While a general theory of reaction-diffusion systems is detailed in the books of Rothe [34] and Grzy- bowski [21]

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu

EXISTENCE OF SOLUTIONS FOR QUASILINEAR ELLIPTIC DEGENERATE SYSTEMS WITH L¹ DATA AND

NONLINEARITY IN THE GRADIENT

ABDELHAQ MOUIDA, NOUREDDINE ALAA, SALIM MESBAHI, WALID BOUARIFI

Abstract. In this article we show the existence of weak solutions for some quasilinear degenerate elliptic systems arising in modeling chemotaxis and an- giogenesis. The nonlinearity we consider has critical growth with respect to the gradient and the data are inL¹.

1. Introduction

Reaction-diffusion systems are important for a wide range of applied areas such as cell processes, drug release, ecology, spread of diseases, industrial catalytic pro- cesses, transport of contaminants in the environment, chemistry in interstellar me- dia, to mention a few. Some of these applications, especially in chemistry and biology, are explained in books by Murray [26, 27] and Baker [10]. While a general theory of reaction-diffusion systems is detailed in the books of Rothe [34] and Grzy- bowski [21]. Various forms of this problems have been proposed in the literature.

Most discussions in the current literature are for linear or nonlinear systems and different methods for the existence problem have been used, see Alaa et al [1]–[9], Baras [11, 12], Boccardo et al [15], Boudiba [16] and Pierre et al [29]-[32]. This is a relatively recent subject of mathematical and applied research. Most of the work that has been done so far is concerned with the exploration of particular aspects of very specific systems and equations. This is because these systems are usually very complex and display a wide range of phenomena remain poorly understood.

Consequently, there is no established program for solving a large class of systems.

For example a system of Chemotaxis, which is a biological phenomenon describ- ing the change of motion of a population densities or of single particles (such as amoebae, bacteria, endothelial cells, any cell, animals, etc.) in response (taxis) to an external chemical stimulus spread in the environment where they reside see for example [28]. The simple mathematical model which describes such a phenomenon

2000Mathematics Subject Classification. 35J55, 35J60, 35J70.

Key words and phrases. Degenerate elliptic systems; auasilinear; chemotaxis;

angiogenesis; weak solutions.

c

2013 Texas State University - San Marcos.

Submitted December 7, 2012. Published June 22, 2013.

1

reads as follows

∂u

∂t −Du∆u+∇(κ(u)∇u+χ(v)∇v) = 0 in Ω×(0, T)

∂v

∂t −Dv∆v+∇(ζ(u)∇u+η(v)∇v) = 0 in Ω×(0, T) u(0) =u₀, v₀(0) =v₀

(1.1)

hereuandv are the population densities. For a simple expansion, we include a(x) = ∂κ(u)

∂u , b(x) =∂χ(v)

∂v , c(x) =∂ζ(u)

∂u , d(x) = ∂η(v)

∂v f =−(κ(u)∆u+χ(v)∆)v, g=−(ζ(u)∆u+η(v)∆v). Then the system can be written as

∂u

∂t −Du∆u+a(x)|∇u|²+b(x)|∇v|²=f in Ω×(0, T)

∂v

∂t −D_v∆v+c(x)|∇u|²+d(x)|∇v|²=g in Ω×(0, T) u(0) =u0, v0(0) =v0.

(1.2)

In this work we are interested in the quasilinear elliplic degenerate problem u−D1∆u+a(x)|∇u|²+b(x)|∇v|^α=f(x) in Ω

v−D₂∆v+c(x)|∇u|^β+d(x)|∇v|²=g(x) in Ω u=v= 0 on∂Ω

(1.3)

where Ω is an open bounded set of R^N, N ≥ 1, with smooth boundary ∂Ω, the diffusion coefficientsD₁ andD₂ are positive constants,a, b, c, d, f, g: Ω→[0,+∞) are a non-negative integrable functions and 1≤α, β≤2.

We are interested in the case where the data are non-regular and where the growth of the nonlinear terms is arbitrary with respect to the gradient. To help understanding the situation, let us mention some previous works concerning the problem whena, b, c, d∈L^∞(Ω).

•iff, gare regular enough (f, g∈W^1,∞(Ω)) and for allα, β≥1, the method of sub- and super-solution can be used to prove the existence of solutions to (1.3). For instance (0,0) is a subsolution and a solution,w= (w1, w2), of the linear problem

w1−D1∆w2=f(x) in Ω w1−D1∆w2=g(x) in Ω

w1=w2= 0 on∂Ω,

(1.4) is a supersolution. Then (1.3) has a solution (u, v)∈W₀^1,∞(Ω)∩W^2,p(Ω); see Lions [23].

• If f, g∈L²(Ω) and 1 ≤α, β ≤2, then |∇u|^α,|∇v|^β ∈L¹(Ω). Many authors have studied this problem and showed that (1.3) has a solution (u, v)∈H₀¹(Ω)× H₀¹(Ω), see Bensoussan et al [14], Boccardo et al [15] and the references there in.

• Iff, g∈L¹(Ω) and 1≤α, β <2, Alaa and Mesbahi [1] proved that (1.3) has a non negative solution (u, v)∈W₀^1,1(Ω)×W₀^1,1(Ω).

• The case wheref, g ∈M_B⁺(Ω) (f, g are a finite non negative measures on Ω) has treated by Alaa and Pierre [9]. They proved that if 1 ≤ α, β ≤ 2 and the supersolution w= (w₁, w₂)∈ H₀¹(Ω)×H₀¹(Ω), then the problem (1.3) has a non negative solution (u, v)∈H₀¹(Ω)×H₀¹(Ω).

We are particularly interested in the case of a system (1.3) when a, b, c, d, f, g are not regular, more precisely,a, b, c, d, f, gare in L¹(Ω).

Let us make some specifications on the model problem u−D1∆u+b(r)|∇v|^α=f in B v−D₂∆v+c(r)|∇u|^β=g inB

u=v= 0 on∂B

(1.5)

whereB is the unit ball inR^N,r=kxk and b(r) =c(r) =

(−lnr ifN = 2

r^2−N ifN ≥3. (1.6)

In this case,b(r), c(r) are inL¹_loc(B) but not in L^∞(B). As a consequence the techniques usually used to prove existence and based on a prioriL^∞-estimates on u and ∇u fail. To overcome this difficulty, we will develop a new method which differ completely of the previous approach.

We have organized this article as follows. In section 2 we give the precise setting of the problem and state the main result. In section 3 we present an approximate problem and we give suitable estimates to prove that (1.3) has a solution in the case where the growth of the nonlinearity with respect to the gradient is arbitrary.

2. Assumptions and statement of main results Letf, g, a, b, c, dare functions that satisfies the following assumptions

f, g∈L¹(Ω), f, g≥0 (2.1)

a, b, c, d∈L¹_loc(Ω), a, b, c, d≥0 (2.2) First, we have to clarify in which sense we want to solved problem (1.3).

Definition 2.1. We say that (u, v) is a weak solution of (1.3) if u, v∈W₀^1,1(Ω)

a(x)|∇u|², b(x)|∇v|^α, c(x)|∇u|^β, d(x)|∇v|²∈L¹_loc(Ω) u−D₁∆u+a(x)|∇u|²+b(x)|∇v|^α=f(x) in D⁰(Ω)

v−D2∆v+c(x)|∇u|^β+d(x)|∇v|²=g(x) inD⁰(Ω)

(2.3)

We are interested to proving the existence of weak positive solutions of the problem (1.3). For this, we define the truncation functionTk ∈C², such that

Tk(r) =r if 0≤r≤k Tk(r)≤k+ 1 ifr≥k 0≤T_k⁰(r)≤1 ifr≥0 T_k⁰(r) = 0 ifr≥k+ 1 0≤ −T_k⁰⁰(r)≤C(k).

(2.4)

For example, the functionTk can be defined as Tk(r) =r in [0, k]

T_k(r) =1

2(r−k)⁴−(r−k)³+r in [k, k+ 1]

T_k(r) =1

2(k+ 1) forr > k+ 1.

(2.5)

Then we define the the space τ^1,2(Ω) =

w: Ω→Rmeasurable, such thatTk(w)∈H¹(Ω) for allk >0 This enables us to state the main result of this paper.

Theorem 2.2. Assume that (2.1)and (2.2)hold, and1≤α, β <2. If there exists a function θ∈τ^1,2(Ω) and a sequenceθn ∈L^∞(Ω)such that

0≤a, b, c, d≤θ inΩ θ_n→θ a.e. Ω

∇T_k(θ_n)→ ∇T_k(θ) strongly inL²(Ω)

k→∞lim sup

n

1 k

Z

Ω

|∇Tk(θn)|²

= 0

(2.6)

Then the problem (1.3)has a non negative weak solution.

Remark 2.3. (i) If a, b, c, d ∈L^∞(Ω), then (2.6) is satisfied. Indeed, θ can take the value of any non negative constantC, such that

C≥max

kakL^∞,kbkL^∞,kckL^∞,kdkL^∞ (2.7) (ii) Hypothesis (2.6) holds for the functions ξ =b or c given in (1.6). Indeed

−∆ξ=λis in this case the measure of Dirac which is a finite non negative measure on Ω. By consequent, we takeθ=ξandθ_n solution of

−∆θn=λ_n in Ω

θn = 0 on∂Ω, (2.8)

whereλn∈C₀^∞(Ω),λn→λinL¹(Ω) and λn ≤λ. Then, we can applied Theorem 2.2 and conclude the existence of the non negative weak solution for our model problem (2.1).

3. Proof of theorem 2.2

3.1. An approximation scheme. In this paragraph, we define an approximated system of (1.3). For this, we truncate the functionsa, b, c, d, f, gby introducing the sequencea_n, b_n, c_n, d_n, f_n, g_n defined as follows

a_n= min{a, θn}, b_n= min{b, θn}, c_n = min{c, θn}, d_n= min{d, θn} and

fn ∈C₀^∞(Ω),fn→f in L¹(Ω),fn ≤f

gn∈C₀^∞(Ω),gn→gin L¹(Ω),gn≤g (3.1) Then the approximate problem is

un, vn∈W₀^1,∞(Ω)

un−D1∆un+an(x)|∇un|²+bn(x)|∇vn|^α=fn(x) inD⁰(Ω) v_n−D₂∆v_n+c_n(x)|∇u_n|^β+d_n(x)|∇v_n|²=g_n(x) inD⁰(Ω).

(3.2)

One can see thatan, bn, cn, dn are inL^∞(Ω). On the other hand, (0,0) is a subso- lution of (3.2) and (Un, Vn) a solution of the linear problem

Un−D1∆Un=fn in Ω Vn−D2∆Vn=gn in Ω

Un, Vn∈W₀^1,∞(Ω)

(3.3) is a supersolution, then by the classical results in Amann and Grandall [13] and Lions [23, 24], there exists (u_n, v_n) solution of (3.2) such that

0≤un≤Un for alln 0≤v_n≤V_n for alln

3.2. A priori estimates. To prove theorem 2.2, we propose to sendnto infinity in (3.2). For this we will need some estimates passing to the limit.

Lemma 3.1. Let un, vn, an, bn, cn, dn be sequences defined as above. Then (i) Z

Ω

|∇Tk(un)|²≤kkfk_L1(Ω)

Z

Ω

|∇Tk(vn)|²≤kkgk_L1(Ω)

and (ii)

Z

Ω

bn.|∇Tk(vn)|^α≤kkfk_L1(Ω)

Z

Ω

cn.|∇Tk(un)|^β≤kkgk_L1(Ω)

Proof. (i) By multiplying the first equation of (3.2) by Tk(un) and the second equation byTk(vn) and integrating over Ω, we have

Z

Ω

|Tk(un)|²+D1

Z

Ω

|∇Tk(un)|² +

Z

Ω

anTk(un)|∇Tk(un)|²+ Z

Ω

bnTk(un)|∇Tk(vn)|^α≤ Z

Ω

fnTk(un) and

Z

Ω

|Tk(vn)|²+D2

Z

Ω

|∇Tk(vn)|² +

Z

Ω

cnTk(vn)|∇Tk(un)|^β+ Z

Ω

dnTk(vn)|∇Tk(vn)|²≤ Z

Ω

gnTk(vn)

Thanks to the positivity ofan, bn, cn, dn, the assumptions onfnandgn, the defini- tion of the functionTk, we deduce the result.

(ii) Integrating the first equation of (3.2) over Ω, we obtain Z

Ω

un−D1

Z

Ω

∆un+ Z

Ω

an(x)|∇un|²+ Z

Ω

bn(x)|∇vn|^α= Z

Ω

fn(x) (3.4) On the other hand, it is well know that for every functiony inW₀^1,1(Ω) such that

−∆y=H, H∈L¹(Ω) y≥0

there exists a sequenceyn in C²(Ω)∩C0(Ω) which satisfies y_n→y strongly inW₀^1,1(Ω)

∆y_n →∆y strongly inL¹(Ω) The regularity ofyn allows us to write

Z

Ω

∆y_n = Z

∂Ω

∂yn

∂υdσ,

but yn ≥0 on Ω andyn = 0 in∂Ω. Then ^∂y_∂υⁿ ≤0. We deduce by passing to the limit thatR

Ω∆y≤0. Therefore Z

Ω

∆un≤0 The relation (3.4) yields

Z

Ω

un+ Z

Ω

an(x)|∇un|²+ Z

Ω

bn(x)|∇vn|^α≤ Z

Ω

fn(x). By (3.1); we conclude that

Z

Ω

un+ Z

Ω

an(x)|∇un|²+ Z

Ω

bn(x)|∇vn|^α≤ kfk_L1(Ω).

In the same way, if we integrate the second equation of (3.2) over Ω, we obtain Z

Ω

vn+ Z

Ω

cn(x)|∇un|^β+ Z

Ω

dn(x)|∇vn|²≤ kgkL¹(Ω),

hence the result follows.

Remark 3.2. (1) Using the assertion (ii) of lemma 3.1, and the compactness of the operator

L¹(Ω)→W₀^1,q(Ω) G7→ϑ

where 1≤q < _N−1^N , andϑis the solution of the problem ϑ∈W₀^1,q(Ω)

αϑ−∆ϑ=G inD⁰(Ω)

we conclude the existence ofu, up to a subsequence, still denoted by u_n for sim- plicity, such that

un →u strongly inW₀^1,q(Ω), 1≤q < N N−1, (u_n,∇u_n)→(u,∇u) a.e. in Ω

see Brezis [17]

(2) Assertion (i) implies that

(T_k(u_n), T_k(v_n))→(T_k(u), T_k(v)) weakly inH₀¹(Ω)×H₀¹(Ω) Lemma 3.3. Let (un, vn) be a solution of (3.2), then

h→+∞lim sup

n

1 h

Z

Ω

|∇Th(un)|²dx

= lim

h→+∞sup

n

1 h

Z

Ω

|∇Th(vn)|²dx

= 0

Proof. We first remark thatun satisfies

−∆un≤fn in D⁰(Ω)

If we multiply this inequality by Th(un) and integrate on Ω, we obtain for every 0< M < h,

Z

Ω

|∇Th(un)|²≤ Z

Ω∩{un≤M}

f Th(un) + Z

Ω∩{un>M}

f Th(un)

≤M Z

Ω

f +h Z

Ω

f χ_{un>M}

hence

1 h

Z

Ω

|∇Th(un)|²≤ M h

Z

Ω

f+ Z

Ω

f χ_{un>M}

|{un> M}|= Z

{un>M}

dx≤ 1

Mkunk_L1≤ C M Then limM→+∞ sup_n|{un> M}|

= 0

On other hand, sincef ∈L¹(Ω), we have for eachε >0 there existsδsuch that for for allE⊂Ω,

|E|< δ Z

E

|f| ≤ ε 2.

Taking into account the above limit, we obtain that for eachε >0, there existsMε

such that for allM ≥Mε, sup

n

Z

Ω

f χ_[un>M]

≤ ε 2 TakingM =M_ε and lettinghtend to infinity, we obtain

lim

h→∞sup

n

1 h

Z

Ω

|∇T_h(u_n)|²

= 0.

Lemma 3.4. Let ηn be sequence such that ηn → η, a.e. in Ω and R

Ω|ηn|² ≤C thenηn→η in L^α(Ω) for all1≤α <2.

Proof. We show thatη_n is equi-integrable inL^α(Ω). LetE be a measurable subset of Ω; we have

Z

E

|η_n|^α≤ |E|^(2−α)/2Z

E

|η_n|²α/2

≤C|E|^(2−α)/2

Since 1 ≤α <2 then 0<2−α≤1. We choose |E| = (_C^ε)^2/(2−α), we obtain R

E|ηn|^α≤ε.

3.3. Convergence. The aim of this paragraph is to prove that (u, v) (obtained in the previous section) is in fact a solution of problem (1.3). According to definition 2.1, we have to show only that

u−D1∆u+a(x)|∇u|²+b(x)|∇v|^α=f(x) in D⁰(Ω) v−D₂∆v+c(x)|∇u|^β+d(x)|∇v|²=g(x) in D⁰(Ω)

By lemma 3.1, we know thatan(x)|∇un|²,dn(x)|∇vn|²,bn(x)|∇vn|^α,cn(x)|∇un|^β are uniformly bounded inL¹(Ω). Moreover

a_n(x)|∇un|²≥0, d_n(x)|∇un|²≥0, b_n(x)|∇un|^α≥0, c_n(x)|∇un|^β ≥0 and for almost everyxin Ω, we have

an(x)|∇un(x)|²→a(x)|∇u(x)|² dn(x)|∇vn(x)|²→d(x)|∇v(x)|² bn(x)|∇vn(x)|^α→b(x)|∇v(x)|^α c_n(x)|∇un(x)|^β →c(x)|∇u(x)|^β

Then there existsµ₁, µ₂ non negative measures, see Schwartz [35], such that

n→+∞lim (un−D1∆un+an(x)|∇un|²+bn(x)|∇vn|^α)

=u−D₁∆u+a(x)|∇u|²+b(x)|∇v|^α+µ₁ inD⁰(Ω)

n→+∞lim (vn−D2∆vn+cn(x)|∇un|^β+dn(x)|∇vn|²)

=v−D2∆v+c(x)|∇u|^β+d(x)|∇v|²+µ2 inD⁰(Ω) Consequently,

u−D1∆u+a(x)|∇u|²+b(x)|∇v|^α≤f inD⁰(Ω) v−D2∆v+c(x)|∇u|^β+d(x)|∇v|²≤g in D⁰(Ω)

Therefore, to conclude the proof of Theorem 2.2, we must establish the opposite inequality. For this, LetH be a function inC¹(R), such that

0≤H(s)≤1 H(s) =

(0 if|s| ≥1 1 if|s| ≤ ¹₂ To this end, we introduce the test functions

Φ1=ψ1exp[−θn

D1

un]H(θn

k )H(un

k ), Φ2=ψ2exp[−θn

D2

vn]H(θn

k )H(vn

k )

where H denotes the function defined above and ψ1, ψ2 ≤0, ψ1, ψ2 ∈ H₀¹(Ω)∩ L^∞(Ω). We multiply the first equation in (3.2) by Φ1 and we integrate on Ω, we obtain

Z

Ω

fnΦ1= X

1≤j≤7

Ij, where

I1= Z

Ω

unΦ1, I2=D1

Z

Ω

∇un∇ψ1exp[−θn

D1

un]H(θn

k)H(un

k ) I3=−

Z

Ω

un∇un∇θnΦ1

I₄= D1

k Z

Ω

∇u_n∇θ_nψ₁exp[−θn

D₁u_n]H⁰(θn

k )H(un

k ) I5=

Z

Ω

(an−θn)|∇un|²Φ1

I₆=D1

k Z

Ω

|∇u_n|²ψ₁exp[−θn

D₁u_n]H(θn

k)H⁰(un

k ) I7=

Z

Ω

bn.|∇vn|^αΦ1

Next we study each term. For the first term, we have

n→+∞lim I1= lim

n→+∞

Z

Ω

Tk(un)ψ1exp[−θn

D1

un]H(θn

k )H(un

k )

= Z

Ω

uψ1exp[− θ D1

u]H(θ k)H(u

k) since

ψ1exp[−θ_n D1

un]H(θ_n k )H(u_n

k ) converges strongly inL²(Ω) to

ψ₁exp[− θ D1

u]H(θ k)H(u

k) inL²(Ω)

and∇Tk(un) converges weakly to ∇Tk(u) in L²(Ω), (see [24, lemma 1.3, p 12]).

Concerning the second term, we get

n→+∞lim I2= lim

n→+∞D1

Z

Ω

∇Tk(un)∇ψ1exp[−θ_n D1

un]H(θ_n k )H(u_n

k )

=D1

Z

Ω

∇u∇ψ1exp[− θ D1

u]H(θ k)H(u

k) since

∇ψ1exp[−θ_n D1

un]H(θ_n k)H(u_n

k ) converges strongly inL²(Ω) to

∇ψ1exp[− θ D1

u]H(θ k)H(u

k). ForI3, we first remark that

n→+∞lim I3=− lim

n→+∞

Z

Ω

Tk(un)∇Tk(un)∇Tk(θn)ψ1exp[−θn

D1

un]H(θn

k )H(un

k )

=− Z

Ω

Tk(u)∇Tk(u)∇Tk(θ)ψ1exp[− θ D1

u]H(θ k)H(u

k) since

Tk(un)→Tk(u) weakly inH₀¹(Ω) Tk(θn)→Tk(θ) strongly inH₀¹(Ω) To studyI4andI6 we use Lemma 3.3. ForI4, we have

I4≤D1

h1 k

Z

Ω

|∇Tk(un)|²ψ1exp[−θn

D1

un]H⁰(θn

k)H(un

k )i^1/2

×[1 k

Z

Ω

|∇T_k(θ_n)|²ψ₁exp[−θn

D₁u_n]H⁰(θn

k)H(un

k )]^1/2

≤D1[kψ1k_L∞(Ω)

1 k

Z

Ω

|∇Tk(un)|²]^1/2[kψ1k_L∞(Ω)

1 k

Z

Ω

|∇Tk(θn)|²]¹² since exp[−_D^θn

1u_n]≤1, thus

I4≤D1[kψ1kL^∞δk]^1/2[kψ1kL^∞ρk]^1/2 Where

δk= sup

n

(1 k

Z

Ω

|∇Tk(un)|²) and ρk= sup

n

(1 k

Z

Ω

|∇Tk(θn)|²) By Lemma 3.3, we have

lim

k→∞δ_k= 0, lim

k→∞ρ_k= 0 Then

lim

k→∞sup

n

(I₄) = 0 Similarly, forI₆, we have

I6≤D1kψ1kL^∞δk

Then

lim

k→∞sup

n

(I₆) = 0

Now we investigate the remaining termI5. Sincean≤θn andψ1≤0, we have (an−θn)|∇un|²exp[−θ_n

D1

u]H(θ_n k)H(u_n

k )≥0 in Ω Therefore, by Fatou’s lemma, we obtain

n→+∞lim I5≥ Z

Ω

(a−θ)|∇u|²exp[− θ D₁u]H(θ

k)H(u k) ForI₇, we obtain

n→+∞lim I7= lim

n→+∞

Z

Ω

Tk(bn)|∇Tk(vn)|^αψ1exp[−θn

D1

un]H(θn

k )H(un

k ) By a direct application of Lemma 3.3, we have|∇Tk(vn)|^α→ |∇Tk(v)|^αstrongly in L¹(Ω), then

n→+∞lim I7= Z

Ω

b|∇v|^αψ1exp[− θ D1

u]H(θ k)H(u

k) We have shown that

ω(1 k) +

Z

Ω

uψ₁exp[− θ D1

u]H(θ k)H(u

k) +D₁ Z

Ω

∇u∇ψ1exp[− θ D1

u]H(θ k)H(u

k)

− Z

Ω

u∇u∇θψ1exp[− θ D1

u]H(θ k)H(u

k) + Z

Ω

(a−θ)|∇u|²exp[− θ D1

u]H(θ k)H(u

k) +

Z

Ω

b|∇v|^αψ1exp[− θ D1

u]H(θ k)H(u

k)

≤ Z

Ω

f ψ1exp[− θ D1

u]H(θ k)H(u

k)

whereω(ε) denotes a quantity that tends to 0 whenεtends to 0. Now we choose ψ1=−ϕ1exp[ θ

D₁u]H(θ k)H(u

k)

whereϕ1≥0,ϕ1∈D(Ω) and we replaceψ1by this value in the previous inequality to obtain

w(1 k)−

Z

Ω

uϕ1H²(θ k)H²(u

k)−D1

Z

Ω

∇u∇ϕ1H²(θ k)H²(u

k)

− Z

Ω

ϕ1u∇u∇θH²(θ k)H²(u

k)− Z

Ω

ϕ1|∇u|²θH²(θ k)H²(u

k)

−D1

k Z

Ω

ϕ1∇u∇θH⁰(θ k)H(θ

k)H²(u k)−D1

k Z

Ω

ϕ1|∇u|²H²(θ k)H(u

k)H⁰(u k) +

Z

Ω

u∇u∇θϕ1H²(θ k)H²(u

k)− Z

Ω

b|∇v|^αϕ1H²(θ k)H²(u

k)

− Z

Ω

a|∇u|²ϕ1H²(θ k)H²(u

k) + Z

Ω

θϕ1|∇u|²H²(θ k)H²(u

k)

≤ − Z

Ω

f ϕ1H²(θ k)H²(u

k)

By developing calculations and remarking that the sixth and seventh terms are equivalent toω(¹_k), we can write

− Z

Ω

uϕ₁H²(θ k)H²(u

k)−D₁ Z

Ω

∇u∇ϕ1H²(θ k)H²(u

k)

− Z

Ω

[a|∇u|²+b|∇v|^α]ϕ₁H²(θ k)H²(u

k) +ω(1 k)

≤ − Z

Ω

f ϕ1H²(θ k)H²(u

k)

Finally passing to the limit ask tends to infinity, we use the fact that

k→∞lim H(θ

k) = 1, lim

k→∞H(u k) = 1 to conclude that for everyϕ1≥0, ϕ1∈D(Ω),

Z

Ω

[u−D₁∆u+a(x)|∇u|²+b(x)|∇v|^α]ϕ₁≥ Z

Ω

f ϕ₁

In the same way, we multiply the second equation in (3.2) by Φ2and we integrate on Ω. By studying separately each term as in the previous case still using Lemmas 3.1, 3.3 and 3.4, we choose

ψ₂=−ϕ₂exp[ θ D₂u]H(θ

k)H(v k)

whereϕ₂≥0,ϕ₂∈D(Ω) and we replaceψ₂by this value in the inequality obtained to conclude that for everyϕ₂≥0,ϕ₂∈D(Ω) that

Z

Ω

[v−D2∆v+c(x)|∇u|^β+d(x)|∇v|²]ϕ2≥ Z

Ω

gϕ2

This completes the proof of theorem 2.2.

Acknowledgments. We are grateful to the anonymous referee for the corrections and useful suggestions that have improved this article.

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Abdelhaq Mouida

Laboratory LAMAI, Faculty of Science and Technology of Marrakech, University Cadi Ayyad, B.P. 549, Street Abdelkarim Elkhattabi, Marrakech - 40000, Morocco

E-mail address:[email protected]

Noureddine Alaa

Laboratory LAMAI, Faculty of Science and Technology of Marrakech, University Cadi Ayyad, B.P. 549, Street Abdelkarim Elkhattabi, Marrakech - 40000, Morocco

E-mail address:[email protected]

Salim Mesbahi

Department of Mathematics, Faculty of Science, University Ferhat Abbas, Setif I, Pole 2 - Elbez, Setif - 19000, Algeria

E-mail address:[email protected]

Walid Bouarifi

Department of Computer Science, National School of Applied Sciences of Safi, Cadi Ayyad University, Sidi Bouzid, BP 63, Safi - 46000, Morocco

E-mail address:[email protected]