Now if G(T /2) = 0, we recall from (2.4) that d 1 -d 0 < 0 and d 2 -d 0 > 0 and optimize (3.10) with respect to s. Indeed denote f (s) = e 2s(d2-d0) G(T /2) + e 2s(d1-d0) and g(s) = e 2s(d2-d0) G(T /2) + se 2s(d1-d0) .

We have f (s) ∼ g(s) at infinity. Moreover the function f has a minimum in

s 3 = 1 2(d 2 -d 1 ) ln( d 0 -d 1 (d 2 -d 0 )G(T /2)
) and f (s

3 ) = K G(T /2) κ with κ = d0-d1 d2-d1 and K = ( d0-d1 d2-d0 ) d 2 -d 0 d 2 -d 1 + ( d0-d1 d2-d0 ) d 1 -d 0 d 2 -d 0 . Finally the minimum s 3 is sufficiently large (s 3 ≥ s 2 ) if the following condition G(T /2) ≤ δ 0 , with δ 0 = d0-d1 (d2-d0)e 2s 2 (d 2 -d 1 )
, is satisfied. Then we get our result (3.3) and so we complete the proof of Theorem 3.1.

Remark 1. • Note that the hypothesis (3.2) is quite usual (cf [START_REF] Benabdallah | Inverse problem for a parabolic system with two components by measurements of one component[END_REF][START_REF] Cristofol | Inverse problems for a 2x2 reaction diffusion system using a Carleman estimate with one observation[END_REF] for a parabolic system in a bounded domain) and is removed in [START_REF] Benabdallah | Inverse problem for a parabolic system with two components by measurements of one component[END_REF] by the control theory and in [START_REF] Cristofol | Inverse problems for a 2x2 reaction diffusion system using a Carleman estimate with one observation[END_REF] by conditions on a, b, c, d, u 0 , v 0 , h 1 , h 2 , g 1 , g 2 . In some cases, one can also diagonalise the coupling matrix of the coefficients (see [START_REF] Cosner | Sign-definite solutions in some linear elliptic systems[END_REF]) then use a parabolic positivity result (see [START_REF] Daners | Abstract Evolution Equations, Periodic Problems and Applications[END_REF]Theorem 13.5]) for the decoupling system. Of course we could obtain the same result as (3.3) for any coefficient in each equation of (1.1). But if we want to determine the coefficients b and d for example, we only have to assume that

| v(., T 2 )| ≥ R in Ω L for some R > 0, instead of (3.2). • In fact we can obtain in the right-hand side of (3.3) the term γ L ×(0,T ) 2 k=1 (|∂ ν (∂ k t (u -ũ))| 2 + |∂ ν (∂ k t (v -ṽ))| 2 ) dσ dt instead of γ L ×(0,T ) 2 k=0 (|∂ ν (∂ k t (u -ũ))| 2 + |∂ ν (∂ k t (v -ṽ))| 2
) dσ dt if we slightly modify d 1 (if we define d 1 = sup O L,˜ φ, the inequalities (2.4) still hold and all the terms inside the integrals on γ L with derivatives of η are therefore bounded above by e 2sd1 ).

, v 0 , g 1 , g 2 , h 1 , h 2 ) (resp. ( a, b, c, d, u 0 , v 0 , g 1 , g 2 , h 1 , h 2 ))
. Assume that all the coefficients a, b, c, d, a, b, c, d belong to Λ(M 0 ). For our second main result, first we need the following lemma inspired from Klibanov and Timonov ([11]). Recall that χ and η are defined by (3.4). Lemma 3.2. There exists a positive constant C such that

Q L e 2sφ φχ 2 η 2 t T /2 f (ξ) dξ 2 dx dt ≤ C s e 2sd1 + Q L e 2sφ χ 2 η 2 f 2 dx dt , for all s > 0 and f ∈ L 2 (0, T, L 2 (Ω L )) ∩ L ∞ (Q L ).
Proof. By the Cauchy-Schwarz inequality, we have

Q L φχ 2 η 2 e 2sφ t T /2 f (x, ξ) dξ 2 dx dt ≤ Q L φχ 2 η 2 e 2sφ |t - T 2 | t T /2 f (x, ξ) 2 dξ dx dt ≤ Ω L T /2 0 φχ 2 η 2 e 2sφ ( T 2 -t) t T /2 f (x, ξ) 2 dξ dxdt+ Ω L T T /2 φχ 2 η 2 e 2sφ (t- T 2 ) t T /2 f (x, ξ) 2 dξ dxdt. (3.11) Note that ∂ t (e 2sφ ) = -4sλ(t - T 2 )φe 2sφ .
For the second integral of the right hand side of (3.11), since η(T ) = 0, by integration by parts we have

Ω L T T /2 φχ 2 η 2 e 2sφ (t - T 2 ) t T /2 f (x, ξ) 2 dξ dx dt = - 1 4sλ Ω L T T /2 χ 2 η 2 ∂ t (e 2sφ )( t T /2 f (x, ξ) 2 dξ) dx dt = - 1 4sλ Ω L χ 2 η 2 e 2sφ ( t T /2 f (x, ξ) 2 dξ) t=T t=T /2 dx + 1 4sλ Ω L T T /2 e 2sφ χ 2 η 2 f 2 dx dt + 1 2sλ Ω L T T /2 e 2sφ χ 2 η∂ t η( t T /2 f (x, ξ) 2 dξ) dx dt = 1 2sλ Ω L T T /2 e 2sφ χ 2 η∂ t η( t T /2 f (x, ξ) 2 dξ) dx dt + 1 4sλ Ω L T T /2 e 2sφ χ 2 η 2 f 2 dx dt.
(3.12)

The first integral of (3.12) is bounded above by C s e 2sd1 due to the derivative of η. Therefore

Ω L T T /2 φχ 2 η 2 e 2sφ (t - T 2 ) t T /2 f (x, ξ) 2 dξ dx dt ≤ C s e 2sd1 + Q L e 2sφ χ 2 η 2 f 2 dx dt .
We obtain a similar result for the first integral of (3.11) and this concludes the proof of Lemma 3.2.

Now we can state our second main result in view to obtain a stability estimate of the four coefficients of (1.1) with nearly the same observations that we obtained in Theorem 3.1 (see the right-hand sides of (3.3) and (3.15)).

Theorem 3.2. Let l > 0. Let T > 0, L > l and a ∈ R n \ Ω satisfying the conditions of Proposition 2.1. We make here the following assumptions

| u| ≥ R and |∂ t ( v u )| ≥ R in Q for some R > 0, (3.13)
and

| v| ≥ R and |∂ t ( u v )| ≥ R in Q for some R > 0. (3.14)
Then there exists a sufficiently small number δ 0 such that if δ ∈ (0, δ 0 ),

1 k=0 ∂ k t (u -u)(., T 2 ) 2 H 2 (Ω L ) + 1 k=0 ∂ k t (v -v)(., T 2 ) 2 H 2 (Ω L ) + γ L ×(0,T ) 2 k=0 (|∂ ν (∂ k t (u -ũ))| 2 + |∂ ν (∂ k t (v -ṽ))| 2 dσ dt ≤ δ
then the following Hölder stability estimate holds

a -a 2 L 2 (Ω l ) + b -b 2 L 2 (Ω l ) + c -c 2 L 2 (Ω l ) + d -d 2 L 2 (Ω l ) ≤ Kδ κ for all δ ∈ (0, δ 0 ). ( 3
.15) Here, K > 0 and κ ∈ (0, 1) are two constants depending on R, r, L, l, M 0 , M 1 , M 2 , T and a.

Proof. As in Thereom 3.1 denote y = u -u and z = v -v. Then (y, z) satisfies

   ∂ t y -∆y + ay + bz = ( a -a) u + ( b -b) v in Q, ∂ t z -∆z + cy + dz = ( c -c) u + ( d -d) v in Q, y = z = 0 on Σ. • First step: Let y 1 = y u and z 1 = z u . Then (y 1 , z 1 ) satisfies    ∂ t y 1 -∆y 1 + ay 1 + bz 1 = f 1 + a -a + ( b -b) v u in Q, ∂ t z 1 -∆z 1 + cy 1 + dz 1 = f 2 + c -c + ( d -d) v u in Q, y 1 = z 1 = 0 on Σ, with f 1 := 1 u (-y 1 ∂ t u + y 1 ∆ u + 2∇y 1 • ∇ u) and f 2 := 1 u (-z 1 ∂ t u + z 1 ∆ u + 2∇z 1 • ∇ u). Denote now y 2 = ∂ t y 1 , z 2 = ∂ t z 1 , y 3 = 1 ∂t( v u ) y 2 and z 3 = 1 ∂t( v u ) z 2 . Then    ∂ t y 2 -∆y 2 + ay 2 + bz 2 = ∂ t f 1 + ( b -b)∂ t ( v u ) in Q, ∂ t z 2 -∆z 2 + cy 2 + dz 2 = ∂ t f 2 + ( d -d)∂ t ( v u ) in Q, y 2 = z 2 = 0 on Σ, and    ∂ t y 3 -∆y 3 + ay 3 + bz 3 = f 3 + b -b in Q, ∂ t z 3 -∆z 3 + cy 3 + dz 3 = f 4 + d -d in Q, y 3 = z 3 = 0 on Σ, (3.16)
with

f 3 := 1 ∂ t ( v u ) -y 3 ∂ 2 t ( v u ) + y 3 ∆(∂ t ( v u )) + 2∇y 3 • ∇(∂ t ( v u )) + ∂ t f 1 and f 4 := 1 ∂ t ( v u ) -z 3 ∂ 2 t ( v u ) + z 3 ∆(∂ t ( v u )) + 2∇z 3 • ∇(∂ t ( v u )) + ∂ t f 2 .
Finally let y 4 = ∂ t y 3 , z 4 = ∂ t z 3 , y 5 = χηy 4 and z 5 = χηz 4 . Then (3.18) As in Thereom 3.1, all the terms in Q L e 2sφ ((Ay 5 ) 2 + (Az 5 ) 2 ) dx dt with derivatives of η or χ will be bounded above by Ce 2sd1 . So since φ ≥ 1

∂ t y 5 -∆y 5 + ay 5 + bz 5 = χη∂ t f 3 + f 5 in Q L , ∂ t z 5 -∆z 5 + cy 5 + dz 5 = χη∂ t f 4 + f 6 in Q L , (3.17
Q L e 2sφ ((Ay 5 ) 2 + (Az 5 ) 2 ) dx dt ≤ C Q L e 2sφ (y 2 5 + z 2 5 ) dx dt + Ce 2sd1 +C Q L e 2sφ χ 2 η 2 (|∂ t f 3 | 2 + |∂ t f 4 | 2 ) dx dt ≤ C Q L e 2sφ (y 2 5 + z 2 5 )dxdt + Ce 2sd1 + C Q L φe 2sφ χ 2 η 2 4 i=1 (y 2 i + |∇y i | 2 + z 2 i + |∇z i | 2 )dxdt. (3.19)
Since χηy 4 = y 5 and χηz 4 = z 5 , (3.19) implies

Q L e 2sφ ((Ay 5 ) 2 + (Az 5 ) 2 ) dx dt ≤ C Q L e 2sφ (y 2 5 + z 2 5 + |∇y 5 | 2 + |∇z 5 | 2 ) dx dt + Ce 2sd1 +C Q L φe 2sφ χ 2 η 2 3 i=1 (y 2 i + |∇y i | 2 + z 2 i + |∇z i | 2 ) dx dt. (3.20)
From (3.18)-(3.20), we get for s sufficiently large

I(y 5 ) + I(z 5 ) ≤ Cs 3 e 2sd1 + C Q L φe 2sφ χ 2 η 2 3 i=1 (y 2 i + |∇y i | 2 + z 2 i + |∇z i | 2 ) dx dt +Cs γ L ×(0,T ) e 2sφ (|∂ ν y 5 | 2 + |∂ ν z 5 | 2 ) dσ dt. ( 3

.21)

Using now Lemma 3.2 we have

Q L φe 2sφ χ 2 η 2 y 2 1 dx dt = Q L φe 2sφ χ 2 η 2 t T /2 ∂ t y 1 (ξ)dξ + y 1 (T /2) 2 dx dt ≤ C s e 2sd1 + C s Q L e 2sφ χ 2 η 2 y 2 2 dx dt + C Q L φe 2sφ χ 2 η 2 y 1 (T /2) 2 dx dt ≤ C s e 2sd1 + C s Q L φe 2sφ χ 2 η 2 y 2 3 dx dt + C Q L φe 2sφ χ 2 η 2 y 1 (T /2) 2 dx dt ≤ C s e 2sd1 + C s Q L φe 2sφ χ 2 η 2 t T /2 ∂ t y 3 (ξ)dξ + y 3 (T /2) 2 dx dt+C Q L φe 2sφ χ 2 η 2 y 1 (T /2) 2 dx dt ≤ C s e 2sd1 + C s 2 e 2sd1 + Q L φe 2sφ χ 2 η 2 y 2 4 dx dt + C Q L φe 2sφ χ 2 η 2 (y 1 (T /2) 2 + y 3 (T /2) 2 ) dx dt ≤ C s e 2sd1 + C s 2 Q L e 2sφ y 2 5 dx dt + Ce 2sd2 Ω L (y 1 (T /2) 2 + y 2 (T /2) 2 ) dx. (3.22) Doing the same for Q L φe 2sφ χ 2 η 2 y 2 i dx dt, Q L φe 2sφ χ 2 η 2 z 2 i dx dt, Q L φe 2sφ χ 2 η 2 |∇y i | 2 dx dt and Q L φe 2sφ χ 2 η 2 |∇z i | 2 dx dt, for i = 1, 2,
γ L ×(0,T ) e 2sφ (|∂ ν y 5 | 2 + |∂ ν z 5 | 2 ) dσ dt +Ce 2sd2 Ω L 2 i=1 (y i (T /2) 2 + z i (T /2) 2 + |∇y i (T /2)| 2 + |∇z i (T /2)| 2 ) dx. (3.23)
Note that (3.23) can be rewritten on the following form

I(y 5 ) + I(z 5 ) ≤ Cs 3 e 2sd1 + Cse 2sd2 γ L ×(0,T ) 2 k=0 (|∂ ν ∂ k t y| 2 + |∂ ν ∂ k t z| 2 ) dσ dt +Ce 2sd2 Ω L 1 k=0 (∂ k t y(T /2) 2 + ∂ k t z(T /2) 2 + |∇∂ k t y(T /2)| 2 + |∇∂ k t z(T /2)| 2 ) dx
and so

I(y 5 ) + I(z 5 ) ≤ Cs 3 e 2sd1 + Cse 2sd2 F 1 (T /2) (3.24) with F 1 (T /2) = γ L ×(0,T ) 2 k=0 (|∂ ν ∂ k t y| 2 + |∂ ν ∂ k t z| 2 ) dσ dt + 1 k=0 ( ∂ k t y(T /2) 2 H 1 (Ω L ) + ∂ k t z(T /2) 2 H 1 (Ω L ) ).
• Second step: Now we evaluate (3.16) at T /2. We have

Ω L e 2sφ(T /2) χ 2 (| b-b| 2 +| d-d| 2 ) dx ≤ C Ω L e 2sφ(T /2) χ 2 (|∂ t y 3 (T /2)| 2 +|∂ t z 3 (T /2)| 2 ) dx+Ce 2sd2 F 2 (T /2) with F 2 (T /2) = 2 i=1 ( y i (T /2) 2 H 2 (Ω L ) + z i (T /2) 2 H 2 (Ω L ) ).
So, since η(T /2) = 1,

Ω L e 2sφ(T /2) χ 2 (| b-b| 2 +| d-d| 2 ) dx ≤ C Ω L e 2sφ(T /2) (|y 5 (T /2)| 2 +|z 5 (T /2)| 2 ) dx+Ce 2sd2 F 2 (T /2).
(3.25) Now let ψ 1 = e sφ y 5 and ψ 2 = e sφ z 5 . Calculate

J 1 = Ω L T /2 0 ∂ t ψ 1 (t)ψ 1 (t) dx dt and J 2 = Ω L T /2 0 ∂ t ψ 2 (t)ψ 2 (t) dx dt. Since η(0) = 0, we get J 1 = 1 2 Ω L ψ 1 (T /2) 2 dx = 1 2 Ω L e 2sφ(T /2) y 5 (T /2) 2 dx and J 2 = 1 2 Ω L e 2sφ(T /2) z 5 (T /2) 2 dx.
Therefore (3.25) becomes

Ω L e 2sφ(T /2) χ 2 (| b -b| 2 + | d -d| 2 ) dx ≤ Ce 2sd2 F 2 (T /2) +C Ω L T /2 0 1 s ∂ t ψ 1 (t)sψ 1 (t) dx dt + C Ω L T /2 0 1 s ∂ t ψ 2 (t)sψ 2 (t) dx dt. (3.26)
Using Young inequality, we deduce from (3.26)

Ω L e 2sφ(T /2) χ 2 (| b -b| 2 + | d -d| 2 ) dx ≤ C s (I(y 5 ) + I(z 5 )) + Ce 2sd2 F 2 (T /2). (3.27)
From (3.24) and (3.27) we get

Ω L e 2sφ(T /2) χ 2 (| b -b| 2 + | d -d| 2 ) dx ≤ Cs 2 e 2sd1 + Ce 2sd2 (F 1 (T /2) + F 2 (T /2)). (3.28)
Proceeding as in Theorem 3.1, we obtain from (3.28)

Ω l (| b -b| 2 + | d -d| 2 ) dx ≤ Cs 2 e 2s(d1-d0) + Ce 2s(d2-d0) F 3 (T /2) (3.29) with F 3 (T /2) = 1 k=0 ( ∂ k t y(T /2) 2 H 2 (Ω L ) + ∂ k t z(T /2) 2 H 2 (Ω L ) ) + γ L ×(0,T ) 2 k=0 (|∂ ν ∂ k t y| 2 + |∂ ν ∂ k t z| 2 ) dσ dt.
Notice that in the first and second steps of this proof, we have only used the hypothesis (3.13).

• Third step: Finally using the hypothesis (3.14), we can proceed exactly as before and obtain

Ω l (| a -a| 2 + | c -c| 2 ) dx ≤ Cs 2 e 2s(d1-d0) + Ce 2s(d2-d0) F 3 (T /2). ( 3
Ω L e 2sφ( T 2 ) χ 2 |z(T /2)| 2 dx ≤ Cs Q L e 2sφ φ 2 χ 2 η 2 |z| 2 dx dt + C s e 2sd1 + C s Q L e 2sφ χ 2 η 2 |∂ t z| 2 dx dt, for all z ∈ H 1 (0, T ; L 2 (Ω L )).
Moreover, if we did so, since we had to give up the end of the first step of the proof of Theorem 3.2, we'd rather follow the ideas of the proof of Theorem 3.1. Therefore, when in the second step we evaluated (3.16) for t = T /2, with the above inequality we would have to estimate

Ω L e 2sφ(T /2) χ 2 |∂ t y 3 (T /2)| 2 dx and Ω L e 2sφ(T /2) χ 2 |∂ t z 3 (T /2)| 2 dx; thus we could obtain Q L e 2sφ χ 2 η 2 |∂ t y 4 | 2 dx dt and Q L e 2sφ χ 2 η 2 |∂ t z 4 | 2 dx
dt in the right-hand side of the estimates. Then we would have to apply the Carleman estimates for χηy 4 , χηz 4 , χη∂ t y 4 , χη∂ t z 4 and so we would obtain a third derivative in time for the observation terms.

• Third the assumptions (3.13) and (3.14) are equivalent to

| u| ≥ R, | v| ≥ R and |det( u ∂ t u v ∂ t v | ≥
R with R a positive constant. For example, if n = 2 and ω = (r 1 , r 2 ) with r 1 > 0, let α(x 1 ) be a positive and bounded function in C 2 (R) such that min x1∈R α(x 1 ) > 2r 2 2 . Then u(x, t) = α(x 1 )t+x 2 and v(x, t) = tx 2 + 1 are solutions of the system (1.1) with

g 1 = g 2 = 0, a(x) = α (x1)+α(x1)x2 α(x1)-x 2 2 , b(x) = -x2α (x1)-α(x1) 2 α(x1)-x 2 2 , c(x) = x 2 2 α(x1)-x 2 2 , d(x) = -x2α(x1) α(x1)-x 2 2
, and satisfy the conditions (3.13)-(3.14).

• Finally note that the above results remain valid for the system (1.2) when all the coefficients a, b, c, d, A 1 , A 2 , A 3 , A 4 are bounded (a, b, c, d ∈ Λ(M 0 ) and A 1 , A 2 , A 3 , A 4 ∈ (Λ(M 0 )) n ). We obtain a stability result of at least two coefficients between a, b, c, d with the same observations in the right-hand sides of (3.3) or (3.15). In the next section we study the inverse problem of determining at least one of the coefficient A 1 , A 2 , A 3 , A 4 , for example A 3 if we assume that this coefficient has the form A 3 = ∇g.

3.3. The third result. Consider now (u, v) (resp. ( u, v)) a strong solution of (1.2) associated with (a, b, c, d, A 1 , A 2 , A 3 , A 4 , u 0 , v 0 , g 1 , g 2 , h 1 , h 2 ) (resp. (a, b, c, d, A 1 , A 2 , A 3 , A 4 , u 0 , v 0 , g 1 , g 2 , h 1 , h 2 )). Assume that all the coefficients a, b, c, d belong to Λ(M 0 ), A 1 , A 2 , A 3 , A 4 , A 3 belong to (Λ(M 0 )) n ∩ (H 1 (Ω)
) n and that there exist functions g, g such that

A 3 = ∇g, A 3 = ∇ g in Ω.
(3.31)

The Assumption (3.31) implies conditions on A 3 , A 3 :

if t A 3 = (c 1 , • • • , c n ), it means that for all i, j = 1, • • • , n, ∂ xi c j = ∂ xj c i , in other words rot(A 3 ) = 0 if n = 3.
Now following an idea developed in [START_REF] Yu | An inverse problem for the dynamical Lamé system with two set of boundary data[END_REF] for Lamé system in bounded domains, also used for example in [START_REF] Cardoulis | A stability result for the diffusion coefficient of the the heat operator defined on an unbounded guide[END_REF], we obtain the following result Lemma 3.3. Assume that the following assumption

|∇d • ∇ u(T /2)| ≥ R in Ω L for some R > 0 (3.32)
holds. Consider the first order partial differential operator P f = ∇f • ∇ u(T /2). Then there exist positive constants s 4 > 0 and C > 0 such that for all s ≥ s 4 ,

s 2 Ω L e 2sφ(T /2) |f | 2 dx ≤ C Ω L e 2sφ(T /2) |P f | 2 dx,
for all f ∈ H 1 0 (Ω L ). Proof. The proof follows [START_REF] Cardoulis | A stability result for the diffusion coefficient of the the heat operator defined on an unbounded guide[END_REF]. Let f ∈ H 1 0 (Ω L ). Denote w = e sφ(T /2) f and Qw = e sφ(T /2) P (e -sφ(T /2) w). So we get Qw = P w -sw∇φ(T /2) • ∇ u(T /2). Therefore we have

Ω L |Qw| 2 dx ≥ s 2 Ω L w 2 |∇φ(T /2) • ∇ u(T /2)| 2 dx -2s Ω L (P w)w(∇φ(T /2) • ∇ u(T /2)) dx Ω L |Qw| 2 dx ≥ s 2 λ 2 Ω L w 2 (φ(T /2)) 2 |∇d • ∇ u(T /2)| 2 dx -2sλ Ω L (∇w • ∇ u(T /2))wφ(T /2)(∇d • ∇ u(T /2)) dx. So Ω L |Qw| 2 dx ≥ s 2 λ 2 Ω L w 2 (φ(T /2)) 2 |∇d • ∇ u(T /2)| 2 dx -sλ Ω L φ(T /2)(∇d • ∇ u(T /2))(∇(w 2 ) • ∇ u(T /2)) dx.

Thus integrating by parts

Ω L |Qw| 2 dx ≥ s 2 λ 2 Ω L w 2 (φ(T /2)) 2 |∇d • ∇ u(T /2)| 2 dx +sλ Ω L w 2 ∇ • (φ(T /2)(∇d • ∇ u(T /2))∇ u(T /2)) dx and Ω L e 2sφ(T /2) |P f | 2 dx = Ω L |Qw| 2 dx ≥ s 2 λ 2 Ω L e 2sφ(T /2) f 2 (φ(T /2)) 2 |∇d • ∇ u(T /2)| 2 dx +sλ Ω L e 2sφ(T /2) f 2 ∇ • (φ(T /2)(∇d • ∇ u(T /2))∇ u(T /2)) dx.
And we can conclude for s sufficiently large.

The strong positivity assumption (3.32) is frequently involved in inverse problems and is removed in [START_REF] Cardoulis | A stability result for the diffusion coefficient of the the heat operator defined on an unbounded guide[END_REF] for one equation by the construction of an adapted control. Now we state the third result. Theorem 3.3. Let l > 0. Let T > 0, L > l and a ∈ R n \ Ω satisfying the conditions of Proposition 2.1. Assume that Assumptions (3.31) and (3.32) hold. We also make the following hypothesis

| v(., T 2 )| ≥ R in Ω L for some R > 0. (3.33)
If g = g and A 3 = A 3 on ∂Ω ∩ ∂Ω L , then there exists a sufficiently small number δ 0 such that if δ ∈ (0, δ 0 ),

1 k=0 ∂ k t (u -u)(., T 2 ) 2 H 2 (Ω L ) + 1 k=0 ∂ k t (v -v)(., T 2 ) 2 H 3 (Ω L ) + γ L ×(0,T ) 2 k=0 (|∂ ν (∂ k t (u -ũ))| 2 + |∂ ν (∂ k t (v -ṽ))| 2 dσ dt ≤ δ then the following Hölder stability estimate holds b -b 2 L 2 (Ω l ) + A 3 -A 3 2 (L 2 (Ω l )) n ≤ Kδ κ for all δ ∈ (0, δ 0 ). (3.34)
Here, K > 0 and κ ∈ (0, 1) are two constants depending on R, r, L, l, M 0 , M 1 , M 2 , T and a.

Proof. As in Theorem 3.1 denote

y = u -u, y 0 = χηy, y 1 = ∂ t y 0 , y 2 = ∂ t y 1 , z = v -v, z 0 = χηz, z 1 = ∂ t z 0 and z 2 = ∂ t z 1 .
Then (y 0 , z 0 ) satisfies

   ∂ t y 0 -∆y 0 + ay 0 + bz 0 + A 1 • ∇y 0 + A 2 • ∇z 0 = ξ 1 in Q L , ∂ t z 0 -∆z 0 + cy 0 + dz 0 + A 3 • ∇y 0 + A 4 • ∇z 0 = ξ 2 in Q L ,
ξ 2 := χη( A 3 -A 3 ) • ∇ u + (∂ t η)χz -(∆χ)ηz -2∇χ • ∇(ηz) + ηyA 3 • ∇χ + ηzA 4 • ∇χ. Then ξ 2 = η∇(χ( g -g)) • ∇ u -η( g -g)∇χ • ∇ u + (∂ t η)χz -(∆χ)ηz -2∇χ • ∇(ηz) + ηyA 3 • ∇χ + ηzA 4 • ∇χ.
• First step: We evaluate (3.35) for t = T 2 and we get ∂ t y 0 (T /2) -∆y 0 (T /2) + ay 0 (T /2)

+ bz 0 (T /2) + A 1 • ∇y 0 (T /2) + A 2 • ∇z 0 (T /2) = χ( b -b) v(T /2) -(∆χ)y(T /2) -2∇χ • ∇y(T /2) + y(T /2)A 1 • ∇χ + z(T /2)A 2 • ∇χ (3.36) and ∂ t z 0 (T /2) -∆z 0 (T /2) + cy 0 (T /2) + dz 0 (T /2) + A 3 • ∇y 0 (T /2) + A 4 • ∇z 0 (T /2) = P (χ( g -g))-( g -g)∇χ•∇ u(T /2)-(∆χ)z(T /2)-2∇χ•∇z(T /2)+y(T /2)A 3 •∇χ+z(T /2)A 4 •∇χ
(3.37) with P the operator defined in Lemma 3.3. From (3.36) we have

Ω L e 2sφ( T 2 ) χ 2 |b -b| 2 | v( T 2 )| 2 dx ≤ C Ω L e 2sφ( T 2 ) |∂ t y 0 ( T 2 )| 2 dx +Ce 2sd2 ( z 0 (T /2) 2 H 1 (Ω L ) + y 0 (T /2) 2 H 2 (Ω L ) + y(T /2) 2 H 1 (Ω L ) + z(T /2) 2 L 2 (Ω L ) ). So Ω L e 2sφ( T 2 ) χ 2 |b -b| 2 | v( T 2 )| 2 dx ≤ Ce 2sd2 F 1 (T /2) + C Ω L e 2sφ( T 2 ) |∂ t y 0 ( T 2 )| 2 dx with F 1 (T /2) = y(T /2) 2 H 2 (Ω L ) + z(T /2) 2 H 1 (Ω L ) . Then, applying Lemma 3.1 we get Ω L e 2sφ( T 2 ) χ 2 |b-b| 2 | v( T 2 )| 2 dx ≤ Ce 2sd2 F 1 (T /2)+Cs Q L e 2sφ φ 3 |y 1 | 2 dx dt+ C s Q L e 2sφ φ 3 |y 2 | 2 dx dt.
(3.38) Moreover using Lemma 3.3 for (3.37) we have

s 2 Ω L e 2sφ(T /2) χ 2 ( g -g) 2 dx ≤ C Ω L e 2sφ(T /2) |P (χ( g -g))| 2 dx ≤ Ce 2sd1 + C Ω L e 2sφ( T 2 ) |∂ t z 0 ( T 2 )| 2 dx +Ce 2sd2 ( z 0 (T /2) 2 H 2 (Ω L ) + y 0 (T /2) 2 H 1 (Ω L ) + y(T /2) 2 L 2 (Ω L ) + z(T /2) 2 H 1 (Ω L ) ). Applying again Lemma 3.1 we get s 2 Ω L e 2sφ(T /2) χ 2 ( g -g) 2 dx ≤ Ce 2sd1 + Ce 2sd2 F 2 (T /2) + Cs Q L e 2sφ φ 3 |z 1 | 2 dx dt + C s Q L e 2sφ φ 3 |z 2 | 2 dxdt (3.39) with F 2 (T /2) = y(T /2) 2 H 1 (Ω L ) + z(T /2) 2 H 2 (Ω L ) . From (3.38)-(3.39) we obtain Ω L e 2sφ( T 2 ) χ 2 |b -b| 2 | v( T 2 )| 2 dx + Ω L e 2sφ(T /2) χ 2 ( g -g) 2 dx ≤ C s 2 e 2sd1 + Ce 2sd2 F 3 (T /2) + Cs Q L e 2sφ φ 3 (|y 1 | 2 + |z 1 | 2 )dxdt + C s Q L e 2sφ φ 3 (|y 2 | 2 + |z 2 | 2 )dxdt (3.40) with F 3 (T /2) = y(T /2) 2 H 2 (Ω L ) + z(T /2) 2 H 2 (Ω L )
. Using now Assumption (3.33), we get from (3.40) and for s sufficiently large

Ω L e 2sφ( T 2 ) χ 2 ((b -b) 2 + ( g -g) 2 ) dx ≤ C s 2 e 2sd1 + Ce 2sd2 F 3 (T /2) +Cs Q L e 2sφ φ 3 (|y 1 | 2 + |z 1 | 2 ) dx dt + C s Q L e 2sφ φ 3 (|y 2 | 2 + |z 2 | 2 ) dx dt. (3.41)
• Second step: As in Theorem 3.1, now we use the Carleman inequalities (2.5) for y i and z i , i = 1, 2. Recall that φ ≤ φ(T /2) so we get for s sufficiently large

I(y i ) + I(z i ) ≤ C Ω L e 2sφ(T /2) (|∇(χ( g -g))| 2 + χ 2 |b -b| 2 ) dx + Cs 3 e 2sd1 +Cs γ L ×(0,T ) e 2sφ (|∂ ν y i | 2 + |∂ ν z i | 2 ) dσ dt. Thus s 3 Q L e 2sφ φ 3 (|y i | 2 + |z i | 2 ) dx dt ≤ C Ω L e 2sφ(T /2) (|∇(χ( g -g))| 2 + χ 2 |b -b| 2 ) dx +Cs 3 e 2sd1 + Cs γ L ×(0,T ) e 2sφ (|∂ ν y i | 2 + |∂ ν z i | 2 ) dσ dt. (3.42)
Therefore, from (3.41) and (3.42), we get for s sufficiently large

Ω L e 2sφ( T 2 ) χ 2 ((b -b) 2 + ( g -g) 2 ) dx ≤ Ce 2sd2 F 3 (T /2) + Cse 2sd1 + C s 2 Ω L e 2sφ(T /2) (|∇(χ( g -g))| 2 + χ 2 |b -b| 2 ) dx + C s γ L ×(0,T ) e 2sφ 2 i=1 (|∂ ν y i | 2 + |∂ ν z i | 2 ) dσ dt.
Thus we have for s sufficiently large

Ω L e 2sφ( T 2 ) χ 2 ((b-b) 2 +( g-g) 2 ) dx ≤ Ce 2sd2 F 4 (T /2)+Cse 2sd1 + C s 2 Ω L e 2sφ(T /2) |∇(χ(g-g))| 2 | dx (3.43) with F 4 (T /2) = F 3 (T /2) + γ L ×(0,T ) 2 k=0 (|∂ ν ∂ k t y| 2 + |∂ ν ∂ k t z| 2 ) dσ dt.
• Third step: We apply the same ideas for ∇(χ( g -g)). For any integer 1 ≤ i ≤ n, taking the space derivative with respect to x i in (3.37), we obtain

3. 2 .

 2 The second result. Consider now (u, v) (resp. ( u, v)) a strong solution of (1.1) associated with (a, b, c, d, u 0

) with f 5 =

 5 (∂ t η)χy 4 -(∆χ)ηy 4 -2η∇χ • ∇y 4 and f 6 = (∂ t η)χz 4 -(∆χ)ηz 4 -2η∇χ • ∇z 4 . Due to the truncation functions, we can apply the Carleman estimates for y 5 and z 5 and now we estimate I(y 5 ) + I(z 5 ) with (2.5). We have I(y 5 )+I(z 5 ) ≤ C Q L e 2sφ ((Ay 5 ) 2 +(Az 5 ) 2 )dxdt+Cs 3 e 2sd1 +Cs γ L ×(0,T ) e 2sφ (|∂ ν y 5 | 2 +|∂ ν z 5 | 2 )dσdt.

  3 we get from (3.21)-(3.22) and for s sufficienlty large I(y 5 ) + I(z 5 ) ≤ Cs 3 e 2sd1 + Cs

2 .

 2 .30) From (3.29)-(3.30) we end the proof of Thereom 3.Remark 2. • First note that our stability results (3.3) and (3.15) are obtained on Ω l for the lefthand term while the observation data G(T /2) and F 3 (T /2) are required on Ω L for the right-hand term of (3.3), (3.15). • Second we have used Lemma 3.2 instead of Lemma 3.1 in the proof of Theorem 3.2 in order to avoid a third derivative with respect to t in the observation terms. Indeed, if we no longer used Lemma 3.2 in the proof of Theorem 3.2, we could use a modified version of Lemma 3.1: applying (3.1) with w = e sφ χηz, we could obtain the following inequality

  y 0 = z 0 = 0 on ∂Ω L × (0, T ) (3.35) with ξ 1 := χη( b -b) v + (∂ t η)χy -(∆χ)ηy -2∇χ • ∇(ηy) + ηyA 1 • ∇χ + ηzA 2 • ∇χ and

∂ 2 Ω 2 Ω 2 Ω(|∇z j | 2 +L e 2sφ 2 j=1 (z 2 j + y 2 j2sφ 2 j=1(|∂ ν z j | 2 + 2 Ω L e 2sφ(T / 2 )) e 2sφ 2 j=1(|∂ ν z j | 2 +Therefore for s sufficienlty large s 2 Ω2sφ 2 j=1(|∂ ν z j | 2 +

 2222222222222222 t ∂ xi z 0 (T /2) -∆∂ xi z 0 (T /2) + ∂ xi (cy 0 (T /2) + dz 0 (T /2) + A 3 • ∇y 0 (T /2) + A 4 • ∇z 0 (T /2)) = P (∂ xi (χ( g -g))) + ∇(χ( g -g)) • ∇(∂ xi u(T /2)) -∂ xi (( g -g)∇χ • ∇ u(T /2))-∂ xi ((∆χ)z(T /2) -2∇χ • ∇z(T /2) + y(T /2)A 3 • ∇χ + z(T /2)A 4 • ∇χ).(3.44) We can apply again Lemma 3.3: there exists a positive constant C such that for s sufficiently large,s L e 2sφ(T /2) ∂ xi (χ( g -g)) 2 dx ≤ C Ω L e 2sφ(T /2) (P (∂ xi (χ( g -g)))) 2 dx.Thus, using (3.44) we obtains L e 2sφ(T /2) (∂ xi (χ( g -g))) 2 dx ≤ Ce 2sd2 F 5 (T /2) + Ce 2sd1 + C Ω L e 2sφ(T /2) |∂ xi z 1 (T /2)| 2 dx +C Ω L e 2sφ(T /2) |∇(χ(g -g))| 2 | dx with F 5 (T /2) = z(T /2) 2 H 3 (Ω L ) + y(T /2) 2 H 2 (Ω L ) . So using Lemma 3.1 we get s L e 2sφ(T /2) (∂ xi (χ( g -g))) 2 dx ≤ Ce 2sd2 F 5 (T /2) + Ce 2sd1 + C Ω L e 2sφ(T /2) |∇(χ(g -g))| 2 | dx +Cs Q L e 2sφ (∂ xi z 1 ) 2 dx dt + C s Q L e 2sφ (∂ xi z 2 ) 2 dx dt. (3.45)Moreover by the Carleman inequality (2.5), we have for j = 1, 2,s Q L e 2sφ (z 2 j + |∇z j | 2 ) dx dt ≤ C Q L e 2sφ |Az j | 2 dx dt + Cs 3 e 2sd1 z j 2 H 2,1 (Q L ) +Cs γ L ×(0,T ) |∂ ν z j | 2 e 2sφ dσ dt. Thus s Q L e 2sφ (z 2 j +|∇z j | 2 ) dx dt ≤ C Q L e 2sφ (y 2 j +|∇y j | 2 +z 2 j +|∇z j | 2 ) dx dt+ Q L e 2sφ |∇(χ( g-g))| 2 dx dt +Cs 3 e 2sd1 + Cs γ L ×(0,T ) |∂ ν z j | 2 e 2sφ dσ dt. (3.46)By the same way we obtains Q L e 2sφ (y 2 j +|∇y j | 2 ) dx dt ≤ C Q L e 2sφ (y 2 j +|∇y j | 2 +z 2 j +|∇z j | 2 ) dx dt+ Q L e 2sφ (χ( b-b)) 2 dxdt +Cs 3 e 2sd1 + Cs γ L ×(0,T ) |∂ ν y j | 2 e 2sφ dσ dt. (3.47) From (3.46) and (3.47) we deduces Q L e 2sφ (z 2 j + y 2 j + |∇z j | 2 + |∇y j | 2 )dxdt ≤ C Q L e 2sφ (y 2 j + |∇y j | 2 + z 2 j + |∇z j | 2 )dxdt + Cs 3 e 2sd1 +C Q L e 2sφ (|∇(χ( g -g))| 2 + (χ( b -b)) 2 ) dx dt + Cs γ L ×(0,T ) e 2sφ (|∂ ν z j | 2 + |∂ ν y j | 2 ) dσ dt. (3.48)Since φ ≤ φ(T /2), (3.48) implies for s sufficiently larges Q L e 2sφ (z 2 j + y 2 j + |∇z j | 2 + |∇y j | 2 ) dx dt ≤ Cs 3 e 2sd1 +C Ω L e 2sφ(T /2) (|∇(χ( g -g))| 2 + (χ( b -b)) 2 ) dx + Cs γ L ×(0,T ) e 2sφ (|∂ ν z j | 2 + |∂ ν y j | 2 ) dσ dt |∇y j | 2 ) dx dt ≤ s Q + |∇z j | 2 + |∇y j | 2 ) dx dt ≤ Cs 3 e 2sd1 + Cs γ L ×(0,T ) e |∂ ν y j | 2 ) dσ dt +C Ω L e 2sφ(T /2) (|∇(χ( g -g))| 2 + (χ( b -b)) 2 ) dx. (3.49) Using inequalities (3.45) for 1 ≤ i ≤ n and (3.49), we get s |∇(χ( g -g))| 2 dx ≤ Ce 2sd2 F 5 (T /2)+C Ω L e 2sφ(T /2) [|∇(χ(g -g))| 2 |+|χ(b-b)| 2 ] dx +Cs 3 e 2sd1 + Cs γ L ×(0,T |∂ ν y j | 2 ) dσ dt. L e 2sφ(T /2) |∇(χ( g -g))| 2 dx ≤ Ce 2sd2 F 5 (T /2) + C Ω L e 2sφ(T /2) (χ(b -b)) 2 dx +Cs 3 e 2sd1 + Cs γ L ×(0,T ) e |∂ ν y j | 2 ) dσ dt. (3.50)

• Fourth step: Now we gather (3.43) and (3.50) and we get for s sufficiently large

with F 6 (T /2) = F 4 (T /2) + F 5 (T /2). Moreover, since e 2sd0 ≤ e 2sφ(T /2) in Ω l and χ = 1 in Ω l , we deduce that

). This concludes the proof of Theorem 3.3.

Remark 3. In Theorem 3.3 we have presented the case of determining the coefficients b and A 3 . Of course we could obtain similar results for at least two coefficients between a, b, c, d, A 1 , A 2 , A 3 , A 4 . If we want to determine A 1 and A 3 , we only have to assume that Assumption (3.32) holds intead of (3.32)- (3.33). If we want to estimate the coefficients A 2 and A 3 , we still have to assume the hypothesis (3.32) satisfied but in this case, we should also assume that the following hypothesis

holds. Note also that the last item of Remark 1 still holds for (3.34). To conclude, if we would like to determine more than two coefficients, we could procede with the same method used in Theorem 3.2.


Laure Cardoulis
email: laure.cardoulis@univ-amu.fr

EXISTENCE OF SOLUTIONS FOR A SYSTEM INVOLVING THE (2,Q)-LAPLACIAN OPERATOR IN A BOUNDED DOMAIN

Keywords: 1991 Mathematics Subject Classification. (AMS) 35J10. Key words and phrases. (2, q)-Laplacian operator, system, existence of solutions

In this paper we study the existence of a non trivial weak solution for a system involving the Laplacian operator and the q-Laplacian operator in a bounded domain Ω of R N with sufficiently smooth boundary.

Introduction

We consider in this paper the following system for i = 1, • • • , m,

-∆u i -∆ q u i + w i |u i | q-2 u i + m j=1 a ij u j = g i (., u 1 , • • • , u m ) in Ω, u i = 0 on ∂Ω.
(S, q, g)

where Ω is a bounded domain with sufficiently smooth boudary, Ω ⊂ R N .

We recall that the q-Laplacian operator is defined by ∆ q φ = div(|∇φ| q-2 ∇φ) and we suppose q > 2 in the whole paper. We study the existence of a weak non-trivial solution u = (u 1 , • • • , u m ) ∈ W for the system (S, q, g) where the variational space is W = (W 1,q 0 (Ω)) m , W 1,q 0 (Ω) being the usual Sobolev space endowed with the norm φ 1,q 0 (Ω) = ( Ω |∇φ| q ) 1/q . We also denote H = (W 1,2 0 (Ω)) m and . W , . H , the norms on W and H ( u W = ( m i=1 u i q W 1,q 0 (Ω) ) 1/q ). We assume throughout all the paper that the bounded functions a ij , w i (for i, j = 1, • • • , m) satisfy the following hypothesis Assumption 1.1.

i): a ij , w i ∈ L ∞ (Ω), a ii ≥ 0, w i ≥ 0 a. e. on Ω. ii): The matrix A = (a ij ) is symmetric and satisfies t ξAξ ≥ 0 for all t ξ = (ξ 1 ,

• • • , ξ m ) ∈ R m .
Note that the above Assumption 1.1ii) is satisfied when the matrix A is a positive definite one. Introduce now the following functionals for u = (u

1 , • • • , u m ) ∈ W H 1 (u) = m i=1 Ω (|∇u i | 2 + a ii u 2 i + m j=1,i =j a ij u j u i ), (1.1)
and

H 2 (u) = m i=1 Ω (|∇u i | q + w i |u i | q ). (1.2) Since A is symmetric then H 1 (u) = m i=1 Ω (|∇u i | 2 + a ii u 2 i + 2 m j=1
,i<j a ij u j u i ). Note that (H 1 (u)) 1/2 and (H 2 (u)) 1/q define norms on H and W equivalent to the norms . H and . W respectively.

We consider different cases for the functions g i : in the second section we deal with g i (., u 1 , • • • , u m ) := h i ∈ W -1,q (Ω) the dual space of W 1,q 0 (Ω) with 1 q + 1 q = 1. In the third section, we define g i (., u 1 , • • • , u m ) := m i |u i | q-2 u i where the functions m i are bounded and indefinite. In the fourth section we consider the case g i (., u 1 , • • • , u m ) := λf i |u i | γ-2 u i where the functions f i are still bounded and undefinite, λ is a positive real parameter and the coefficient γ satisfies some hypotheses in which γ < q.

In each of the precedent cases, the system (S, q, g) will be rewritten under a variational form with I(u) an adapted Euler functional defined in W and the existence of weak solutions for the system (S, q, g) will be equivalent to the existence of critical points for this functional I. In the second and third sections, we will mimimize the Euler functional I using either standard arguments (cf. Theorem I.1.2 in [START_REF] Struwe | Variational Methods, Application to nonlinear partial differential equations and Hamiltonian systems[END_REF]) or the Moutain-Pass Theorem. In the third section, we will use the principal eigenvalue λ 1,q,ρ of the q-Laplacian operator associated with a weight ρ whereas in the fourth section we will define a characteristic value λ + 1 (see (4.7)).

Equations and systems with the p-Laplacian have been widely studied for the existence of solutions or the maximum and antimaximum principles (see for examples [START_REF] Boccardo | Existence of solutions for some nonlinear cooperative systems[END_REF][START_REF] Drábek | Existence and uniqueness of positive solutions for some quasilinear elliptic problems[END_REF][START_REF] Clément | Existence of positive solutions for a nonvariational quasilinear elliptic system[END_REF][START_REF] Fleckinger | Antimaximum principle in R N : local versus global[END_REF][START_REF] Fleckinger | Uniqueness and positivity for solutions of equations with the p-Laplacian[END_REF][START_REF] Giacomoni | Régularité höldérienne pour des équations quasilinéaires elliptiques singulières[END_REF], see also [START_REF] Il'yasov | Hopf maximum principle violation for elliptic equations with non-Lipschitz nonlinearity[END_REF] for the fibering procedure). These last few years, equations with the (p,q)-Laplacian have been studied (see for examples [START_REF] Bobkov | On positive solutions for (p,q)-Laplace equations with two parameters[END_REF][START_REF] Candito | On a class of critical (p,q)-Laplacian problems[END_REF][START_REF] Kang | Positive solutions to the weighted critical quasilinear problems[END_REF][START_REF] Tanaka | Generalized eigenvalue problems for (p,q)-Laplacian with indefinite weight[END_REF][START_REF] Yin | A class of pq-Laplacian type equation with concaveconvex nonlinearities in bounded domain[END_REF] in a bounded domain and [START_REF] Chaves | Existence of a nontrivial solutions for the (p,q)-Laplacian in R N without the Ambrosetti-Rabinowitz condition[END_REF] in R N ). Authors study the existence of solutions (sometimes the sign of these solutions and generalized eigenvalue problems) mainly by minimization of the energy functional either by standard arguments or the mountain-pass geometry, also by using the method of sub-and super-solutions. The case of the (2,q)-Laplacian arises in quantum physics (see [START_REF] Benci | Solitons in several space dimensions: Derricks problem and infinitely many solutions[END_REF]). A few systems with two equations have been studied (see for example [START_REF] Li | Multiple solutions with constant sign for a (p, q)-elliptic system Dirichlet problem with product nonlinear term[END_REF] for a system with two equations, one with the p-Laplacian and the other one with the q-Laplacian ; see also [START_REF] Yin | Existence of multiple positive solutions for a pq-Laplacian system with critical nonlinearities[END_REF] for a system of two equations with the (p,q)-Laplacian with critical nonlinearities) but as far as we know, there is no system with m equations for the (2,q)-Laplacian studied yet. This paper is organised as follows: in section 2, we use standard arguments for minimizing the functional I when we consider the case where

g i (., u 1 , • • • , u m ) := h i ∈ W -1,q (Ω). In section 3 (in the case of g i (., u 1 , • • • , u m ) := m i |u i | q-2 u i and q < 2 * where 2 * = 2N N -2 if N > 2 and 2 * = ∞ if N ≤ 2)
, first we recall some results of the existence of the principal eigenvalue for the q-Laplacian operator associated with a bounded weight (and the existence of a positive eigenfunction associated with). Then we use the Mountain-Pass Theorem in order to get the existence of a non-trivial solution for our system. Finally in section 4 (when

g i (., u 1 , • • • , u m ) := λf i |u i | γ-2 u i with 2 < γ < q and γ < 2 * where 2 * = 2N N -2 if N > 2 and 2 * = ∞ if N ≤ 2)
, first we follow a method introduced by Cherfils-Il'Yasov in [START_REF] Cherfils | On the stationary solutions of generalized reaction diffusion equations with p-q Laplacian[END_REF] for one equation involving the (p-q)-Laplacian operator to define a characteristic value λ + 1 . Then we get the existence of a non-trivial solution by means of global minimization of the Euler functional.

First case: g

i (., u 1 , • • • , u m ) := h i ∈ W -1,q (Ω)
In this case the system (S, q, g) is rewritten under the following form

-∆u i -∆ q u i + w i |u i | q-2 u i + m j=1 a ij u j = h i in Ω, u i = 0 on ∂Ω, (2.1)
with

h i ∈ W -1,q (Ω) for each i = 1, • • • , m.
Recall that -∆ q may be seen acting from W 1,q 0 (Ω) into W -1,q (Ω) with 1 q + 1 q = 1 by

< -∆ q φ, ψ > q ,q = Ω |∇φ| q-2 ∇φ • ∇ψ for all φ, ψ ∈ W 1,q 0 (Ω)
(see [START_REF] Dinca | Variational and topological methods for Dirichlet problems with p-Laplacian[END_REF][START_REF] Lions | Quelques méthodes de résolution des problèmes aux limites non linéaires[END_REF]) where < ., . > q ,q denotes the duality mapping between W -1,q (Ω) and W 1,q 0 (Ω). Therefore the Euler functional is, for u

= (u 1 , • • • , u m ) ∈ W, I(u) = 1 2 H 1 (u) + 1 q H 2 (u) - m i=1 < h i , u i > q ,q . (2.
2)

The result of the existence of solution for the system (2.1) is the following.

Theorem 2.1. Assume that Assumption 1.1 is satisfied and that h i ∈ W -1,q (Ω)

for each i = 1, • • • , m.
Then the system (2.1) has a unique solution.

Proof. The functional I : W → R defined by (2.2) is weakly lower semi-continuous by the compactness of the embedding of W to (L q (Ω)) m and (L 2 (Ω)) m and of class C 1 on W. Moreover this functional I is also coercive. Indeed by the Young's inequality we have

| < h i , u i > q ,q | ≤ h i W -1,q (Ω) u i W 1,q 0 (Ω) ≤ 1 2q u i q W 1,q 0 (Ω) + C h i q W -1,q (Ω)
with C > 0, C independent of u. And since H 1 (u) ≥ 0 and H 2 (u) ≥ u W we get that

I(u) ≥ 1 2q u W -C m i=1 h i q W -1,q (Ω) .
Therefore the functional I has a gobal minimizer (cf.[18, Theorem I.1.2]) and the system (2.1) has a solution. Let us prove now the uniqueness of the solution. Suppose on the contrary that there exist two distinct solutions u

= (u 1 , • • • , u m ) ∈ W and v = (v 1 , • • • , v m ) ∈ W for (2.1), so there exists k such that u k = v k . Since (I (u) -I (v)) • (u -v) = I (u) • u -I (v) • u -I (u) • v + I (v) • v = 0, we have m i=1 Ω |∇u i | 2 + m i,j=1 Ω a ij u j u i + m i=1 Ω (|∇u i | q + w i |u i | q ) - m i=1 Ω ∇v i • ∇u i - m i,j=1 Ω a ij v j u i - m i=1 Ω (|∇v i | q-2 ∇v i • ∇u i + w i |v i | q-2 v i u i ) - m i=1 Ω ∇u i • ∇v i - m i,j=1 Ω a ij u j v i - m i=1 Ω (|∇u i | q-2 ∇u i • ∇v i + w i |u i | q-2 u i v i ) + m i=1 Ω |∇v i | 2 + m i,j=1 Ω a ij v j v i + m i=1 Ω (|∇v i | q + w i |v i | q ) = 0. So we get m i=1 Ω ∇u i • (∇u i -∇v i ) + m i,j=1 Ω a ij u j (u i -v i ) + m i=1 Ω |∇u i | q-2 ∇u i • (∇u i -∇v i ) + m i=1 Ω w i |u i | q-2 u i (u i -v i ) - m i=1 Ω ∇v i • (∇u i -∇v i ) - m i,j=1 Ω a ij v j (u i -v i ) - m i=1 Ω |∇v i | q-2 ∇v i • (∇u i -∇v i ) - m i=1 Ω w i |v i | q-2 v i (u i -v i ) = 0. Thus m i=1 Ω |∇u i -∇v i | 2 + m i=1 Ω (|∇u i | q-2 ∇u i -|∇v i | q-2 ∇v i ) • (∇u i -∇v i ) + m i,j=1 Ω a ij (u j -v j )(u i -v i ) + m i=1 Ω w i (|u i | q-2 u i -|v i | q-2 v i )(u i -v i ) = 0.
The last equality can be rewritten under the following form with the duality product

m i=1 < -∆u i + ∆v i , u i -v i > 2,2 + m i=1 < -∆ q u i + ∆ q v i , u i -v i > q ,q + m i,j=1 < a ij (u j -v j ), u i -v i > 2,2 + m i=1 < w i (|u i | q-2 u i -|v i | q-2 v i ), u i -v i > q ,q = 0.
Moreover a consequence of the strict convexity of the spaces W 1,2 0 (Ω) and W 1,q 0 (Ω) is that the duality mappings -∆ and -∆ q are strictly monotone. So from

u k = v k we get < -∆u k + ∆v k , u k -v k > 2,2 > 0, and
< -∆ q u k +∆ q v k , u k -v k > q ,q ≥ ( u k q-1 W 1,q 0 (Ω) -v k q-1 W 1,q 0 (Ω) )( u k W 1,q 0 (Ω) -v k W 1,q 0 (Ω) ) ≥ 0 since x → x q-1 is increasing on [0, ∞) (and even < -∆ q u k + ∆ q v k , u k -v k > q ,q > 0 from [8, Proposition 1]). Thus m i=1 < -∆u i + ∆v i , u i -v i > 2,2 + m i=1 < -∆ q u i + ∆ q v i , u i -v i > q ,q > 0.
Furthermore, since the function x → |x| q-2 x is increasing and w i ≥ 0, we have

m i=1 < w i (|u i | q-2 u i -|v i | q-2 v i ), u i -v i > q ,q ≥ 0. Finally from Assumption 1.1, m i,j=1 < a ij (u j -v j ), u i -v i > 2,2 ≥ 0.
Therefore we get a contradiction.

Remark: We can generalize Theorem 2.1 replacing the 2-Laplacian operator by the p-Laplacian with 2 < p < q, that for the following system

-∆ p u i -∆ q u i + w i |u i | q-2 u i + m j=1 a ij u j = h i in Ω, u i = 0 on ∂Ω,
and even for

-∆ p u i -∆ q u i + b i |u i | p-2 u i + w i |u i | q-2 u i + m j=1 a ij u j = h i in Ω, u i = 0 on ∂Ω,
under the additional hypothesis that the bounded functions b

i , i = 1, • • • , m are non-negative. 3. Second case: g i (., u 1 , • • • , u m ) := m i |u i | q-2 u i
In this section we assume that

Assumption 3.1. q < 2 * where 2 * = 2N N -2 if N > 2 and 2 * = ∞ if N ≤ 2
, and we rewrite the system (S, q, g) under the following form:

for i = 1, • • • , m, -∆u i -∆ q u i + w i |u i | q-2 u i + m j=1 a ij u j = m i |u i | q-2 u i in Ω, u i = 0 on ∂Ω. (3.1)
Note that the decomposition with the weights c i := m i -w i does not necessarily coincide with the decomposition

c i = c i+ -c i-where c i+ = max(c i , 0) and c i-= max(-c i , 0). Define now for u = (u 1 , • • • , u m ) ∈ W the functional M (u) = m i=1 Ω m i |u i | q . (3.2)
The Euler functional associated with (3.1) is consequently for u

= (u 1 , • • • , u m ) ∈ W, I(u) = 1 2 H 1 (u) + 1 q H 2 (u) - 1 q M (u). (3.3)
First let us recall the usual weighted eigenvalue problem for the q-Laplacian:

-∆ q u = λρ|u| q-2 u in Ω, u = 0 on ∂Ω, (3.4)
with a bounded weight function ρ and a real parameter λ. It is said that λ is an eigenvalue of the q-Laplacian associated with the weight ρ if (3.4) has a non-trivial solution u which is called an eigenfunction associated with λ. It is well known (see [START_REF] Anane | Simplicité et isolation de la première valeur propre du p-Laplacien avec poids[END_REF]) that if the Lebesgue measure of {x ∈ Ω, ρ(x) > 0} is positive, then the first positive eigenvalue λ 1,q,ρ of -∆ q with weight function ρ is obtained by the Rayleight quotient

λ 1,q,ρ = inf{ Ω |∇u| q Ω ρ|u| q ; u ∈ W 1,q 0 (Ω), Ω ρ|u| q > 0}. (3.5)
Moreover, λ 1,q,ρ has a positive eigenfunction φ 1,q,ρ ∈ C 1,αq 0

(Ω) (for some α q ∈ (0, 1)). Assume in this section that Assumption 3.2. i):

For all i = 1, • • • , m, m i ∈ L ∞ (Ω), ii): For all i = 1, • • • , m,
the real 1 is not an eigenvalue of the q-Laplacian with the weight m i -w i .

Assume also in this section that either Assumption 3.3 or Assumption 3.4 holds

Assumption 3.3. There exists k ∈ {1, • • • , m} such that: meas{x ∈ Ω, (m k -w k )(x) > 0} = 0 and λ 1,q,m k -w k < 1. Assumption 3.4. There exist k, l ∈ {1, • • • , m}, k = l such that: meas{x ∈ Ω, (m k -w k )(x) > 0} = 0 and λ 1,q,m k -w k + Ω (w l -m l )|φ 1,q,m k -w k | q < 0 with φ 1,q,m k -w k the normalized eigenfunction associated with λ 1,q,m k -w k .
Note that Assumption 3.4 is satisfed when

λ 1,q,m k -w k (m k -w k ) + w l -m l < 0 a. e. in Ω.
Our aim is to study the existence of a weak solution for the system (3.1) by minimizing the functional I defined by (3.3). As in section 2, the functional I is weakly lower semi-continous on W but may be no more coercive so we cannot use standard arguments for minimizing I. First, we prove that any Palais-Smale sequence is bounded in W and has a strong convergent subsequence. Then we are able to apply the Mountain-Pass Lemma and Assumptions 3.3 or 3.4 allow us to get a non-trivial solution.

We say that (u

n ) ⊂ W, u n = (u 1n , • • • , u mn ), is a Palais-Smale sequence if it satisfies the following conditions |I(u n )| ≤ D for all n ∈ N and I (u n ) W * → 0 as n → ∞ (3.6)
with some constant D > 0, W * being the dual space of W.

Lemma 3.1. Assume that Assumptions 1.1 and 3.2 are satisfied.

If (u n ) ⊂ W, u n = (u 1n , • • • , u mn ), is a Palais-Smale sequence, then (u n ) is bounded in W. Proof. Let (u n ) ⊂ W, u n = (u 1n , • • • , u mn
), be a Palais-Smale sequence. We want to prove that ( u n W ) n is bounded or equivalently that (H

2 (u n )) n is bounded. But 1 q H 2 (u n ) = I(u n )- 1 2 H 1 (u n )+ 1 q M (u n ) ≤ D+ 1 q M (u n ) ≤ D+C u n q (L q (Ω)) m (3.7)
with C a positive constant, C independent of u n (since the functions m i are bounded in the functional M (u) defined by (3.2)). So it is sufficient to show that ( u n (L q (Ω)) m ) is bounded. We adapt ideas from [START_REF] Tanaka | Generalized eigenvalue problems for (p,q)-Laplacian with indefinite weight[END_REF]. Assume on the contrary that

α n := u n (L q (Ω)) m → n→∞ ∞ (for a subsequence) and denote v n = 1 αn u n = (v 1n , • • • , v mn ). From (3.7), we deduce that ( v n W ) is bounded and from the com- pact embedding of W into (L q (Ω)) m we get the existence of v 0 = (v 01 , • • • , v 0m ) ∈ W such that (v n ) converges to v 0 ,
strongly in (L q (Ω)) m and weakly in W (for a subsequence).

• Now we prove that (v n ) converges strongly to v 0 in W. Indeed by taking

φ n := 1 α q-1 n (v n -v 0 ), we obtain I (u n ).φ n = 1 α q-1 n m i=1 Ω (∇u in .∇(v in -v 0i ) + a ii u in (v in -v 0i )) + 1 α q-1 n m i=1 Ω (|∇u in | q-2 ∇u in .∇(v in -v 0i ) + w i |u in | q-2 u in (v in -v 0i )) + 1 α q-1 n i,j;i =j Ω a ij u jn (v in -v 0i ) - 1 α q-1 n m i=1 Ω m i |u in | q-2 u in (v in -v 0i ). (3.8) But u n = α n v n so (3.8) becomes I (u n ).φ n = 1 α q-2 n m i=1 Ω (∇v in .∇(v in -v 0i ) + a ii v in (v in -v 0i )) + m i=1 Ω (|∇v in | q-2 ∇v in .∇(v in -v 0i ) + w i |v in | q-2 v in (v in -v 0i )) + 1 α q-2 n i,j;i =j Ω a ij v jn (v in -v 0i ) - m i=1 Ω m i |v in | q-2 v in (v in -v 0i ). (3.9) Note that |I (u n ).φ n | ≤ I (u n ) W * φ n W = I (u n ) W * 1 α q-1 n v n -v 0 W so I (u n ).φ n → n→∞ 0 from (3.6), α n → n→∞ ∞ and ( v n W ) bounded.
Moreover, since the functions a ij , w i , m i are bounded there exists a positive constant, denoting C at each step, such that

| Ω a ij v jn (v in -v 0i ))| ≤ C v jn L 2 (Ω) v in -v 0i L 2 (Ω) ≤ C v n W v n -v 0 (L q (Ω)) m and therefore Ω a ij v jn (v in -v 0i )) → 0 as n → ∞. (3.10)
By the same way, for

b i = w i or b i = m i , | Ω b i |v in | q-2 v in (v in -v 0i ))| ≤ C( Ω |v in | q ) q-1 q ( Ω |v in -v 0i | q ) 1/q ≤ C v n q-1 W v n -v 0 (L q (Ω)) m so Ω b i |v in | q-2 v in (v in -v 0i )) → 0 as n → ∞. (3.11)
Recall that < ., . > q ,q is the duality product between W -1,q (Ω) and W 1,q 0 (Ω) with 1 q + 1 q = 1. From (3.9), (3.10), (3.11), we deduce that

1 α q-2 n m i=1 < -∆v in , v in -v 0i > 2,2 + m i=1 < -∆ q v in , v in -v 0i > q ,q → 0 as n → ∞.
(3.12) Moreover we have (see also the proof of Theorem 2.1)

< -∆ q v in +∆ q v 0i , v in -v 0i > q ,q ≥ ( v in q-1 W 1,q 0 (Ω) -v 0i q-1 W 1,q 0 (Ω) )( v in W 1,q 0 (Ω) -v 0i W 1,q 0 (Ω) ) ≥ 0 (3.13) and < -∆v in +∆v 0i , v in -v 0i > 2,2 = v in -v 0i 2 W 1,2 0 (Ω) ≥ ( v in W 1,2 0 (Ω) -v 0i W 1,2 0 (Ω) ) 2 . ( 3
≤ m i=1 ( v in q-1 W 1,q 0 (Ω) -v 0i q-1 W 1,q 0 (Ω) )( v in W 1,q 0 (Ω) -v 0i W 1,q 0 (Ω) ) + 1 α q-2 n m i=1 ( v in W 1,2 0 (Ω) -v 0i W 1,2 0 (Ω) ) 2 ≤ 1 α q-2 n m i=1 < -∆v in , v in -v 0i > 2,2 + m i=1 < -∆ q v in , v in -v 0i > q ,q + m i=1 < ∆ q v 0i , v in -v 0i > q ,q + 1 α q-2 n m i=1 < ∆v 0i , v in -v 0i > 2,2 .
Because the right-hand side of the above estimate tends to 0 as n tends to infinity (from (3.12) and the weak convergence of (v n ) to v 0 in W ) we obtain that for

i = 1, • • • , m, v in W 1,q 0 (Ω) → v 0i W 1,q 0 (Ω)
as n → ∞ and therefore (v n ) strongly converges to v 0 in W.

• Finally, we prove that v 0i is a non-trivial solution of the eigenvalue problem of the q-Laplacian with weight m i -w i for at least one i.

Let φ = (φ 1 , • • • , φ m ) ∈ W. Taking 1 α q-1 n φ as a test function, since u n = α n v n , we have I (u n ). 1 α q-1 n φ = 1 α q-2 n m i=1 Ω (∇v in .∇φ i + m j=1 Ω a ij v jn φ i ) + m i=1 Ω (|∇v in | q-2 ∇v in .∇φ i + w i |v in | q-2 v in φ i -m i |v in | q-2 v in φ i ).
Letting n → ∞, we see that

for each i = 1, • • • , m, v 0i is a weak solution to -∆ q v 0i + w i |v 0i | q-2 v 0i = m i |v 0i | q-2 v 0i in Ω v 0i = 0 on ∂Ω . (3.15)
Since v n (L q (Ω)) m = 1 and (v n ) converges strongly to v 0 in W we get that v 0 W ≥ 1. Therefore there exists i such that v 0i is a non-trivial solution of (3.15). This contradicts Assumption 3.2. Proof. Let (u n ) be a Palais-Smale sequence in W , u n = (u 1n , • • • , u mn ). By Lemma 3.1, the sequence (u n ) is bounded in W. From the compact embedding of W 1,q (Ω) into L q (Ω) we get the existence of u 0 = (u 01 , • • • , u 0m ) ∈ W such that (u n ) converges to u 0 strongly in (L q (Ω)) m and weakly in W (for a subsequence still denoted by (u n )). We want to prove that u n W → u 0 W as n → ∞ and we proceed as in the proof of Lemma 3.1.

Since

|I (u n ).(u n -u 0 )| ≤ I (u n ) W * ( u n W + u 0 W ) we deduce that I (u n ).(u n -u 0 ) → 0 as n → ∞. (3.16)
But

I (u n ).(u n -u 0 ) = m i=1 Ω (∇u in • ∇(u in -u 0i ) + m j=1 a ij u jn (u in -u 0i )) + m i=1 Ω (|∇u in | q-2 ∇u in .∇(u in -u 0i ) + (w i -m i )|u in | q-2 u in (u in -u 0i )).
As in Lemma 3.1, denoting b i either w i or m i , we have for i

, j = 1, • • • , m,, Ω b i |u in | q-2 u in (u in -u 0i )) → 0 as n → ∞, Ω a ij u jn (u in -u 0i )) → 0 as n → ∞.
(3.17)

From (3.16) and (3.17), we get that

m i=1 < -∆u in , u in -u 0i > 2,2 + m i=1 < -∆ q u in , u in -u 0i > q ,q → 0 as n → ∞.
Moreover we have 0

≤ m i=1 ( u in q-1 W 1,q 0 (Ω) -u 0i q-1 W 1,q 0 (Ω) )( u in W 1,q 0 (Ω) -u 0i W 1,q 0 (Ω) ) + m i=1 ( u in W 1,2 0 (Ω) -u 0i W 1,2 0 (Ω) ) 2 ≤ m i=1 < -∆u in , u in -u 0i > 2,2 + m i=1 < -∆ q u in , u in -u 0i > q ,q + m i=1 < ∆ q u 0i , u in -u 0i > q ,q + m i=1 < ∆u i , u in -u 0i > 2,2 .
As in Lemma 3.1 we deduce that for i = 1, • • • , m, u in W 1,q 0 (Ω) → u 0i W 1,q 0 (Ω) as n → ∞ and therefore (u n ) strongly converges to u 0 in W.

So we can state the main result of this section Theorem 3.1. Assume that Assumptions 1.1, 3.1 and 3.2 are satisfied. Assume also that either Assumption 3.3 or 3.4 holds. Then the system (3.1) has a nontrivial solution in W.

Proof. The C 1 -functional I satisfies the Palais-Smale conditions and I(0) = 0.

• First, we claim that there exist positive constants ρ * > 0 and δ > 0 such that

I(u) ≥ δ for any u = (u 1 , • • • , u m ) ∈ W satisfying u W = ρ * . Let u = (u 1 , • • • , u m ) ∈ W. Put ρ = u W and note that H 1 (u) ≥ u 2 H and H 2 (u) ≥ ρ q . Moreover, since q < 2 * , for i = 1, • • • , m, | Ω m i |u i | q | ≤ ( Ω |m i | r ) 1/r ( Ω |u i | qt ) 1/t with 1 r + 1 t = 1 and s := qt < 2 * .
From the continous embedding of W 1,2 (Ω) ⊂ L s (Ω) we deduce the existence of a positive constant

C 1 such that | Ω m i |u i | q | ≤ C 1 u i q W 1,2 (Ω) . Thus |M (u)| ≤ C 1 u q H and I(u) ≥ 1 q ρ q + 1 2 u 2 H (1 - 2C 1 q u q-2 H
). Recall also that there exists a positive constant

C 2 > 0 such that u H ≤ C 2 u W for all u ∈ W. Therefore if ρ ≤ ρ * := 1 C2 q 2C1 1 q-2 , then 1 -2C1 q u q-2 H ≥ 1 -2C1
q (C 2 ρ) q-2 ≥ 0 and I(u) ≥ 1 q ρ q := δ.

• Assume here that Assumption 3.3 is satisfied with k = 1 for simplicity. Let φ 1,q,m1-w1 be the normalized eigenfunction associated with λ 1,q,m1-w1 (i. e. be such that Ω (m 1 -w 1 )|φ 1,q,m1-w1 | q = 1, we may choose such φ 1,q,m1-w1 because the equation (3.4) is homogeneous). Denote Φ q = (φ 1,q,m1-w1 , 0, • • • , 0) and take R sufficiently large such that RΦ q W > ρ * . We have from (3.4) and (3.5)

I(RΦ q ) = R 2 2 H 1 (Φ q ) + R q q Ω (|∇φ 1,q,m1-w1 | q + (w 1 -m 1 )|φ 1,q,m1-w1 | q ) = R 2 2 H 1 (Φ q ) + R q q Ω (λ 1,q,m1-w1 (m 1 -w 1 ) + w 1 -m 1 )|φ 1,q,m1-w1 | q .
So, since λ 1,q,m1-w1 < 1,

I(RΦ q ) = R 2 2 H 1 (Φ q ) + R q q (λ 1,q,m1-w1 -1) < 0
for R sufficiently large.Therefore we can apply the mountain-pass theorem to deduce that I has a non-trivial critical point which is a non-trivial weak solution of the system (3.1).

• Assume now that Assumption 3.4 is satisfied with k = 1 and l = 2 for simplicity. Denote again φ 1,q,m1-w1 the normalized eigenfunction associated with λ 1,q,m1-w1 such that Ω (m 1 -w 1 )|φ 1,q,m1-w1 | q = 1 and denote here Ψ q = (0, φ 1,q,m1-w1 , 0, • • • , 0). Take R sufficiently large such that RΨ q W > ρ * . We have here

I(RΨ q ) = R 2 2 H 1 (Ψ q ) + R q q Ω (|∇φ 1,q,m1-w1 | q + (w 2 -m 2 )|φ 1,q,m1-w1 | q ) = R 2 2 H 1 (Ψ q ) + R q q Ω (λ 1,q,m1-w1 (m 1 -w 1 ) + w 2 -m 2 )|φ 1,q,m1-w1 | q .
From Assumption 3.4, we get that I(RΨ q ) < 0 for R sufficiently large.Therefore, as in the precedent case, we apply the mountain-pass theorem and deduce that I has a non-trivial critical point.

Remark: As in section 2, we can generalize Theorem 3.1 replacing the 2-Laplacian operator by the p-Laplacian with 2 ≤ p < q for the following system

-∆ p u i -∆ q u i + b i |u i | p-2 u i + λw i |u i | q-2 u i + m j=1 a ij u j = λm i |u i | q-2 u i in Ω, u i = 0 on ∂Ω,
under the additional hypotheses that the bounded functions b i , i = 1, • • • , m are non-negative and λ is a real parameter. Then the hypothesis ii) in Assumption 3.2 is replaced by λ is not an eigenvalue of -∆ q associated with m i -w i for each i. Moreover the hypothesis λ 1,q,m k -w k < 1 in Assumption 3.3 is replaced by λ 1,q,m k -w k < λ.

Third case:

g i (., u 1 , • • • , u m ) := λf i |u i | γ-2 u i
In this section we rewrite the system (S, q, g) under the following form:

for i = 1, • • • , m, -∆u i -∆ q u i + w i |u i | q-2 u i + m j=1 a ij u j = λf i |u i | γ-2 u i in Ω, u i = 0 on ∂Ω. (4.1)
We assume throughout all this section that the indefinite bounded functions f i and the coefficients γ and q satisfy the following hypotheses

Assumption 4.1. i): 2 < γ < q, ii): γ < 2 * where 2 * = 2N N -2 if 2 < N and 2 * = ∞ if 2 ≥ N, iii): For each i = 1, • • • , m, f i ∈ L ∞ (Ω) and meas{x ∈ Ω, f i (x) > 0} = 0.
We also define the functionals

F (u) = m i=1 Ω f i |u i | γ (4.2)
and

I λ (u) = 1 2 H 1 (u) + 1 q H 2 (u) - λ γ F (u) (4.3)
where H 1 and H 2 are respectively defined by (1.1) and (1.2). We recall that we study here the existence of a weak non-trivial solution u = (u 1 , • • • , u m ) ∈ W for the system (4.1) with respect to the real positive parameter λ and that the existence of weak solutions for the system (4.1) is equivalent to the existence of critical points for the Euler functional I λ . The main result is the existence of a weak non-trivial solution for the system (4.1) associated with λ > λ + 1 where λ + 1 is defined by (4.7). For the first part of this section we follow a method developed by Cherfils-Il'Yasov in [START_REF] Cherfils | On the stationary solutions of generalized reaction diffusion equations with p-q Laplacian[END_REF] for one equation with the (p,q)-Laplacian operator. This method is based on proving the existence of solution for λ = λ + 1 then on applying the mountain-pass theorem for λ > λ + 1 . Although we also could apply the mountain-pass theorem for our case, we will use in fact standard arguments to minimize the functional I λ .

In section 4.1 we present some preliminary results: we define λ + 1 and we prove the existence of a solution for the system (4.1) for λ = λ + 1 . The section 4.2 is devoted to the main theorem of the existence of a solution for the system (4.1) associated with λ > λ + 1 .

4.1. Some preliminaries results. As in [START_REF] Cherfils | On the stationary solutions of generalized reaction diffusion equations with p-q Laplacian[END_REF] we define for λ > 0, t > 0 and u ∈ W, Ĩλ (t, u) = I λ (tu).

Lemma 4.1. Assume that Assumptions 1.1, 4.1 i), 4.1 iii) are satisfied. For given u in W, u = 0 such that F (u) = 0, the unique solution (t(u), λ(u)) of the system

∂ ∂t Ĩλ (t, u) = 0 ∂ 2 ∂t 2 Ĩλ (t, u) = 0 is given by t(u) = γ -2 q -γ 1 q-2 H 1 (u) H 2 (u) 1 q-2 > 0, λ(u) = C q,γ H 1 (u) α H 2 (u) 1-α F (u) (4.4) with α = q -γ q -2 , C q,γ = q -2 (q -γ) α (γ -2) 1-α . (4.5)
Proof. The system (S)

∂ ∂t Ĩλ (t, u) = 0 ∂ 2 ∂t 2 Ĩλ (t, u) = 0 is equivalent to the system tH 1 (u) + t q-1 H 2 (u) -λt γ-1 F (u) = 0 H 1 (u) + (q -1)t q-2 H 2 (u) -λ(γ -1)t γ-2 F (u) = 0
and to the following system

H 1 (u) + t q-2 H 2 (u) -λt γ-2 F (u) = 0 H 1 (u) + (q -1)t q-2 H 2 (u) -λ(γ -1)t γ-2 F (u) = 0 . Therefore (q -2)t q-2 H 2 (u) -λ(γ -2)t γ-2 F (u) = 0. (4.6)
Note that the system (S) is not solvable in the case where u ∈ W, u = 0 satisfies F (u) = 0 (since if u = 0, then H 2 (u) = 0 and from (4.6) we deduce F (u) = 0). We deduce that

λ = (q -2)t q-2 H 2 (u) (γ -2)t γ-2 F (u) .
Replacing λ by (q-2)t q-2 H2(u) (γ-2)t γ-2 F (u) in H 1 (u) + t q-2 H 2 (u) -λt γ-2 F (u) = 0, we get that t q-2 = ( γ-2 q-γ ) H1(u) H2(u) . And we obtain (4.4) associated with (4.5).

Thus we can define the following characteristic points (recall that F is defined by (4.2))

Λ + 1 = inf{λ(u), u ∈ W, F (u) > 0} and λ + 1 = γ 2 α q 1-α Λ + 1 . (4.7)
Lemma 4.2. Assume that Assumptions 1.1 and 4.1 are satisfied. We have

0 < Λ + 1 < λ + 1 . Proof. Let u = (u 1 , • • • , u m ) ∈ W be such that F (u) > 0. First from γ < 2 * , let (t, l) be such that γ < t < 2 * and 1 l + γ t = 1. Since W 1,2 0 (Ω) ⊂ L t (Ω)
with a continuous embedding and since the functions f i are bounded, there exist positive constants still denoting C at each step and depending on some Sobolev constants, such that for i

= 1, • • • , m | Ω f i |u i | γ | ≤ ( Ω |f i | l ) 1/l ( Ω |u i | t ) γ/t ≤ C u i γ L t (Ω) ≤ C u i γ W 1,2 0 (Ω) . Then F (u) ≤ CH 1 (u) γ/2 .
By the same way, from γ < q, let s = q γ and r be such that 1 s + 1 r = 1. Then we have

| Ω f i |u i | γ | ≤ m( Ω |f i | r ) 1/r ( Ω |u i | γs ) 1/s ≤ C u i γ L q (Ω) ≤ C u i γ W 1,q 0 (Ω) and F (u) ≤ CH 2 (u) γ/q .
Therefore there exists a positive constant C , independent of u, such that

λ(u) = C q,γ H 1 (u) α H 2 (u) 1-α F (u) ≥ C C q,γ F (u) 2α γ F (u) q(1-α) γ F (u) = C C q,γ since 2α γ + q(1-α) γ = 1. Thus Λ + 1 > 0.
Finally we prove that Λ +

1 < λ + 1 . Indeed note that λ + 1 > Λ + 1 ⇔ γ 2 α q 1-α Λ + 1 > Λ + 1 ⇔ ( γ 2 ) q-2 > ( q 2 ) γ-2 . Denote µ = q-2
2 > 0 and η = γ-2 2 > 0. Since 2 < γ < q we have µ > η. Moreover the function f defined by f (x) = (1 + x) 1/x , is strictly decreasing on (0, ∞). Then (1 + µ) 1/µ < (1 + η) 1/η . And we get that ( q 2 ) γ-2 < ( γ 2 ) q-2 . So Λ + 1 < λ + 1 .

We obtain now the following result that will enable us to get the existence of a non-trivial solution for the system (4.1) associated with λ + 1 .

Proposition 4.1. Assume that Assumptions 1.1 and 4.1 are satisfied. Assume that u = (u 1 , • • • , u m ) ∈ W satisfies F (u) = 0 and λ (u) = 0 (i.e. u is a critical point of λ(u)). Then ũ = ( ũ1 , • • • , ũm ) ∈ W is a non-trivial solution of the system (4.1) associated with λ = γ 2 α q 1-α λ(u) where for all i = 1, • • • , m, ũi = 1 s u i and

1 s = ( q 2 ) 1 q-2 t(u) > 0. Moreover I λ(ũ) = 0. Proof. Let u = (u 1 , • • • , u m ) ∈ W which satisfies F (u) = 0 and λ (u) = 0.
For all test function φ, we have

∂λ ∂u 1 (u) • φ = 0. So 2C q,γ α(H 1 (u)) α-1 (H 2 (u)) 1-α (F (u)) -1 Ω (∇u 1 • ∇φ + a 11 u 1 φ + m j=2 a 1j u j φ) +qC q,γ (1 -α)(H 1 (u)) α (H 2 (u)) -α (F (u)) -1 Ω (|∇u 1 | q-2 ∇u 1 • ∇φ + w 1 |u 1 | q-2 u 1 φ) -γC q,γ α(H 1 (u)) α (H 2 (u)) 1-α (F (u)) -2 Ω f 1 |u 1 | γ-2 u 1 φ = 0. And 2C q,γ α H 1 (u) H 2 (u) α-1 Ω (∇u 1 • ∇φ + a 11 u 1 φ + m j=2 a 1j u j φ) +qC q,γ (1 -α) H 1 (u) H 2 (u) α Ω (|∇u 1 | q-2 ∇u 1 • ∇φ + w 1 |u 1 | q-2 u 1 φ) -λ(u)γ Ω f 1 |u 1 | γ-2 u 1 φ = 0. Define ũi = 1 s u i for i = 1, • • • , m, s > 0 and H(u) = H1(u) H2(u) . Then 2C q,γ α(H(u)) α-1 s Ω (∇ ũ1 • ∇φ + a 11 ũ1 φ + m j=2 a 1j ũj φ) +C q,γ (1 -α)(H(u)) α qs q-1 Ω (|∇ ũ1 | q-2 ∇ ũ1 • ∇φ + w 1 | ũ1 | q-2 ũ1 φ) -λ(u)γs γ-1 Ω f 1 | ũ1 | γ-2 ũ1 φ = 0.
And equivalently

2C q,γ α(H(u)) α-1 s 2-γ Ω (∇ ũ1 • ∇φ + a 11 ũ1 φ + m j=2 a 1j ũj φ) +C q,γ (1 -α)(H(u)) α qs q-γ Ω (|∇ ũ1 | q-2 ∇ ũ1 • ∇φ + w 1 | ũ1 | q-2 ũ1 φ) -λ(u)γ Ω f 1 | ũ1 | γ-2 ũ1 φ = 0.
Multiplying this last equation by

1 2 α q 1-α and denoting λ = γ 2 α q 1-α λ(u) we get 2(q -γ) q(γ -2)H(u) γ-2 q-2 s 2-γ Ω (∇ ũ1 • ∇φ + a 11 ũ1 φ + m j=2 a 1j ũj φ) + q(γ -2)H(u) 2(q -γ) q-γ q-2 s q-γ Ω (|∇ ũ1 | q-2 ∇ ũ1 • ∇φ + w 1 | ũ1 | q-2 ũ1 φ) -λ Ω f 1 | ũ1 | γ-2 ũ1 φ = 0. Choosing s = 2(q-γ) q(γ-2)H(u) 1 q-2 we obtain Ω (∇ ũ1 • ∇φ + a 11 ũ1 φ + m j=2 a 1j ũj φ) + Ω (|∇ ũ1 | q-2 ∇ ũ1 • ∇φ + w 1 | ũ1 | q-2 ũ1 φ) -λ Ω f 1 | ũ1 | γ-2 ũ1 φ = 0.
Doing the same for ũi , i = 2, • • • , m, we get that ũ = ( ũ1 , • • • , ũm ) is a weak solution of (4.1) associated with λ. Now we prove that I λ(ũ) = 0. Recall that ũ = 1 s u and λ = γ 2 α q 1-α λ(u). Then we have

I λ(ũ) = H 1 (u) 2s 2 + H 2 (u) qs q - C q,γ H 1 (u) α H 2 (u) 1-α 2 α q 1-α s γ . Denoting r = Cq,γ H1(u) α H2(u) 1-α 2 α q 1-α s γ , since 1 s = ( q 2 )
1 q-2 t(u) > 0 and α = q-γ q-2 we get

I λ(ũ) = r 1 C q,γ qH 1 (u) 2H 2 (u) 1-α q 2 α-1 (t(u)) 2-γ + 1 C q,γ 2H 2 (u) qH 1 (u) α q 2 α (t(u)) q-γ -1 .
But t(u) = γ-2 q-γ 1 q-2 H1(u) H2(u) 1 q-2 > 0 and C q,γ = q-2 (q-γ) α (γ-2) 1-α so

I λ(ũ) = r q -γ q -2 + γ -2 q -2 -1 = 0.
Proposition 4.2. Assume that Assumptions 1.1 and 4.1 are satisfied and 0 < λ < Λ + 1 . Then the system (4.1) has no non-trivial solution in W associated with λ.

Proof. Assume that 0 < λ < Λ + 1 . Assume also that the system (4.1) has a nontrivial solution u = (u 1 , • • • , u m ) ∈ W associated with λ. Then we have

H 1 (u) + H 2 (u) = λF (u). Note that this is impossible if F (u) ≤ 0. Therefore assume now that F (u) > 0. Recall that Ĩλ (t, v) = I λ (tv) = t 2 2 H 1 (v) + t q q H 2 (v) -λt γ γ F (v) for all t > 0 and v ∈ W. We have ∂ ∂t Ĩλ (t, v) = tH 1 (v) + t q-1 H 2 (v) -λt γ-1 F (v)
and in particular, since u is a weak solution of (4.1), note that

∂ ∂t Ĩλ ( u , 1 u u) = I λ (u) • 1 u u = 0. Moreover we have ∂ ∂t Ĩλ (t, v) = t γ-1 R λ (t, v) with R λ (t, v) = t 2-γ H 1 (v) + t q-γ H 2 (v) -λF (v).
Let v ∈ W be such that v = 0 and F (v) > 0. Note that from Lemma 4.1 we have R λ(v) (t(v), v) = 0. Moreover we can prove that R λ (t, v) ≥ R λ (t(v), v) for all t > 0. Indeed let f (t) = t 2-γ H 1 (v) + t q-γ H 2 (v). The function f admits a global mini-

mum on t(v) on (0, ∞) so f (t) ≥ f (t(v)) = H1(v) α α H2(v) 1-α 1-α > 0. Therefore R λ (t, v) ≥ R λ (t(v), v) for all t > 0. Finally since λ < Λ + 1 ≤ λ(v), we get that R λ (t, v) > R λ(v) (t, v) for all t > 0. Thus R λ (t, v) ≥ R λ (t(v), v) > R λ(v) (t(v), v) = 0
v = (v 1 , • • • , v m ) ∈ W such that λ(v) = Λ + 1 . Proof. First note that λ(tu) = λ(u) for all t > 0 and u ∈ W. Define t(u) = 1 ((H1(u)) α (H2(u)) 1-α ) 1 γ for u ∈ W \ {0} and note that (H 1 ( t(u)u) α (H 2 ( t(u))) 1-α = 1.
Therefore we can derive that

Λ + 1 = inf{λ(u), u ∈ W such that F (u) > 0 and H 1 (u) α H 2 (u) 1-α = 1}. Then we consider a minimizing sequence (v n ) of Λ + 1 . We have γ = 2α + q(1 -α), so v n γ H = v n 2α H v n q(1-α) H
and since W ⊂ H with a continuous embedding, there exists a positive constant C such that

v n γ H ≤ C v n 2α H v n q(1-α) W
.

But H 1 and H 2 are equivalent norms respectively in H and W so we get that

v n γ H ≤ C(H 1 (v n )) α (H 2 (v n )) 1-α = C (for a positive constant C). We deduce that (v n ) is a bounded sequence in H. By the compact embedding W 1,2 0 (Ω) ⊂ L γ (Ω) (for γ < 2 * ), we get the existence of v = (v 1 , • • • , v m ) ∈ H such that (v n ) converges to v, strongly in (L γ (Ω)) m and
weakly in H (for a subsequence).

Afterwards we prove that F (v) > 0, v ∈ W and since H 1 and H 2 are weakly lower semi-continous in H and W respectively, we get that λ(v) = Λ + 1 .

Indeed, since F is a continuous function and

F (v n ) > 0, F (v n ) → n→∞ F (v), we have F (v) ≥ 0. Moreover, if F (v) = 0, then λ(v n ) = Cq,γ F (vn) → n→∞ ∞. This contradicts λ(v n ) → n→∞ Λ + 1 . So F (v) > 0 and v = 0.
Now we prove that v ∈ W. Recall that (v n ) is a bounded sequence in H and that (v n ) converges to v = 0 strongly in (L γ (Ω)) m . So there exists a positive constant C such that v n (L γ (Ω)) m ≥ C > 0 for n large enough. Therefore, from the continuous embedding H ⊂ (L γ (Ω)) m , we get that v n H ≥ C > 0 for n large enough. 2 and weakly in W. Thus v ∈ W.

Finally from v n H ≥ C > 0 and v n 2α H v n (1-α)q W ≤ C we obtain that (v n ) is a bounded sequence in W . Therefore (v n ) admits a subsequence, still denoted (v n ) such that (v n ) converges to v strongly in (L γ (Ω))
Finally we prove that λ(v) = Λ + 1 . From the weakly semi-continuousness of H 1 and H 2 respectively on H and W we have

H 1 (v) ≤ lim inf H 1 (v n ) and H 2 (v) ≤ lim inf H 2 (v n ). But λ(v n ) = Cq,γ F (vn) = Cq,γ (H1(vn)) α (H2(vn)) 1-α F (vn)
→ n→∞ Λ + 1 . Passing to the limit inf as n tends to ∞ we get that Λ

+ 1 ≥ Cq,γ (H1(v)) α (H2(v)) 1-α F (v)
= λ(v). We deduce that λ(v) = Λ + 1 . This concludes the proof.

Contrary to [START_REF] Cherfils | On the stationary solutions of generalized reaction diffusion equations with p-q Laplacian[END_REF], we are not able to prove that the minimizer v is non-negative because of the coupling terms Proof. Even if we could follow [START_REF] Cherfils | On the stationary solutions of generalized reaction diffusion equations with p-q Laplacian[END_REF] for proving this result using the mountainpass theorem, we use here standard arguments by global minimization of the C 1functional I λ . Note that I λ is weakly lower semi-continuous by the compact embedding of W into (L q (Ω)) m and (L 2 (Ω)) m . Moreover I λ is coercive: indeed for any u ∈ W,

a ij v j v i in H 1 (v).
I λ (u) ≥ 1 q H 2 (u) - λ γ F (u).
Since |F (u)| ≤ C u γ W with C a positive constant, we get that

I λ (u) ≥ 1 q u q W 1 - λCq γ u γ-q W .
Thus I λ is coercive. Furthermore from Proposition 4.4, we have I λ + 1 (u + ) = 0 and F (u + ) > 0. Finally from the hypothesis λ > λ + 1 , we get that I λ (u + ) < I λ + 1 (u + ) = 0. Therefore we deduce that I λ has a non-trivial critical point which is a non-trivial weak solution of the system (4.1) associated with λ.

Remarks: We can get the same results for a larger class of coefficients, assuming that a ij , w i , f i ∈ L r (Ω) for some r > 1 as in [START_REF] Cherfils | On the stationary solutions of generalized reaction diffusion equations with p-q Laplacian[END_REF]. But we have not been able to adapt this method for a system with a (p,q)-Laplacian operator (with p = 2) and even for a non-symmetric system with a (2,q)-Laplacian operator. However in the particular case where the matrix A is not symmetric and has the following form: A = (a ij ) with a j1 = Ka 1j for j = 2, • • • , m for some positive constant K > 0 (K independent of j) and a ij = a ji for i, j ≥ 2, we can generalize all the above results. Indeed we introduce the diagonal matrix D = (d ij ) with d 11 = K, d ii = 1 for i = 2, • • • , m and d ij = 0 if i = j. We replace the functionals H 1 , H 2 and F (defined before by (1.1),(1.2),(4.2)) by

H 2 (u) = m i=1 d ii Ω (|∇u i | q + w i |u i | q ), F (u) = m i=1 d ii Ω f i |u i | γ , H 1 (u) = m i=1 d ii Ω (|∇u i | 2 + a ii u 2 i + m j=1,i =j a ij u j u i ), H 1 (u) = m i=1 d ii Ω |∇u i | 2 + Ω t U DAU with t U = (u 1 , • • • , u m ).
Therefore if we assume that the matrix DA satisfies the following hypothesis t ξDAξ ≥ 0 for all t ξ = (ξ 1 , • • • , ξ m ) ∈ R m , we still derive that the Euler functional I λ defined by (4.3) (with the new functionals H 1 , H 2 and F ) is associated with the system (4.1) and the existence of weak solutions for the system (4.1) is equivalent to the existence of critical points for I λ . Finally note that we obtain the existence of a non-trivial solution for (4.1) in Theorem 4.1 but contrary to [START_REF] Cherfils | On the stationary solutions of generalized reaction diffusion equations with p-q Laplacian[END_REF], due to the coupling terms, we are not able to get the existence of a non-negative solution.

Lemma 3 . 2 .

 32 Assume that Assumptions 1.1 and 3.2 are satisfied. If (u n ) ⊂ W, u n = (u 1n , • • • , u mn ), is a Palais-Smale sequence, then (u n ) has a strong convergent subsequence in W.

1 . 1 ( 1 q

 111 Finally combining Propositions 4.1 and 4.3, since v (defined by Proposition 4.3) is a critical point of λ(u), we derive the existence of a non-trivial weak solution u + = (u + 1 , • • • , u + m ) for the system (4.1) associated with λ + This is the following result Proposition 4.4. Assume that Assumptions 1.1 and 4.1 are satisfied. There exists u + = (u + 1 , • • • , u + m ) ∈ W a non-trivial solution for the system (4.1) associated with λ + 1 . Moreover I λ + u + ) = 0 and F (u + ) > 0. Proof. From Proposition 4.3 we have λ(v) = Λ + 1 = inf{λ(u), u ∈ W, F (u) > 0}. Thus v is a critical point of the function λ on W. From Proposition 4.1 we derive that there exists a non-trivial solution u+ = (u + 1 , • • • , u + m ) of system (4.1) associated with γ 2 α q 1-α λ(v) = λ + 1 where for all i = 1, • • • , m, u + i = 1 s v i and 1 s = ( q 2 ) -2 t(v) >0. Moreover from Proposition 4.1, I λ + 1 (u + ) = 0 and from Proposition 4.3, F (u + ) = 1 s γ F (v) > 0. 4.2. Main result. Theorem 4.1. Assume that Assumptions 1.1 and 4.1 are satisfied. If λ > λ + 1 , then the system (4.1), associated with λ, admits a non-trivial solution in W.

  Assume that Assumptions 1.1 and 4.1 are satisfied. There exists

	and ∂ ∂t Ĩλ (t, v) = ∂ ∂t I λ (tv) > 0 for all
	t > 0.
	So, choosing t = u and v = 1 u u, we get a contradiction since ∂ ∂t Ĩλ ( u , 1 u u) =
	0.
	Now we obtain a minimizer for Λ + 1 .
	Proposition 4.3.