On the existence of bounded solutions of nonlinear elliptic systems

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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)+h2inΩ,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 operatorTand some invariant setKwhere 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:

−∆pu=f (u, v)+h1, −∆qv=g(u, v)+h2 inΩ,

u=v=0 on∂Ω, (1.1)

whereΩis a smooth bounded domain ofRN_{, whereN≥1,p, q >1,f , gare continuous}

functions ofR2_{intoRandh1, h2are the functions given inL}+∞_(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 nonlinearityf andg:

f (u, v)≤a1|u|α0|_v|β0+1+_a2|_u|α1−1+_a3|_v|β1−1_, g(u, v)≤a4|u|α0+1|_v|β0+_a5|_u|α2−1+_a6|_v|β2−1_,

(1.2)

whereai(i=1, . . . ,6)are positive constants andαiandβi(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:

−∆mu=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−sm

m

, (1.6)

andR(Ω)denotes the radius of the smallest open ballB(0, R)containingΩ. The par-ticular casesN=1 andm=2 were considered in [10]. It was shown there that (1.4) is solvable for anyh∈L∞(Ω)if

lim

u→∞inf

2F (u)

u2 < λ1,2, (1.7) whereλ1,2is the first eigenvalue of−∆ andΩ=]a, b[. Observe that forN >1,we haveµ2< λ1,2(Ω). Then, the question naturally arises whetherµmcan be replaced by

λ1,m(Ω)in (1.5), whereλ1,mis 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 constantsµp andµq, defined below,

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

better thanµm , 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 forf andg.

2. Main result. We make the following assumptions:

(H1) (i) The functionf (u,·)is a nonincreasing function onRfor alluinR, (ii) The functiong(·, v)is a nonincreasing function onRfor allvinR. (H2) There exists some unbounded increasing subsequence(mk)k, satisfying

lim

k→+∞

pFm1k/p, m

1/q k

mk

< µp, lim k→+∞

qGm1k/p, m

1/q k

mk

< µq, (2.1)

lim

k→+∞

pF−m1k/p,−m

1/q k

mk

< µp, lim k→+∞

qG−mk1/p,−m

1/q k

mk

whereFandGare 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 byµpandµqthe 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)

withb−a=min(bi−ai)andP=Π[ai, bi]is the smallest cube such thatP⊃Ω.

Observe that forN=1,µpandµqare, respectively, the first eigenvalue of−∆pand −∆qwhenΩ=]a, b[. It is clear thatµpis better thanµpdefined in (1.5). In particular,

it is interesting whenΩis a rectangle or a triangle, becauseµpµpandµp≈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(W01,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_{, g(u, v)=b(x)|u|}γ+1_v|v|δ−1_{. (2.5)}

(1) Assume thata∞< 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)inL+∞(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(u0_{, v}0_{)in the sense of Hernandez’s definition [13]. InSection 4, we consider} some compact operatorT and some invariant setK. And, we look for solutions of problem (1.1) as fixed points of the operatorT. We will be in the conditions of the Schauder fixed point theorem.

3. Construction of sub-supersolutions

u0, v0∈W1,p_(Ω)×W1,q_(Ω)∩L+∞_(Ω)×L+∞_(Ω),

u0_{, v}0_∈W1,p_(Ω)×W1,q_(Ω)∩L+∞_(Ω)×L+∞_(Ω),

−∆pu0−fx, u0, v≤0≤ −∆pu0−fx, u0, v inΩ,∀v∈ v0, v0, −∆qv0−fx, u, v0≤0≤ −∆qv0−fx, u, v0 inΩ,∀u∈ u0, u0,

u0≤u0_{, v0≤v}0 _{inΩ, u0≤0≤u}0_{, v0≤0≤v}0 _on∂Ω.

(3.1)

Similar definitions can be found in Diaz and Hernández [7], and Diaz and Herrero [8]. For allM >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 ifF andGsatisfy the assumption (2.1) of (H2), then the same holds for ˆF and ˆG.

Proposition3.2[6]. Under hypothesis (2.1) of (H2), there exist two sequencesd_k andd_ksuch that

(a)m1k/p≥dk≥0, for allk∈Nand

dk

0

ds

p

pFˆdk, m

1/q k

−pFˆs, mk1/q

> 1

0 ds p √

1−sp µp

−1/p

. (3.3)

(b)m1k/p≥dk≥0, for allk∈Nand

d_k

0

dt

q

qGˆm1_k/p, d_k−qGˆm1_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 someµ >0 such that

lim

k→+∞

pFˆm1k/p, m

1/q k

mk

< µ < µp, (3.6)

then

lim

k→+∞µmk−p

ˆ

Fm1k/p, m

1/q k

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

Hs, mk

=µs−pFˆs1/p_{, m}1/q k

. (3.8)

For allk >0, we have

H0, mk

= −pFˆ0, m1k/q

=0,

Hmk, mk=µmk−pFˆ

m1k/p, m

1/q k

>0.

(3.9)

Then for allk∈Nthere existsdk>0 such thatdkp≤mkand for alls∈[0, dpk], we have

Hs, mk≤Hdpk, mk, (3.10)

that is,

µs−pFˆs1/p, m1k/q

≤µdpk−pF

dk, m

1/q k

, (3.11)

then

pFˆdk, m1k/q

−pFˆs1/p_{, m}1/q k

≤µdp_k−s. (3.12)

Lets=ωp_{, whereω∈[0, d}

k]⊂[0, m1k/p]. We obtain

pFˆdk, m

1/q k

−pFˆω, m1k/q

≤µdpk−ωp

, (3.13)

that is,

1 p

dp_k−ωp[µ]

−1/p_≤ 1

p

pFˆdk, mk1/q

−pFˆω, m1_k/q

. (3.14)

Then integrating on[0, dk]we obtain

1

0 dω p√

1−ωp[µ]

−1/p_≤

dk

0

dω

p

pFˆdk, m1k/q

−pFˆω, m1_k/q

. (3.15)

This proves (a).

3.1. Construction of supersolution(u0_{, v}0_{). In the following step we suppose that} for allk∈Nand for alls∈[0, m1_k/p]

fs, m1k/q

+M≥0. (3.16)

Denote by(fˆk)kthe sequence of functions defined by

ˆ fk(s)=

            

fm1_k/p, m1_k/q+M forsinm1_k/p,+∞, fs, m1_k/q+M forsin0, m1_k/p, f0, m1_k/q+M forsin]−∞,0].

For allk∈N, we associate to the function ˆfk, the following problem:

−|u|p−2u(t)=fˆku(t), u(t)≥0 fortin[a, b]. (3.18)

For allk∈N,we define the nonlinear operatorTksuch 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 ˆfkis a nonnegative function, the operatorTkis well defined.

Lemma_3.4. _{For allk}≥_0,

(i) the operatorTkis completely continuous,

(ii) there exists a fixed point forTk.

Proof. _Letk∈N_,

(1) the continuity is immediate,

(2) let(un)nbe a bounded sequence inC([a, b])such that the sequence(Tk(un))n

is also bounded inC([a, b]).

By the continuity of the function ˆfk, there exists some constantCksuch that

t∈[a, b], ∀t, (3.21)

for alln∈Nwe have

Tkun(t)−Tkunt≤Ckt−t. (3.22)

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

(Tk(un))nis relatively compact inC([a, b]).

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

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

Remark_3.5. _{By definition of the operatorTk, we have} (i) −|u_k|p−2_u

k(t)=

t

afˆk(uk(s))ds,

(ii) u_k(a)=0, (iii) uk(a)=dk.

Since ˆfkis a nonnegative function we have

(iv) u_k(t)≤0 fortin[a, b].

That is,ukis a nonincreasing function on[a, b].

Lemma_3.6. _{From (2.1), choose(dk)ksuch that}

uk(t)≥0 in

a,a+b

2

∀k∈N, (3.23)

Proof. Let(d_k)_kbe some sequence such thatd_k∈[0, m1_k/p]for allk∈N. We denote bytka real number such thatuk(tk)=0 anduk(t)≥0 on[a, tk].

Then, fromRemark 3.5, sinceukis a nonincreasing function anddk∈[0, m1k/p], we

have

m1_k/p≥uk≥0 ∀t∈ a, tk. (3.24)

Consequently, for allt∈[a, tk]we have

−uk p−2

uk

(t)=_fˆ_{kuk(t)=fuk(t), m}1/q k

+M. (3.25)

Multiplying (3.25) byu_kwe obtain

p−1 p

−u_k(t)p=_dtd Fˆuk(t), m1k/q

, (3.26)

where

ˆ

Fu, m1_k/q=F (u, v)+Mu. (3.27) Integrating (3.26) on[a, t]⊂[a, tk], we obtain

−p_p−1u

k(t)=

p

pFˆdk, m1k/q

−pFˆuk(t), m1k/q

. (3.28)

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

p_p−1 tk

a

−u_k(t) p

pFˆdk, m1k/q

−pFˆuk(t), m1k/q

dt≤tk−a. (3.29)

Then, we obtain

p

p−1 dk

0

1

p

pFˆdk, m

1/q k

−pFˆs, m1_k/q

ds≤tk−a. (3.30)

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

such that for allk≥k0we have

b−a 2 <

p_p−1 dk

0

1

p

pFˆdk, m1k/q

−pFˆs, m1_k/q

ds. (3.31)

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

satisfyingtk> (b+a)/2.

Proposition_3.7. _{Suppose that the sequence(mk)ksatisfies (2.1), and that for all} k >0we have

inf

s∈[0,m1/p_k ]

Then, there exists some numberk0∈Nsuch that for allk≥k0the problem

−|u|p−2u=fu, m1k/q

+M in(a, b),

u≥0 on[a, b], (3.33)

has a solutionuˆksatisfyinguˆk∈C1([a, b]),(|uˆk|p−2uˆk)∈C([a, b])andm

1/p

k ≥uˆk≥0

for allk≥k0.

Proof. Let(u_k)_kbe the sequence defined inLemma 3.6. This sequence satisfies that for allk≥k0,

uk∈C1

a,a+b

2

, u_kp−2u_k∈C

a,a+b 2

,

−u_kp−2u_k(t)=fuk(t), m1k/q

+M in

a,a+b 2

, (3.34)

m1_k/p≥uˆk≥0 in

a,a+b

2

,

uk(a)=0. (3.35)

We denote by ˆukthe following function:

ˆ uk(t)=

         uk

_3a+b

2 −t

ift∈

a,a+b 2

,

uk

t−a+b 2

ift∈ _a+b

2 , b

.

(3.36)

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

∀k≥k0, uˆk∈C1

[a, b], uˆ_kp−2uˆ_k∈C[a, b], −uˆk

p−2 ˆ uk

(t)=fuˆk(t), m1k/q

+M in[a, b],

mk1/p≥uˆk≥0 in[a, b].

(3.37)

Then the conclusion holds.

Proposition_3.8. _{LetM >0. From (2.1), there exist somem >0and(ˆum,vˆm)}∈ (C1_{([a, b]))}2_{such that}

ˆ um

p−2 ˆ um

,vˆm p−2

ˆ vm

∈C[a, b]2,

−uˆm p−2

ˆ um

≥fuˆm, m1/q

+M in(a, b),

−vˆm q−2

ˆ vm

≥gm1/p_,vˆ m

+M in(a, b),

m1/p_≥uˆ

m≥0, m1/q≥vˆm≥0 on[a, b].

Proof. We study three cases.

Case₁. _{We suppose that for allk}∈N_{we have}

inf

s∈g[0,m1/p_k ]

fs, m1_k/q+M <0, inf

t∈[0,m1/q_k ]

gm1_k/p, t+M <0. (3.39)

Then for allk∈N, there existsm_k∈[0, mk1/p]andtm_k∈[0, m1k/q]satisfying

fsmk, m 1/q k

+M <0, gm1_k/p, tmk

+M <0. (3.40)

Consequently, form=mk, the couple(uˆm,vˆm)=(sm_k, tm_k)satisfies the result.

Case2. Assume that for allk∈Nwe have,

inf

s∈[0,m1/p_k ]

fs, m1_k/q+M≥0, (3.41)

inf

t∈[0,m1/q_k ]

gm1_k/p, t+M <0. (3.42)

(a) From (3.41) andProposition 3.7 there exist some k0∈Nand some sequence (ˆuk)ksuch that, for allk≥k0, we have

ˆ uk∈C1

[a, b], uˆ_kp−2uˆ_k∈C[a, b], −uˆ_kp−2uˆ_k≥fuˆk, m1k/q

+M in(a, b),

mk1/p≥uˆk≥0 in[a, b].

(3.43)

(b) From (3.42), there exists a sequence(tm_k)ksuch that

m1_k/p≥tmk≥0, g

m1_k/p, tmk

+M <0 ∀k≥k0. (3.44)

Consequently, form=mkwithk > k0, the pair(uˆmk, tmk)satisfies the result. Case₃. _{Assume that for allk}∈N_{,we have}

inf

s∈[0,m_k1/p]

fs, m_k1/q+M≥0, inf

t∈[0,m1/q_k ]

gm1_k/p, t+M≥0. (3.45)

Then, fromProposition 3.7, for allk≥k0, there exists(ˆuk,vˆk)∈(C1([a, b]))2such

that

uˆk p−2

ˆ uk

,vˆk p−2

ˆ vk

∈C[a, b]2,

−uˆ_kp−2uˆ_k≥fuˆk, m1k/q

+M in(a, b),

−vˆk p−2

ˆ vk

≥gm1k/p,vˆk

+M in(a, b),

m1k/p≥uˆk≥0, m

1/q

k ≥vˆk≥0 on[a, b].

(3.46)

This proves the results.

Now, for problem (1.1), we consider a smooth bounded domain ΩinRN_{,and we}

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

non-negative supersolution(u0_{, v}0_)inW1,p_(Ω)×W1,q_(Ω).

Proof. LetM≥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 existm >0 and(uˆm,vˆm)∈

(C1_{([a, b]))}2_{such that}

ˆ um

p−2 ˆ um

,vˆm p−2

ˆ vm

∈C[a, b]2, (3.48)

and(uˆm,vˆm)satisfies

−uˆ_mp−2uˆ_m≥fuˆm, m1/q

+M in(a, b),

−vˆm q−2

ˆ vm

≥gm1/p,vˆm

+M in(a, b),

m1/p_≥uˆ

m≥0, m1/q≥vˆm≥0 on[a, b].

(3.49)

We denote byu0_andv0_{the functions such that for allx∈Ωwithx=(x1, x2, . . . , x}

N)

u0(x)=uˆm

x1, v0(x)=vˆm

x1, (3.50)

where(u0_{, v}0_{)is clearly inW}1,p_(Ω)×W1,q_{(Ω), moreover by hypothesis (H1), we easily}

obtain

−∆pu0≥fu0, v+h1 forv≤v0a.e. onΩ, −∆qv0_{≥gu, v}0_{+h2 foru≤u}0_{a.e. onΩ,}

u0_{≥0, v}0_{≥0 onΩ.}

(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)inW1,p_(Ω)×W1,q_(Ω).

4. Proof ofTheorem 2.1. We proceed in the following steps.

(i) From Propositions3.9and3.10, there exists a pair[(u0, v0);(u0_{, 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 setK=[u0, u0_{]×[v0, v}0_{]. Next we define the following} nonlinear operator T: for all(u1, v1)∈W1,p_(Ω)×W1,q_{(Ω), we associate(u2, v2)=}

T ((u1, v1)), where(u2, v2)is the solution of the system

−∆pu=_fˆ_{x, u, v1, −∆qv=gˆx, u1, v inΩ,}

u=0, v=0 on∂Ω, (4.1)

where ˆ

with

U (x)=u(x)+u0−u₊−u−u0₊, V (x)=v(x)+v0−v₊−v−v0

+.

(4.3)

The functions ˆf and ˆgare bounded, so the operatorT is well defined. Furthermore, Kis an invariant set forT. Let(u1, v1)∈Kand(u2, v2)=T ((u1, v1)).

We show, for example, thatu2≤u0_{. From (3.51), (4.1), and (4.2) we have}

0≥ −∆pu2−_fˆ_{x, u2, v1≥ −∆pu2−fU2, V1−h1(x)}

≥ −∆pu2+∆pu0+ fu0, v1−fU2, V1, (4.4)

multiplying (4.4) by(u2−u0₎

+and integrating overΩ, we obtain

0≥

Ω u2|

p−2_u2−u0p−2

u0u2−u0₊dx

+

Ω f

u0, v1−fU2, V1u2−u0₊dx.

(4.5)

Sincev1∈[v0, v0_{], we haveV1=v1, whereV1is associated withv1as in (4.3).} Denote byΩ+the set

Ω+=x∈Ω; u2−u0>0. (4.6)

We haveU2=u0_inΩ

+. Then

Ω f

u0, v1−fU2, V1u2−u0₊dx =

Ω f

u0, v1−fu0, v1u2−u0₊dx=0.

(4.7)

By the monotonicity of−∆pinLp_{(Ω), we get that 0≥ (u2−u}0₎

+Lp_(Ω).

Thusu2≤u0 _{onΩand similarlyv2≤v}0_{onΩ. So that the property,T (K)⊂K,} holds.

(iii) The operatorT is completely continuous. (a) We prove thatT is compact; let(uj1, v

j

1)j be a bounded sequence inLp(Ω)×

Lq_{(Ω). Let(u}j

2, v

j

2)=T ((u

j

1, v

j

1)), so multiplying (4.1) byu

j

2, we obtain

Ω uj2

p

dx=

Ω ˆ

fx, uj2, v

j

1

uj2dx≤C

Ω uj

2

p

dx 1/p

. (4.8)

Therefore,(uj2)jis bounded inW1,p(Ω)and it possesses a convergent subsequence

inLp_{(Ω). Analogously for(v}j

2)jinLq(Ω).

(b) Now we prove the continuity of the operatorT; from the continuity of the func-tionsf andgassociated at the bounded functions ˆf ,g, and by the dominated con-ˆ vergence theorem, we deduce easily the continuity of the operatorT.

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.

References

[1] A. Anane and O. Chakrone,3ème cycle, thesis, Mohammed I University, Oujda, Morocco, 1996.

[2] C. Baoyao,Nonexistence results and existence theorems of positive solutions of Dirichlet problems for a class of semilinear elliptic systems of second order, Acta Math. Sci. (English Ed.)7(1987), no. 3, 299–309.

[3] L. Boccardo, J. Fleckinger, and F. de Thélin,Elliptic systems with various growth, Reaction Diffusion Systems (Trieste, 1995), Lecture Notes in Pure and Appl. Math., 194, Dekker, New York, 1998, pp. 59–66.

[4] H. Brézis and E. H. Lieb,Minimum action solutions of some vector field equations, Comm. Math. Phys.96(1984), no. 1, 97–113.

[5] P. Clément, R. Manásevich, and E. Mitidieri,Positive solutions for a quasilinear system via blow up, Comm. Partial Differential Equations18(1993), no. 12, 2071–2106. [6] D. G. de Figueiredo and J.-P. Gossez,Nonresonance below the first eigenvalue for a

semi-linear elliptic problem, Math. Ann.281(1988), no. 4, 589–610.

[7] J. I. Diaz and J. Hernández,On the existence of a free boundary for a class of reaction-diffusion systems, SIAM J. Math. Anal.15(1984), no. 4, 670–685.

[8] J. I. Diaz and M. A. Herrero,Estimates on the support of the solutions of some nonlinear elliptic and parabolic problems, Proc. Roy. Soc. Edinburgh Sect. A89(1981), no. 3-4, 249–258.

[9] P. Felmer, R. F. Manásevich, and F. de Thélin,Existence and uniqueness of positive solutions for certain quasilinear elliptic systems, Comm. Partial Differential Equations17 (1992), no. 11-12, 2013–2029.

[10] M. L. C. Fernandes, P. Omari, and F. Zanolin,On the solvability of a semilinear two-point BVP around the first eigenvalue, Differential Integral Equations2(1989), no. 1, 63–79.

[11] A. Fonda, J.-P. Gossez, and F. Zanolin,On a nonresonance condition for a semilinear elliptic problem, Differential Integral Equations4(1991), no. 5, 945–951.

[12] J.-P. Gossez and A. El Hachimi,On a nonresonance condition near the first eigenvalue for a quasilinear elliptic problem, thesis, Univ. Catholique de Louvain, France, 1994. [13] J. Vélin and F. de Thélin,Existence and nonexistence of nontrivial solutions for some

non-linear elliptic systems, Rev. Mat. Univ. Complut. Madrid6(1993), no. 1, 153–194.

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