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**ON THE EXISTENCE OF BOUNDED SOLUTIONS**

**OF NONLINEAR ELLIPTIC SYSTEMS**

**ABDELAZIZ AHAMMOU**

Received 24 March 2000 and in revised form 13 August 2000

We study the existence of bounded solutions to the elliptic system*−∆pu=f (u, v)+h1*
inΩ,*−∆qv=g(u, v)+h2*inΩ,*u=v=*0 on*∂Ω*, non-necessarily potential systems. The
method used is a shooting technique. We are concerned with the existence of a negative
subsolution and a nonnegative supersolution in the sense of Hernandez; then we construct
some compact operator*T*and some invariant set*K*where we can use the Leray Schauder’s
theorem.

2000 Mathematics Subject Classiﬁcation: 35J25, 35J60.

**1. Introduction.** The aim of this paper is to study the existence of solutions for the
following system:

*−*∆p*u=f (u, v)+h1,* *−*∆q*v=g(u, v)+h2* inΩ*,*

*u=v=*0 on*∂*Ω*,* (1.1)

whereΩis a smooth bounded domain ofR*N*_{, where}_{N}_{≥}_{1,}_{p, q >}_{1,}_{f , g}_{are continuous}

functions ofR2_{into}_{R}_{and}_{h1, h2}_{are the functions given in}_{L}+∞_{(}_{Ω}_{).}

System (1.1) results from the study of the nonlinear phenomena, such as the
evo-lution of population, of chemical reaction, and so forth. A great attention was given
to the existence of the solutions for a system of the (1.1) type, by using various
ap-proaches (cf. [3,4,5,7,13]). When the system has a variational structure, the existence
of the solutions for (1.1) can be established by means of the variational approaches
un-der adapted conditions (cf. [9,13]). When (1.1) does not have a variational structure, as
in Vélin and de Thélin [13], where the authors obtained some results for the existence
of solutions to problem (1.1) with the following growth conditions of nonlinearity*f*
and*g:*

*f (u, v)≤a1|u|α*0*| _{v}|β*0

*+*1

*+*1

_{a2}|_{u}|α*−*1

*+*1

_{a3}|_{v}|β*−*1

_{,}*g(u, v)≤a4|u|α*0

*+*1

*|*0

_{v}|β*+*2

_{a5}|_{u}|α*−*1

*+*2

_{a6}|_{v}|β*−*1

_{,}(1.2)

where*ai(i=*1, . . . ,6)are positive constants and*αi*and*βi(i=*0,1,2)satisfy

*α0+*1
*p* *+*

*β0+*1
*q* *<*1,

1*< α1< p;* 0*< β1−*1*<* *q*
*p∗,*

1*< α2−*1*<* *p*

*q∗*; 0*< β2< q.*

Always in the case of a system, we can notice the existence results obtained in Baoyao [2], and Brézis and Lieb [4].

The case of a scalar equation has been studied by many authors, see de Figueiredo and Gossez [6], Fernandes et al. [10] and Fonda et al. [11]. More recently, some in-teresting results have been obtained by Gossez and El Hachimi [12] and Anane and Chakrone [1]. Those authors derived the solvability of the following problem:

*−*∆m*u=f (u)+h* inΩ*,*

*u=*0 on*∂*Ω*,* (1.4)

under the following condition:

lim

*u→∞*inf

*pF (u)*

*um* *< µm* *,* (1.5)

where

*µm=(m−*1)

2
*R(*Ω*)*

1

0
*ds*
*m√*_{1}_{−}_{s}m

*m*

*,* (1.6)

and*R(*Ω*)*denotes the radius of the smallest open ball*B(0, R)*containingΩ. The
par-ticular cases*N=*1 and*m=*2 were considered in [10]. It was shown there that (1.4)
is solvable for any*h∈L∞(*Ω*)*if

lim

*u→∞*inf

2F (u)

*u*2 *< λ1,*2, (1.7)
where*λ1,*2is the ﬁrst eigenvalue of*−*∆ andΩ*=]a, b[. Observe that forN >*1,we
have*µ*2*< λ1,*2(Ω*). Then, the question naturally arises whetherµm*can be replaced by

*λ1,m(*Ω*)*in (1.5), where*λ1,m*is the ﬁrst eigenvalue of*−*∆m. This problem remains open.

The goal of this paper is to show that the same approach in [12] can be applied
for some quasilinear elliptic systems with the constants*µp* and*µq*, deﬁned below,

associated, respectively, with the operators*−*∆pand*−*∆q, and where*µm(m=p, q),*

better than*µm* , is presented in (1.5). In this case, we treat the question of the existence

of the solutions for system (1.1) without imposing variational structures, which is
often the case for system (1.1) and without necessarily the growth conditions for*f*
and*g.*

**2. Main result.** We make the following assumptions:

(H1) (i) The function*f (u,·)*is a nonincreasing function onRfor all*u*inR,
(ii) The function*g(·, v)*is a nonincreasing function onRfor all*v*inR.
(H2) There exists some unbounded increasing subsequence*(mk)k*, satisfying

lim

*k→+∞*

*pFm*1*k/p, m*

1*/q*
*k*

*mk*

*< µp,* lim
*k→+∞*

*qGm*1*k/p, m*

1*/q*
*k*

*mk*

*< µq,* (2.1)

lim

*k→+∞*

*pF−m*1*k/p,−m*

1*/q*
*k*

*mk*

*< µp,* lim
*k→+∞*

*qG−mk*1*/p,−m*

1*/q*
*k*

*mk*

where*F*and*G*are the following functions:

*F (u, v)=*
*u*

0

*f (s, v)ds,* *G(u, v)=*
*v*

0

*g(u, t)dt,* (2.3)

and where we denote by*µp*and*µq*the following constants:

*µp=(p−*1)

2
*b−a*

1

0
*ds*
*p*
*√*

1*−sp*

*p*

*,* *µq=(q−*1)

2
*b−a*

1

0
*dt*
*q*
*√*

1*−tq*

*q*

*,* (2.4)

with*b−a=*min(b*i−ai)*and*P=*Π*[ai, bi]*is the smallest cube such that*P⊃*Ω.

Observe that for*N=*1,*µp*and*µq*are, respectively, the ﬁrst eigenvalue of*−*∆pand
*−*∆qwhenΩ*=]a, b[. It is clear thatµp*is better than*µp*deﬁned in (1.5). In particular,

it is interesting whenΩis a rectangle or a triangle, because*µpµp*and*µp≈λ1,p(*Ω*).*

The main result of this paper is the following statement.

**Theorem2.1.** *Under hypotheses (H1) and (H2). Problem (1.1) has a solution(u, v)*

*in(W*01*,p(*Ω*)×W*
1*,q*

0 *(*Ω*))∩(L+∞(*Ω*)×L+∞(*Ω*))for any(h1, h2)inL+∞(*Ω*)×L+∞(*Ω*).*
**Example _{2.2}.**

_{Consider}

*f (u, v)=a(x)u|u|α−*1* _{|}_{v}_{|}β+*1

_{,}*1*

_{g(u, v)}_{=}_{b(x)}_{|}_{u}_{|}γ+*1*

_{v}_{|}_{v}_{|}δ−

_{.}_{(2.5)}

(1) Assume that*a∞< µp*,*b∞< µq*, and

*α+*1
*p* *+*

*β+*1
*q* *≤*1,

*γ+*1
*p* *+*

*δ+*1

*q* *≤*1. (2.6)

Then we conclude the existence of solutions. (2) If

*α+*1
*p* *+*

*β+*1
*q* *<*1,

*γ+*1
*p* *+*

*δ+*1

*q* *<*1, (2.7)

we have the existence for all*(a, b)*in*L+∞(*Ω*).*

The method used in this paper is a shooting technique. InSection 3, we are
con-cerned with the existence of a negative subsolution*(u0, v0)*and a nonnegative
super-solution*(u*0* _{, v}*0

_{)}_{in the sense of Hernandez’s deﬁnition [}

_{13}

_{]. In}

_{Section 4}

_{, we consider}some compact operator

*T*and some invariant set

*K. And, we look for solutions of*problem (1.1) as ﬁxed points of the operator

*T*. We will be in the conditions of the Schauder ﬁxed point theorem.

**3. Construction of sub-supersolutions**

*u0, v0∈W*1*,p _{(}*

_{Ω}

*1*

_{)}_{×}_{W}*,q*

_{(}_{Ω}

_{)}_{∩}_{L}+∞_{(}_{Ω}

_{)}_{×}_{L}+∞_{(}_{Ω}

_{)}_{,}

*u*0* _{, v}*0

*1*

_{∈}_{W}*,p*

_{(}_{Ω}

*1*

_{)}_{×}_{W}*,q*

_{(}_{Ω}

_{)}_{∩}_{L}+∞_{(}_{Ω}

_{)}_{×}_{L}+∞_{(}_{Ω}

_{)}_{,}*−*∆p*u0−fx, u0, v≤*0*≤ −*∆p*u*0*−fx, u*0*, v* inΩ*,∀v∈* *v0, v*0*,*
*−*∆q*v0−fx, u, v0≤*0*≤ −*∆q*v*0*−fx, u, v*0 inΩ*,∀u∈* *u0, u*0*,*

*u0≤u*0_{,}* _{v0}_{≤}_{v}*0

_{in}

_{Ω}

_{,}

_{u0}_{≤}_{0}

*0*

_{≤}_{u}

_{,}

_{v0}_{≤}_{0}

*0*

_{≤}_{v}_{on}

_{∂}_{Ω}

_{.}(3.1)

Similar deﬁnitions can be found in Diaz and Hernández [7], and Diaz and Herrero [8].
For all*M >*0, we note that

ˆ

*f (u, v)=f (u, v)+M,* *g(u, v)*ˆ *=g(u, v)+M,*

ˆ

*F (u, v)=F (u, v)+Mu,* *G(u, v)*ˆ *=G(u, v)+Mv.* (3.2)

Notice that if*F* and*G*satisfy the assumption (2.1) of (H2), then the same holds for ˆ*F*
and ˆ*G.*

**Proposition3.2**[6]**.** *Under hypothesis (2.1) of (H2), there exist two sequencesd _{k}*

*andd*

_{k}such that(a)*m*1*k/p≥dk≥*0, for all*k∈*N*and*

*dk*

0

*ds*

*p*

*pF*ˆ*dk, m*

1*/q*
*k*

*−pF*ˆ*s, mk*1*/q*

*>*
1

0
*ds*
*p*
*√*

1*−sp* *µp*

*−*1*/p*

*.* (3.3)

(b)*m*1*k/p≥dk≥*0, for all*k∈Nand*

*d _{k}*

0

*dt*

*q*

*qG*ˆ*m*1* _{k}/p, d_{k}−qG*ˆ

*m*1

_{k}/p, t*>*

1

0
*dt*
*q*
*√*

1*−tq* *µq*

*−*1*/q*

*.* (3.4)

**Remark _{3.3}.**

_{We have}

*p*

*p−*1
1

0
*ds*
*p√*

1*−sp* *µp*

*−*1*/p*

*=* *q _{q}_{−}*

_{1}1

0
*dt*
*q*
*√*

1*−tq* *µq*

*−*1*/q*
*=b−a*

2 *.* (3.5)

**Proof ofProposition3.2.** We only prove (a); the proof of (b) is similar.

(1) From (2.1) of hypothesis (H2), there exists some*µ >*0 such that

lim

*k→+∞*

*pF*ˆ*m*1*k/p, m*

1*/q*
*k*

*mk*

*< µ < µp,* (3.6)

then

lim

*k→+∞µmk−p*

ˆ

*Fm*1*k/p, m*

1*/q*
*k*

(2) We consider the functions*[H(·, mk)]k*, where

*Hs, mk*

*=µs−pF*ˆ*s*1*/p _{, m}*1

*/q*

*k*

*.* (3.8)

For all*k >*0, we have

*H*0, m*k*

*= −pF*ˆ0, m1*k/q*

*=*0,

*Hmk, mk=µmk−pF*ˆ

*m*1*k/p, m*

1*/q*
*k*

*>*0.

(3.9)

Then for all*k∈*Nthere exists*dk>*0 such that*dkp≤mk*and for all*s∈[0, dpk], we have*

*Hs, mk≤Hdpk, mk,* (3.10)

that is,

*µs−pF*ˆ*s*1*/p, m*1*k/q*

*≤µdpk−pF*

*dk, m*

1*/q*
*k*

*,* (3.11)

then

*pF*ˆ*dk, m*1*k/q*

*−pF*ˆ*s*1*/p _{, m}*1

*/q*

*k*

*≤µdp _{k}−s.* (3.12)

Let*s=ωp*_{, where}_{ω}_{∈}_{[0, d}

*k]⊂[0, m*1*k/p]. We obtain*

*pF*ˆ*dk, m*

1*/q*
*k*

*−pF*ˆ*ω, m*1*k/q*

*≤µdpk−ωp*

*,* (3.13)

that is,

1
*p*

*dp _{k}−ωp[µ]*

*−*1*/p _{≤}* 1

*p*

*pF*ˆ*dk, mk*1*/q*

*−pF*ˆ*ω, m*1_{k}/q

*.* (3.14)

Then integrating on*[0, dk]*we obtain

1

0
*dω*
*p√*

1*−ωp[µ]*

*−*1*/p _{≤}*

*dk*

0

*dω*

*p*

*pF*ˆ*dk, m*1*k/q*

*−pF*ˆ*ω, m*1_{k}/q

*.* (3.15)

This proves (a).

**3.1. Construction of supersolution***(u*0* _{, v}*0

_{)}

_{.}_{In the following step we suppose that}for all

*k∈*Nand for all

*s∈[0, m*1

_{k}/p]*fs, m*1*k/q*

*+M≥*0. (3.16)

Denote by*(f*ˆ*k)k*the sequence of functions deﬁned by

ˆ
*fk(s)=*

*fm*1* _{k}/p, m*1

*for*

_{k}/q+M*s*in

*m*1

_{k}/p,+∞,*fs, m*1

*for*

_{k}/q+M*s*in0, m1

_{k}/p,*f*0, m1

*for*

_{k}/q+M*s*in

*]−∞,0].*

For all*k∈*N, we associate to the function ˆ*fk*, the following problem:

*−|u|p−*2*u(t)=f*ˆ*ku(t),* *u(t)≥*0 for*t*in*[a, b].* (3.18)

For all*k∈*N*,*we deﬁne the nonlinear operator*Tk*such that

*Tk*:*C*

*[a, b]→C[a, b]* (3.19)

in the following way:

*Tk(u)(t)=dk−*

*t*

*a*

*r*

*a*

ˆ
*fk*

*u(s)ds*

1*/(p−*1*)*

*dr .* (3.20)

Since ˆ*fk*is a nonnegative function, the operator*Tk*is well deﬁned.

**Lemma _{3.4}.**

_{For all}_{k}≥_{0,}

(i) *the operatorTkis completely continuous,*

(ii) *there exists a ﬁxed point forTk.*

**Proof.** _{Let}* _{k}∈*N

_{,}

(1) the continuity is immediate,

(2) let*(un)n*be a bounded sequence in*C([a, b])*such that the sequence*(Tk(un))n*

is also bounded in*C([a, b]).*

By the continuity of the function ˆ*fk*, there exists some constant*Ck*such that

*t∈[a, b],* *∀t,* (3.21)

for all*n∈*Nwe have

*Tkun(t)−Tkunt≤Ckt−t.* (3.22)

So *(Tk(un))n* is uniformly equicontinuous and by Ascoli theorem the sequence

*(Tk(un))n*is relatively compact in*C([a, b]).*

(3) Using the Leray-Schauder theorem we deduce that *Tk* has a ﬁxed point*uk∈*

*C([a, b]), that is,Tk(uk)=uk*.

**Remark _{3.5}.**

_{By deﬁnition of the operator}

_{T}_{k}_{, we have}(i)

*−|u*2

_{k}|p−

_{u}*k(t)=*

*t*

*af*ˆ*k(uk(s))ds,*

(ii) *u _{k}(a)=*0,
(iii)

*uk(a)=dk*.

Since ˆ*fk*is a nonnegative function we have

(iv) *u _{k}(t)≤*0 for

*t*in

*[a, b].*

That is,*uk*is a nonincreasing function on*[a, b].*

**Lemma _{3.6}.**

_{From (}_{2.1}_{), choose}_{(d}_{k}_{)}_{k}_{such that}*uk(t)≥*0 *in*

*a,a+b*

2

*∀k∈*N*,* (3.23)

**Proof.** Let*(d _{k})_{k}*be some sequence such that

*d*1

_{k}∈[0, m*for all*

_{k}/p]*k∈*N. We denote by

*tk*a real number such that

*uk(tk)=*0 and

*uk(t)≥*0 on

*[a, tk].*

Then, fromRemark 3.5, since*uk*is a nonincreasing function and*dk∈[0, m*1*k/p], we*

have

*m*1* _{k}/p≥uk≥*0

*∀t∈*

*a, tk.*(3.24)

Consequently, for all*t∈[a, tk]*we have

*−uk*
*p−*2

*uk*

*(t)= _{f}*ˆ

*1*

_{k}_{u}_{k}_{(t)}_{=}_{f}_{u}_{k}_{(t), m}*/q*

*k*

*+M.* (3.25)

Multiplying (3.25) by*u _{k}*we obtain

*p−*1
*p*

*−u _{k}(t)p=_{dt}d*

*F*ˆ

*uk(t), m*1

*k/q*

*,* (3.26)

where

ˆ

*Fu, m*1* _{k}/q=F (u, v)+Mu.* (3.27)
Integrating (3.26) on

*[a, t]⊂[a, tk], we obtain*

*−p _{p}_{−}*

_{1u}

*k(t)=*

*p*

*pF*ˆ*dk, m*1*k/q*

*−pF*ˆ*uk(t), m*1*k/q*

*.* (3.28)

Integrating (3.28) again on*[a, tk]*we deduce that

*p _{p}_{−}*

_{1}

*tk*

*a*

*−u _{k}(t)*

*p*

*pF*ˆ*dk, m*1*k/q*

*−pF*ˆ*uk(t), m*1*k/q*

*dt≤tk−a.* (3.29)

Then, we obtain

*p*

*p−*1
*dk*

0

1

*p*

*pF*ˆ*dk, m*

1*/q*
*k*

*−pF*ˆ*s, m*1_{k}/q

*ds≤tk−a.* (3.30)

It follows fromProposition 3.2andRemark 3.3that one can choose the sequence*(dk)*

such that for all*k≥k0*we have

*b−a*
2 *<*

*p _{p}_{−}*

_{1}

*dk*

0

1

*p*

*pF*ˆ*dk, m*1*k/q*

*−pF*ˆ*s, m*1_{k}/q

*ds.* (3.31)

Consequently, from (3.30) and (3.31), we obtain that for all*k≥k0, there existstk*

satisfying*tk> (b+a)/2.*

**Proposition _{3.7}.**

_{Suppose that the sequence}_{(m}_{k}_{)}_{k}_{satisﬁes (}_{2.1}_{), and that for all}*k >*0

*we have*

inf

*s∈[*0*,m*1/p_{k}*]*

*Then, there exists some numberk0∈*N*such that for allk≥k0the problem*

*−|u|p−*2*u=fu, m*1*k/q*

*+M* *in(a, b),*

*u≥*0 *on[a, b],* (3.33)

*has a solutionu*ˆ*ksatisfyingu*ˆ*k∈C*1*([a, b]),(|u*ˆ*k|p−*2*u*ˆ*k)∈C([a, b])andm*

1*/p*

*k* *≥u*ˆ*k≥*0

*for allk≥k0.*

**Proof.** Let*(u _{k})_{k}*be the sequence deﬁned inLemma 3.6. This sequence satisﬁes
that for all

*k≥k0,*

*uk∈C*1

*a,a+b*

2

*,* *u _{k}p−*2

*u*

_{k}∈C*a,a+b*
2

*,*

*−u _{k}p−*2

*u*1

_{k}(t)=fuk(t), m*k/q*

*+M* in

*a,a+b*
2

*,* (3.34)

*m*1* _{k}/p≥u*ˆ

*k≥*0 in

*a,a+b*

2

*,*

*uk(a)=*0. (3.35)

We denote by ˆ*uk*the following function:

ˆ
*uk(t)=*

*uk*

_{3a}_{+}_{b}

2 *−t*

if*t∈*

*a,a+b*
2

*,*

*uk*

*t−a+b*
2

if*t∈*
_{a}_{+}_{b}

2 *, b*

*.*

(3.36)

Then, from (3.34), it is easy to see that

*∀k≥k0,* *u*ˆ*k∈C*1

*[a, b],* *u*ˆ* _{k}p−*2

*u*ˆ

_{k}∈C[a, b],*−u*ˆ

*k*

*p−*2
ˆ
*uk*

*(t)=fu*ˆ*k(t), m*1*k/q*

*+M* in*[a, b],*

*mk*1*/p≥u*ˆ*k≥*0 in*[a, b].*

(3.37)

Then the conclusion holds.

**Proposition _{3.8}.**

_{Let}_{M >}_{0. From (}

_{2.1}_{), there exist some}_{m >}_{0}

_{and}_{(ˆ}_{u}_{m}_{,}_{v}_{ˆ}

_{m}_{)}∈*(C*1

*2*

_{([a, b]))}

_{such that}_{}
ˆ
*um*

*p−*2
ˆ
*um*

*,v*ˆ*m*
*p−*2

ˆ
*vm*

*∈C[a, b]*2*,*

*−u*ˆ*m*
*p−*2

ˆ
*um*

*≥fu*ˆ*m, m*1*/q*

*+M* *in(a, b),*

*−v*ˆ*m*
*q−*2

ˆ
*vm*

*≥gm*1*/p _{,}_{v}*

_{ˆ}

*m*

*+M* *in(a, b),*

*m*1*/p _{≥}_{u}*

_{ˆ}

*m≥*0, *m*1*/q≥v*ˆ*m≥*0 *on[a, b].*

**Proof.** We study three cases.

**Case _{1}.**

_{We suppose that for all}

*N*

_{k}∈_{we have}

inf

*s∈g[*0*,m*1/p_{k}*]*

*fs, m*1* _{k}/q+M <*0, inf

*t∈[*0*,m*1/q_{k}*]*

*gm*1* _{k}/p, t+M <*0. (3.39)

Then for all*k∈*N, there exist*sm _{k}∈[0, mk*1

*/p]*and

*tm*1

_{k}∈[0, m*k/q]*satisfying

*fsmk, m*
1*/q*
*k*

*+M <*0, *gm*1_{k}/p, tmk

*+M <*0. (3.40)

Consequently, for*m=mk*, the couple*(u*ˆ*m,v*ˆ*m)=(sm _{k}, tm_{k})*satisﬁes the result.

**Case2.** Assume that for all*k∈*Nwe have,

inf

*s∈[*0*,m*1/p_{k}*]*

*fs, m*1* _{k}/q+M≥*0, (3.41)

inf

*t∈[*0*,m*1/q_{k}*]*

*gm*1* _{k}/p, t+M <*0. (3.42)

(a) From (3.41) andProposition 3.7 there exist some *k0∈*Nand some sequence
*(ˆuk)k*such that, for all*k≥k0, we have*

ˆ
*uk∈C*1

*[a, b],* *u*ˆ* _{k}p−*2

*u*ˆ

_{k}∈C[a, b],*−u*ˆ

*2*

_{k}p−*u*ˆ

*ˆ*

_{k}≥fu*k, m*1

*k/q*

*+M* in*(a, b),*

*mk*1*/p≥u*ˆ*k≥*0 in*[a, b].*

(3.43)

(b) From (3.42), there exists a sequence*(tm _{k})k*such that

*m*1* _{k}/p≥tmk≥*0,

*g*

*m*1_{k}/p, tmk

*+M <*0 *∀k≥k0.* (3.44)

Consequently, for*m=mk*with*k > k0, the pair(u*ˆ*mk, tmk)*satisﬁes the result.
**Case _{3}.**

_{Assume that for all}

*N*

_{k}∈

_{,}_{we have}

inf

*s∈[*0*,m _{k}*1/p

*]*

*fs, m _{k}*1

*/q+M≥*0, inf

*t∈[*0*,m*1/q_{k}*]*

*gm*1* _{k}/p, t+M≥*0. (3.45)

Then, fromProposition 3.7, for all*k≥k0, there exists(ˆuk,v*ˆ*k)∈(C*1*([a, b]))*2such

that _{}

*u*ˆ*k*
*p−*2

ˆ
*uk*

*,v*ˆ*k*
*p−*2

ˆ
*vk*

*∈C[a, b]*2*,*

*−u*ˆ* _{k}p−*2

*u*ˆ

*ˆ*

_{k}≥fu*k, m*1

*k/q*

*+M* in*(a, b),*

*−v*ˆ*k*
*p−*2

ˆ
*vk*

*≥gm*1*k/p,v*ˆ*k*

*+M* in*(a, b),*

*m*1*k/p≥u*ˆ*k≥*0, *m*

1*/q*

*k* *≥v*ˆ*k≥*0 on*[a, b].*

(3.46)

This proves the results.

Now, for problem (1.1), we consider a smooth bounded domain ΩinR*N _{,}*

_{and we}

**Proposition3.9.** *Under hypotheses (H1) and (2.1) of (H2), problem (1.1) has a *

*non-negative supersolution(u*0* _{, v}*0

*1*

_{)}_{in}_{W}*,p*

_{(}_{Ω}

*1*

_{)}_{×}_{W}*,q*

_{(}_{Ω}

_{).}**Proof.** Let*M≥h1 _{∞}+ h2_{∞}·P=[a_{i}, b_{i}]*is a cube containingΩand

*b−a=* inf

1*≤i≤Nbi−ai=b1−a1.* (3.47)

From (2.1) of hypothesis (H2) andProposition 3.8, there exist*m >*0 and*(u*ˆ*m,v*ˆ*m)∈*

*(C*1* _{([a, b]))}*2

_{such that}

_{}

ˆ
*um*

*p−*2
ˆ
*um*

*,v*ˆ*m*
*p−*2

ˆ
*vm*

*∈C[a, b]*2*,* (3.48)

and*(u*ˆ*m,v*ˆ*m)*satisﬁes

*−u*ˆ* _{m}p−*2

*u*ˆ

*ˆ*

_{m}≥fu*m, m*1

*/q*

*+M* in*(a, b),*

*−v*ˆ*m*
*q−*2

ˆ
*vm*

*≥gm*1*/p,v*ˆ*m*

*+M* in*(a, b),*

*m*1*/p _{≥}_{u}*

_{ˆ}

*m≥*0, *m*1*/q≥v*ˆ*m≥*0 on*[a, b].*

(3.49)

We denote by*u*0_{and}* _{v}*0

_{the functions such that for all}

_{x}_{∈}_{Ω}

_{with}

_{x}_{=}_{(x1, x2, . . . , x}*N)*

*u*0*(x)=u*ˆ*m*

*x1,* *v*0*(x)=v*ˆ*m*

*x1,* (3.50)

where*(u*0* _{, v}*0

_{)}_{is clearly in}

*1*

_{W}*,p*

_{(}_{Ω}

*1*

_{)}_{×}_{W}*,q*

_{(}_{Ω}

_{), moreover by hypothesis (H1), we easily}obtain

*−*∆p*u*0*≥fu*0*, v+h1* for*v≤v*0a.e. onΩ*,*
*−*∆q*v*0* _{≥}_{g}_{u, v}*0

_{+}_{h2}_{for}

*0*

_{u}_{≤}_{u}_{a.e. on}

_{Ω}

_{,}*u*0_{≥}_{0,} * _{v}*0

_{≥}_{0}

_{on}

_{Ω}

_{.}(3.51)

Then the result follows.

**3.2. Construction of a subsolution***(u0, v0)***.** Similar to the construction of a
super-solution we can prove the following result.

**Proposition** **3.10.** *Under hypotheses (H1) and (2.2) of (H2), problem (1.1) has a*

*subsolution(u0, v0)inW*1*,p _{(}*

_{Ω}

*1*

_{)}_{×}_{W}*,q*

_{(}_{Ω}

_{).}**4. Proof ofTheorem 2.1.** We proceed in the following steps.

(i) From Propositions3.9and3.10, there exists a pair*[(u0, v0);(u*0* _{, v}*0

_{)]}_{of }sub-supersolution of problem (1.1).

(ii) Construction of an invariant set. In order to apply Schauder’s ﬁxed point
theorem, we introduce the set*K=[u0, u*0* _{]}_{×}_{[v0, v}*0

*nonlinear operator*

_{]. Next we deﬁne the following}*T*: for all

*(u1, v1)∈W*1

*,p*

_{(}_{Ω}

*1*

_{)}_{×}_{W}*,q*

_{(}_{Ω}

_{), we associate}_{(u2, v2)}_{=}*T ((u1, v1)), where(u2, v2)*is the solution of the system

*−*∆p*u= _{f}*ˆ

_{x, u, v1}_{,}

_{−}_{∆q}

_{v}_{=}_{g}_{ˆ}

_{x, u1, v}_{in}

_{Ω}

_{,}*u=*0, *v=*0 on*∂*Ω*,* (4.1)

where ˆ

with

*U (x)=u(x)+u0−u _{+}−u−u*0

_{+},*V (x)=v(x)+v0−v*0

_{+}−v−v*+.*

(4.3)

The functions ˆ*f* and ˆ*g*are bounded, so the operator*T* is well deﬁned. Furthermore,
*K*is an invariant set for*T*. Let*(u1, v1)∈K*and*(u2, v2)=T ((u1, v1)).*

We show, for example, that*u2≤u*0_{. From (}_{3.51}_{), (}_{4.1}_{), and (}_{4.2}_{) we have}

0*≥ −*∆p*u2− _{f}*ˆ

_{x, u2, v1}_{≥ −}_{∆p}

_{u2}_{−}_{f}_{U2, V1}_{−}_{h1(x)}*≥* *−*∆p*u2+*∆p*u*0*+* *fu*0*, v1−fU2, V1,* (4.4)

multiplying (4.4) by*(u2−u*0_{)}

*+*and integrating overΩ, we obtain

0*≥*

Ω *u2|*

*p−*2* _{}_{u2}_{−}_{}_{u}*0

*p−*2

*u*0*u2−u*0_{+}dx

*+*

Ω *f*

*u*0*, v1−fU2, V1u2−u*0_{+}dx.

(4.5)

Since*v1∈[v0, v*0_{], we have}_{V1}_{=}_{v1, where}_{V1}_{is associated with}_{v1}_{as in (}_{4.3}_{).}
Denote byΩ*+*the set

Ω*+=x∈*Ω; *u2−u*0*>*0*.* (4.6)

We have*U2=u*0_{in}_{Ω}

*+*. Then

Ω *f*

*u*0*, v1−fU2, V1u2−u*0_{+}dx*=*

Ω *f*

*u*0*, v1−fu0, v1u2−u*0* _{+}dx=*0.

(4.7)

By the monotonicity of*−*∆pin*Lp _{(}*

_{Ω}

*0*

_{), we get that 0}_{≥ }_{(u2}_{−}_{u}

_{)}*+Lp _{(}*

_{Ω}

*.*

_{)}Thus*u2≤u*0 _{on}_{Ω}_{and similarly}* _{v2}_{≤}_{v}*0

_{on}

_{Ω}

_{. So that the property,}

*holds.*

_{T (K)}_{⊂}_{K,}(iii) The operator*T* is completely continuous.
(a) We prove that*T* is compact; let*(uj*1*, v*

*j*

1*)j* be a bounded sequence in*Lp(*Ω*)×*

*Lq _{(}*

_{Ω}

_{). Let}_{(u}j2*, v*

*j*

2*)=T ((u*

*j*

1*, v*

*j*

1*)), so multiplying (*4.1) by*u*

*j*

2, we obtain

Ω
*uj*2

*p*

*dx=*

Ω ˆ

*fx, uj*2*, v*

*j*

1

*uj*2*dx≤C*

Ω
u*j*

2

*p*

*dx*
1*/p*

*.* (4.8)

Therefore,*(uj*2*)j*is bounded in*W*1*,p(*Ω*)*and it possesses a convergent subsequence

in*Lp _{(}*

_{Ω}

_{). Analogously for}_{(v}j2*)j*in*Lq(*Ω*).*

(b) Now we prove the continuity of the operator*T*; from the continuity of the
func-tions*f* and*g*associated at the bounded functions ˆ*f ,g, and by the dominated con-*ˆ
vergence theorem, we deduce easily the continuity of the operator*T.*

**Acknowledgments.** A part of this work was made when the author was visiting

the Laboratory MIP of the University of Toulouse-3.

The author also thanks Professor François de Thélin for useful discussions.

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Abdelaziz Ahammou: Département des Mathématiques et Informatique, Faculté des Sciences, Université Chouaib Doukkali, BP20, 24000 El Jadida, Morocco