EXISTENCE OF WEAK SOLUTIONS FOR NONLINEAR ... - EMIS

Electronic Journal of Differential Equations, Vol. 2006(2006), No. 69, pp. 1–10. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp)

EXISTENCE OF WEAK SOLUTIONS FOR NONLINEAR ELLIPTIC SYSTEMS ON RN EADA A. EL-ZAHRANI, HASSAN M. SERAG

Abstract. In this paper, we consider the nonlinear elliptic system −∆p u = a(x)|u|p−2 u − b(x)|u|α |v|β v + f, −∆q v = −c(x)|u|α |v|β u + d(x)|v|q−2 v + g, lim u =

|x|→∞

lim v = 0

|x|→∞

u, v > 0

on a bounded and unbounded domains of RN , where ∆p denotes the pLaplacian. The existence of weak solutions for these systems is proved using the theory of monotone operators

1. Introduction The generalized (the so-called weak) formulation of many stationary boundaryvalue problems for partial differential equations leads to operator equation of type A(u) = f on a Banach space. Indeed, the weak formulation consists in looking for an unknown function u from a Banach space V such that an integral identity containing u holds for each test function v from the space V . Since the identity is linear in v, we can take its sides as values of continuous linear functionals at the element v ∈ V . Denoting the terms containing unknown u as the value of an operator A, we obtain (A(u), v) = (f, v) ∀v ∈ V, which is equivalent to equality of functionals on V , i.e. the equality of elements of V 0 (the dual space of V ): A(u) = f . Functional analysis yields tools for proving existence of generalized (weak) solutions to a relatively wide class of differential equations that appear in mathematical physics and industry. In our work, we consider nonlinear systems with model A of the form A{u, v} = {−∆p u−a(x)|u|p−2 u+b(x)|u|α |v|β v, −∆q v+c(x)|u|α |v|β u−d(x)|v|q−2 v} These nonlinear systems involving p-Laplacian appear in many problems in pure and applied mathematics e.g. in quasiconformal mappings, non-Newtonian fluids, and nonlinear elasticity [3, 4, 9]. 2000 Mathematics Subject Classification. 35B45, 35J55. Key words and phrases. Weak solutions; nonlinear elliptic systems; p-Laplacian; monotone operators. c

2006 Texas State University - San Marcos. Submitted February 9, 2006. Published July 6, 2006. 1

2

E. A. EL-ZAHRANI, H. M. SERAG,

EJDE-2006/69

The existence of solutions for such systems was proved, using the method of sub and super solutions in [7, 8, 20]. Here, we use another technique for proving the existence of weak solutions. We use the theory of monotone operators. First, we consider the following system defined on a bounded domain Ω of RN with boundary ∂Ω: −∆p u = a(x)|u|p−2 u − b(x)|u|α |v|β v + f (x), q−2

−∆q v = d(x)|v|

β

α

v − c(x)|v| |u| u + g(x),

u = v = 0,

in Ω in Ω

on ∂Ω .

Then, we generalize the discussion to system defined on the whole space RN . This article is organized as follows: Some technical results and definitions are introduced in section two concerning the theory of nonlinear monotone operators, also, the scalar case is discussed. Section three, is devoted to study the existence of solutions for nonlinear systems defined on a bounded domain. In section four, the existence of solutions for nonlinear systems defined on unbounded domain is proved. 2. Scalar case First, we introduce some technical results [6, 8, 21]. Definitions. Let A : V → V 0 be an operator on a Banach space V . We say that the operator A is: = ∞; Coercive if limkuk→∞ hA(u),ui kuk Monotone if hA(u1 ) − A(u2 ), u1 − u2 i ≥ 0 for all u1 , u2 ; w Strongly continuous if un → u implies A(un ) → A(u); w w Weakly continuous if un → u implies A(un ) → A(u); w Demicontinuous if un → u implies A(un ) → A(u). w w The operator A is said to satisfy the Mo -condition if un → u, A(un ) → f , and [hA(un ), un i → hf, ui] imply A(u) = f . Theorem 2.1. Let V be a separable reflexive Banach space and A : V → V 0 an operator which is: coercive, bounded, demicontinuous, and satisfying Mo condition. Then the equation A(u) = f admits a solution for each f ∈ V 0 . Now, we prove the existence of a weak solution u ∈ W01,p (Ω) for the scalar case −∆p u = m(x)|u|p−2 u + f (x), u=0

x ∈ Ω,

on ∂Ω

(2.1)

where 0 < a(x) ∈ L∞ (Ω) and Ω is a bounded domain of RN . In this case, the operator A is Au = −∆p u − m(x)|u|p−2 u. It is proved in [2], that if m(x) is a positive function in L∞ (Ω), then the first eigenvalue λp (m) of the Dirichlet p-Laplacian problem −∆p u = λm(x)|u|p−2 u u(x) = 0

on ∂Ω

in Ω

(2.2)

is simple, isolated and it is the unique positive eigenvalue having a nonnegative eigenfunction. Moreover it is characterized by Z Z p |∇u| ≥ λp (m) m(x)|u|p (2.3) Ω

Ω

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EXISTENCE OF WEAK SOLUTIONS

3

We prove that (2.1) admits a weak solution if λp (m) > 1 . First, we prove that A is a bounded operator: Z Z (Au, v) = |∇u|p−2 ∇u∇v − m(x)|u|p−2 uv Ω

Ω

Using H¨ older’s inequality, we obtain Z p−1 Z p1 Z Z p1 p−1 p p |(Au, v)| ≤ |∇u|p |∇v|p + m(x)|v|p m(x)|u|p ≤

Ω p−1 kvk1,p kuk1,p

Ω

Ω

Ω

To prove that A is continuous, let us assume that un → u in W01,p (Ω). Then kun − uk1,p → 0 So that k∇un − ∇ukp → 0 Applying Dominated convergence theorem, we obtain k|∇un |p−2 ∇un − |∇u|p−2 ∇ukp → 0 hence kAun − Aukp ≤ k|∇un |p−2 ∇un − |∇u|p−2 ∇ukp + k|un |p−2 un − |u|p−2 ukp → 0 Operator A is strictly monotone: Z Z p−2 (Au1 − Au2 , u1 − u2 ) = |∇u1 | ∇u1 ∇u1 + |∇u2 |p−2 ∇u2 ∇u2 Ω Ω Z Z − |∇u1 |p−2 ∇u1 ∇u2 − |∇u2 |p−2 ∇u2 ∇u1 Ω Z Ω Z Z p−1/p Z 1/p p p ≥ |∇u1 | + |∇u2 | − |∇u1 |p |∇u2 |p Ω Ω Ω Ω Z p−1/p Z 1/p p p |∇u2 | |∇u1 | − = =

Ω Ω p p p−1 ku1 kp + ku2 kp − ku1 kp ku2 kp − ku2 kp−1 ku1 kp p p−1 p−1 ku1 k1,p − ku2 k1,p ku1 k1,p − ku2 k1,p > 0 .

Also, A is a coercive operator, since from (2.3), we have Z Z (Au, u) = |∇u|p − m|u|p Ω ZΩ Z 1 p ≥ |∇u| − |∇u|p λp (m) Ω Ω Z 1 = 1− |∇u|p . λp (m) Ω Then (Au, u) p−1 = kuk1,p → ∞ as kukp

kuk1,p → ∞

which proves the existence of a weak solution for (2.1).

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3. Nonlinear systems on bounded domains In this section, we consider the system −∆p u = a(x)|u|p−2 u − b(x)|u|α |v|β v + f (x), q−2

−∆q v = d(x)|v|

β

in Ω

α

v − c(x)|v| |u| u + g(x),

u = v = 0, 1 p

on ∂Ω

where Ω is a bounded domain of R , + = 1, 1q + a(x), b(x), c(x), d(x) are positive functions in L∞ (Ω). N

(3.1)

in Ω

1 p0

1 q0

= 1, α + β + 2 < N and

Theorem 3.1. For (f, g) ∈ Lp (Ω) × Lq (Ω), there exists a weak solution (u, v) ∈ W01,p (Ω) × W01,q (Ω) for system (3.1) if the following condition is satisfied: λp (a) > 1,

and

λq (d) > 1 .

(3.2)

Proof. We transform the weak formulation of the system (3.1) to the operator form A(u, v) − B(u, v) = F where, A, B and F are operators defined on W01,p (Ω) × W01,q (Ω) by Z Z (A(u, v), (Φ1 , Φ2 )) = |∇u|p−2 ∇u∇Φ1 + |∇v|q−2 ∇v∇Φ2 , Ω

Ω

Z (B(u, v), (Φ1 , Φ2 )) =

Z

a(x)|u|p−2 uΦ1 + d(x)|v|q−2 vΦ2 Ω Z Z − b(x)|u|α |v|β vΦ1 − c(x)|v|β |u|α uΦ2 Ω

Ω

and

Ω

Z (F, Φ) = ((f1 , f2 ), (Φ1 , Φ2 )) =

Z f1 Φ1 +

Ω

f2 Φ2 Ω

We can write the operator A(u, v) as the sum of the two operators J2 (v), J1 (u), where Z Z (J2 (v), (Φ2 )) = |∇v|q−2 ∇v∇Φ2 and (J1 (u), (Φ1 )) = |∇u|p−2 ∇u∇Φ1 . Ω

Ω

Operators J1 and J2 are bounded, continuous, and strictly monotone; so their sum, the operator A, will be the same. For the operator B(u, v), 0

0

B(u, v) : W01,p (Ω) × W01,q (Ω) → Lp (Ω) × Lq (Ω) ⊂ W0−1,p (Ω) × W0−1,q (Ω), using Dominated convergence theorem and compact imbedding property [1] for the space W01,p (Ω) inside the space Lp (Ω) and the space W01,q (Ω) inside Lq (Ω), when Ω is a bounded domain of RN , we can prove that it is a strongly continuous operator. w w To prove that let us assume that vn → v in W01,q (Ω) and un → u in W01,p (Ω). Then p q (un , vn ) → (u, v) in L (Ω)×L (Ω). Also, (∇un , ∇vn ) → (∇u, ∇v) in Lp (Ω)×Lq (Ω). By the Dominated Convergence Theorem, we have: a(x)|un |p−2 un → a(x)|u|p−2 u

in Lp (Ω)

d(x)|vn |q−2 vn → d(x)|v|q−2 v

in Lq (Ω)

−b(x)|un |α |vn |β vn → −b(x)|u|α |v|β v

in Lp (Ω)

−c(x)|vn |β |un |α un → −c(x)|v|β |u|α u

in Lq (Ω) .

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Since (B(un , vn ) − B(u, v), (w1 , w2 )) Z Z p−2 p−2 = a(x)(|un | un − |u| u)w1 + d(x)(|vn |q−2 vn − |v|q−2 v)w2 Ω Ω Z Z − b(x)(|un |α |vn |β vn − |u|α |v|β v)w1 − c(x)(|vn |β |un |α un − |v|β |u|α u)w2 , Ω

it follows that kB(un , vn ) − B(u, v)k ≤ ka(x)(|un |p−2 un − |u|p−2 u)kp + kd(x)(|vn |q−2 vn − |v|q−2 v)kq + kb(x)(|un |α |vn |β+1 − |u|α |v|β+1 )kp + kc(x)(|un |α+1 |vn |β − |u|α+1 |v|β )kq → 0 . This proves that −B(u, v) is a strongly continuous operator. So A(u, v) − B(u, v) will be an operator satisfying the Mo -condition. Now, it remains to prove that A(u, v) − B(u, v) is a coercive operator: |(A(u, v) − B(u, v), (u, v))| Z Z Z Z = |∇u|p + |∇v|q − a(x)|u|p − d(x)|v|q Ω Ω Z Z α+1 β+1 + b(x)|u| |v| + c(x)|u|α+1 |v|β+1 Ω Ω Z Z Z Z 1 1 p p q |∇u| − |∇v|q ≥ |∇u| + |∇v| − λp (a) Ω λq (d) Ω Ω Ω Z Z 1 1 |∇u|p + 1 − |∇v|q = 1− λp (a) Ω λq (d) Ω From (3.2), we deduce (A(u, v) − B(u, v), (u, v)) ≥ c(kukp1,p + kvkq1,q ) = c|(u, v)kW 1,p ×W 1,q 0

0

So that hA(u, v) − B(u, v), (u, v)i → ∞

as

k(u, v)kW 1,p ×W 1,q → ∞ . 0

0

This proves the coercive condition and so, the existence of a weak solution for system (3.1). 4. Nonlinear systems defined on Rn We consider the nonlinear system −∆p u = a(x)|u|p−2 u − b(x)|u|α |v|β v + f, −∆q v = −c(x)|u|α |v|β u + d(x)|v|q−2 v + g, lim u = lim v = 0

|x|→∞

|x|→∞

(4.1)

u, v > 0

which is defined on RN . We assume that 1 ≤ N2N +1 < p, q < N and the coefficients a(x), b(x), c(x), d(x) are smooth positive functions such that a(x), d(x) ∈ Lp/N (Rn ) ∩ L∞ (Rn ), α+1 β+1 + = 1, p q

α + β + 2 < N,

(4.2)

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and b(x) < (a(x))α+1/p (d(x))β+1/q

(4.3)

c(x) < (a(x))α+1/p (d(x))β+1/q

To prove our theorem, we need the following results which are studied in [12] and that we recall briefly: Let us introduce the Sobolev space D1,p (RN ) defined as the completion of C0∞ (RN ) with respect to the norm Z 1/p kukD1,p = |∇u|p . RN

It can be shown that Np D1,p (RN ) = u ∈ L N −p (RN ) : ∇u ∈ (Lp (RN ))N and that there exists k > 0 such that for all u ∈ D1,p (RN ), kukLN p/(N −p) ≤ KkukD1,p (RN ) .

(4.4)

Clearly, the space D1,p (RN ) is a reflexive Banach space embedded continuously in the space LN p/(N −p) (RN ). Lemma 4.1. The eigenvalue problem −∆p u = λa(x)|u|p−2 u u(x) → 0

in RN

(4.5)

as |x| → ∞

admits a positive principal eigenvalue Λa (p) which is associated with a positive eigenfunction φ ∈ D1,p (RN ); moreover Λa (p) is characterized by Z Z Λa (p) a(x)|u|p ≤ |∇u|p , ∀u ∈ D1,p (RN ) (4.6) RN

RN Np N (p−1)+p

Nq

Theorem 4.2. For (f, g) ∈ L (RN ) × L N (q−1)+q (RN ), there exists a weak solution (u, v) ∈ D1,p (RN ) × D1,q (RN ) for system (4.1) if the following conditions are satisfied: Λp (a) > 1, and Λq (d) > 1 . (4.7) Proof. By transforming the weak formulation for the system to the operator formulation, we will get the bounded operators A, B, F on the space D1,p (RN )×D1,q (RN ) which take the same previous definitions in Theorem 3.1. To distinguish that: let us assume that (Φ1 , Φ2 ) in D1,p (RN ) × D1,q (RN ), then applying H¨older inequality, we get |(A(u, v), (Φ1 , Φ2 ))| Z Z p−1 ≤ |∇u| |∇Φ1 | + RN

≤ = ≤ =

Z

|∇v|q−1 |∇Φ2 |

RN p

|∇u|

Z p−1 p

p

|∇Φ1 |

p1

+

Z

RN RN RN p−1 q−1 kukD1,p kΦ1 kD1,p + kvkD1,q kΦ2 kD1,q q−1 1,p + kΦ2 kD 1,q ) (kukp−1 D 1,p + kvkD 1,q )(kΦ1 kD p−1 q−1 kukD1,p + kvkD1,q k(Φ1 , Φ2 )kD1,p ×D1,q

q

|∇v|

q−1 Z q RN

|∇Φ2 |q

q1

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For the operator B(u, v), we have |(B(u, v), (Φ1 , Φ2 ))| Np Z Z N ≤ (a(x)) p

Np

|u(x)| N −p

−p) Z (p−1)(N Np

RN

+

Z

+

Nq

|v| N −q

(d(x)) q

N

× ×

Np

|u| N −p

α(NN−p) Z p

RN

|Φ1 |

Np N −p

NN−p p

+

Z

Nq

|v| N −q

β(NN−q) Z q

RN

NN−q q Nq

|v| N −q

−q) (β+1)(N Nq

RN N

(c(x)) α+β+2

Z α+β+2 N

Np

|u| N −p

−p) (α+1)(N Np

RN

RN

RN

Z

Nq

|Φ2 | N −q

RN

α+β+2 Z N

RN

Z

−q) Z (q−1)(N Nq

RN

(b(x)) α+β+2

NN−p p

RN

Nq Z N

RN

Z

Np

|Φ1 | N −p

Np

|Φ2 | N −p

NN−q q

RN

q−1 1,p + k2 kvk 1,q kΦ2 kD 1,q ≤ k1 kukp−1 D 1,p kΦ1 kD D β+1 β α+1 + k3 kukα D 1,p kvkD 1,p kΦ1 kD 1,p + k4 kukD 1,p kvkD 1,q kΦ2 kD 1,q q−1 β+1 β α+1 α ≤ k1 kukp−1 D 1,p + k2 kvkD 1,q + k3 kukD 1,p kvkD 1,p + k4 kukD 1,p kvkD 1,q

× k(Φ1 , Φ2 )kD1,p ×D1,q , this proves the boundedness of the operator B(u, v). For F , we have |(F, Φ)| = |((f1 , f2 ), (Φ1 , Φ2 ))| N (p−1)+p Z Z Np Np N (p−1)+p ≤ (|f1 |)

Np

|Φ1 | N −p

NN−p p

RN

RN

Z

Nq

+ (|f2 |) N (q−1)+q RN ≤ kf1 k N p + kf2 k N (p−1)+p

N (q−1)+q Z Nq

Nq

|Φ2 | N −q

NN−q q

RN

Nq N (q−1)+q

k(Φ1 , Φ2 )kD1,p ×D1,q .

Now, the operator A(u, v) = J1 (u) + J2 (v) is continuous and strictly monotone on D1,p × D1,q , since p−1 1,p − ku2 kD 1,q ) > 0, (J1 (u1 ) − J1 (u2 ), u1 − u2 ) ≥ (ku1 kp−1 D 1,p − ku2 kD 1,p )(ku1 kD q−1 1,q − ku2 kD 1,q ) > 0 (J2 (u1 ) − J2 (u2 ), u1 − u2 ) ≥ (ku1 kq−1 D 1,q − ku2 kD 1,q )(ku1 kD

For the operator B(u, v), we can prove that it is a strongly continuous operator by using Dominated convergence theorem and continuous imbedding property for the Np Nq w space D1,p (RN )×D1,q (RN ) into L N −p (RN )×L N −q (RN ): let us assume that vn → v 1,q N w 1,p N p N q in D (R ) and un → u in D (R ). Then (un , vn ) → (u, v) in L (R ) × L (RN ) and (∇un , ∇vn ) → (∇u, ∇v) in Lp (RN ) × Lq (RN ). Now, the sequence (un ) is bounded in D1,p (RN ), then it is containing a subsequence again denoted by (un ) Np converges strongly to u in L N −p (Br0 ) for any bounded ball Br0 = {x ∈ RN : kxk ≤ Nq Np r0 } . Similarly (vn ) converges strongly to v in L N −q (Br0 ). Since un , u ∈ L N −p (Br0 )

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EJDE-2006/69

Nq

and vn , v ∈ L N −q (Br0 ). Then using the dominated convergence theorem, we have ka(x)(|un |p−2 un − |u|p−2 u)k kd(x)(|vn |q−2 vn − |v|q−2 v)k

Np N (p−1)+p

Nq N (q−1)+q

kb(x)(|un |α−1 |vn |β+1 un − |u|α−1 |v|β+1 u)k kc(x)(|un |α+1 |vn |β−1 un − |u|α+1 |v|β−1 u)k

→ 0,

(4.8)

→ 0,

(4.9)

Np N (p−1)+p Nq N (q−1)+q

→ 0,

(4.10)

→ 0.

(4.11)

Then kB(un , vn ) − B(u, v)kD1,p (Br0 )×D1,q (Br0 ) ≤ ka(x)(|un |p−2 un − |u|p−2 u)k

Np N (p−1+p)

+ kd(x)(|vn |q−2 vn − |v|q−2 v)k

+ kb(x)(|un |α |vn |β+1 un − |u|α |v|β+1 v)k

Nq N (q−1+q)

Np N (p−1)+p

+ kc(x)(|un |α+1 |vn |β−1 un − |u|α+1 |v|β−1 v)k

Nq N (q−1)+q

→ 0.

It remains to study the norm kB(un , vn ) − B(u, v)kD1,p (RN −Br0 )×D1,q (RN −Br0 ) It is sufficient to study the norms in the inequalities (4.8)–(4.11) and try to make it as small as possible. We will study the norm in (4.8) only because the others will be the same. Since, (un ) converges weakly in the space D1,p (RN ), using Sobelev inequalNp ity, (un ) will be bounded in the space L N −p (RN ), so |un |p−1 will be bounded in Np Np L N (p−1)+p (RN −Br0 ) and (|un |p−2 un −|u|p−2 u) is bounded in L N (p−1)+p (RN −Br0 ). R N N Since, a(x) ∈ L p (RN ), we can make the integral (RN −Br ) |a(x)| p as small as 0 possible by choosing r0 big as possible, this means that there exists r0 > 0 such that Np < .M = ka(x)(|un |p−2 un − |u|p−2 u)k N (p−1)+p N 4 4 (R −Br0 ) L for all n ≥ N0 , r ≥ r0 . Since ka(x)(|un |p−2 un − |u|p−2 u)k

Np

L N (p−1)+p (RN )

= ka(x)(|un |p−2 un − |u|p−2 u)k

Np

L N (p−1)+p (Br0 )

+ ka(x)(|un |p−2 un − |u|p−2 u)k

,

Np

L N (p−1)+p (RN −Br0 )

it follows that ka(x)(|un |p−2 un − |u|p−2 u)k

Np

→ 0.

L N (p−1)+p (RN )

By repeating the previous steps on the remaining terms in kB(un , vn ) − B(u, v)kD1,p (RN )×D1,q (RN ) , we can prove that this norm tending strongly to zero and then the operator B(u, v) is strongly continuous. It remains to justify that the operator A(u, v) − B(u, v) is

EJDE-2006/69


9

a coercive operator. From (4.3), (4.6) and (4.7), we obtain (A(u, v) − B(u, v), (u, v)) Z Z Z Z p q p = |∇u| + |∇v| − a(x)|u| − d(x)|v|q N N N N R R R Z Z R α+1 β+1 α+1 β+1 + b(x)|u| |v| + b(x)|u| |v| Z Z Z Z 1 1 p q p ≥ |∇u| + |∇v| − |∇u| − |∇v|q Λa (p) RN Λd (q) RN RN RN Z Z 1 1 |∇u|p + 1 − |∇v|q = 1− Λa (p) RN Λd (q) RN > c kukpD1,p + kvkqD1,q . So that (A(u, v) − B(u, v), (u, v)) → ∞ as

k(u, v)kD1.p ×D1,q → ∞

The coercive condition for the operator completes the proof of the existence of a weak solution for system (4.1). References [1] R. A. Adams; Sobolev Spaces, Academic Press, New York, 1975. [2] A. Anane; Siomlicite et Isolution de la Premiere Valeur Propre du p-Laplaian aves Poids, Comptes Rendus Acad. Sc. Paris, vol. 305 (1987), 725-728. [3] C. Atkinson, K. El-Ali; Some boundeary value problems for the Bingham model , J. nonNewtonian Fluid Mech., vol. 41 (1992), 339-363. [4] C. Atkinson, Champion, C. R.; On some boundary-value problems for the equation ∇.(F |∇w|∇w) = 0, Proc. R. Soc. London A, vol. 448 (1995), 269-279. [5] G. Barles; Remarks on Uniqueness Results of the First Eigenvalue of the p-Laplacian , Ann. Fac. Sc. Toulouse t., vol. 9 (1988), 65-75. [6] M. Berger; Nonlinear and Functional Analysis, Academic Press, New York 1977. [7] L. Boccardo, J. Fleckinger, F. De Thelin; Existence of Solutions for Some Non-Linear Cooperative Systems and Some Applications, Diff. and Int. Eqn., vol. 7 no. 3 (1994), 689-698. [8] M. Bouchekif, H. Serag, F. and De Thelin; On Maximum Principle and Existence of Solutions for Some Nonlinear Elliptic Systems, Rev. Mat. Apl., vol. 16 (1995), 1-16. [9] J. I. Diaz; Nonlinear Partial Differential Equations and free Boundaries, Pitman, Program (1985). [10] P. Drabek, A. Kufner, F. Nicolosi; Quasilinear Elliptic Equations with Degenerations and Singularities, Walter de Gruyter, New York, 1997. [11] J. Fleckinger, J. Hernandes, P. Takac, F. De Thelin; On Uniqueness and positivity for Solutions of equations with the p-Laplacian, J. Diff. and Int. Eqns., vol. 8 (1996), 69-85. [12] J. Fleckinger, R. Manasevich, N. Stavrakakies, F. De Thelin; Principal Eigenvalues for some Quasilinear Elliptic Equations on Rn , Advances in Diff. Eqns., vol. 2 no. 6 (1997), 981-1003. [13] J. Fleckinger, H. Serag; Semilinear Cooperative Elliptic Systems on Rn , Rend. di Mat., vol. Seri VII, 15 Roma (1995), 89-108. [14] J. Fleckinger, P. Takac; Uniqueness of Positive Solutions for Non-Linear Cooperative Systems with the p-Laplacian, Indian Univ. Math. J., vol. 34 no. 4 (1994), 1227-1253. [15] J. Fleckinger, R. F. Mansevich, N. M. Stravrakakis, F. de Thelin; Principal Eigenvalues for some Quasilinear Elliptic systems on RN Advances in Diff.Eq., vol. 2, no. 6 (1997), p981-1003 [16] J. Francu; Solvability of Operator Equations, Survey Directed to Differential Equations, Lecture Notes of IMAMM 94, Proc. of the Seminar “Industrial Mathematics and Mathematical Modelling”, Rybnik, Univ. West Bohemia in Pilsen, Faculty of Applied Sciences, Dept. of Math., July, 4-8, 1994. [17] S. Fucik, A. Kufner; Nonlinear Differential Equations, Czeeh edition SNTL, Prague, (1978), English translation, Elsevier, Amsterdam 1980.

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[18] S. Fucik, J. Necas, J. Soucek, V. Soucek; Spectral Analysis of Nonlinear Operators, SpringerVerlag, New York, 1973. [19] D. Pascali, S. Sburlan; Nonlinear mappings of monotone type, Editura Academiei, Bucuresti, Romania, 1978. [20] H. M. Serag, E. A. El-Zahrani; Maximum Principle and Existence of Positive Solution for Nonlinear Systems on RN , Electron. J. Diff. Eqns., vol. 2005 (2005) no. 85, 1-12. [21] E. Zeidler; Nonlinear Functional Analysis and its Applications I, II, Springer Verlag, New York, 1986. Eada A. El-Zahrani Mathematics Department, Faculty of Science for Girls,, Dammam, P. O. Box 838, Pincode 31113, Saudi Arabia Hassan M. Serag Mathematics Department, Faculty of Science, Al-Azhar University, Nasr City (11884), Cairo, Egypt E-mail address: [email protected]