Electronic Journal of Differential Equations, Vol. 2007(2007), No. 66, pp. 1–14.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu (login: ftp)
MAXIMUM PRINCIPLE AND EXISTENCE OF POSITIVE
SOLUTIONS FOR NONLINEAR SYSTEMS INVOLVING
DEGENERATE P-LAPLACIAN OPERATORS
SALAH A. KHAFAGY, HASSAN M. SERAG
Abstract. We study the maximum principle and existence of positive solutions for the nonlinear system
−∆p,P u = a(x)|u|p−2 u + b(x)|u|α |v|β v + f
in Ω,
−∆Q,q v = c(x)|u|α |v|β u + d(x)|v|q−2 v + g
in Ω,
u=v=0
on ∂Ω,
where the degenerate p-Laplacian defined as ∆p,P u = div[P (x)|∇u|p−2 ∇u].
We give necessary and sufficient conditions for having the maximum principle
for this system and then we prove the existence of positive solutions for the
same system by using an approximation method.
1. Introduction
One of the most useful and best known tools employed in the study of partial
differential equations is the maximum principle, since they are an useful tool to
prove many results such as existence, multiplicity and qualitative properties for
their solutions.
The maximum principle have been studied for linear elliptic systems. In particular, de Figueiredo and Mitidieri [6, 7, 8] gave a necessary and sufficient conditions
for the maximum principle. In [12, 13] the authors proved sufficient and necessary
conditions for having the maximum principle and the existence of positive solutions for linear systems involving Laplace operator with variable coefficients. These
results have been extended in [11], to the nonlinear system
−∆p ui =
n
X
aij |uj |p−2 uj + fi (x)
in Ω,
j=1
ui = 0,
(1.1)
i = 1, 2, . . . n on ∂Ω.
Boushkief, Serag and de Thélin [5], proved the validity of the maximum principle
and the existence of positive solutions for the following nonlinear elliptic system of
2000 Mathematics Subject Classification. 35B50, 35J67, 35J55.
Key words and phrases. Maximum principle; existence of positive solution;
nonlinear elliptic system; degenerated p-Laplacian.
c
2007
Texas State University - San Marcos.
Submitted February 2, 2007. Published May 9, 2007.
1
2
S. A. KHAFAGY, H. M. SERAG,
EJDE-2007/66
two equations involving different operators ∆p , ∆q defined on bounded domain Ω
of Rn , with constant coefficients a, b, c and d
−∆p u = a|u|p−2 u + b|u|α |v|β v + f
α
β
q−2
−∆q v = c|u| |v| u + d|v| v + g
u = u = 0 on Ω.
in Ω,
(1.2)
in Ω,
These results have been extended in [16] to the following nonlinear system defined
on unbounded domain with variable coefficients
−∆p u = a(x)|u|p−2 u + b(x)|u|α |v|β v + f x ∈ Rn ,
−∆q v = c(x)|u|α |v|β u + d(x)|v|q−2 v + g
lim u(x) = lim v(x) = 0,
|x|→∞
|x|→∞
x ∈ Rn ,
u, v > 0
(1.3)
in Rn .
Here, we consider nonlinear system involving degenerated p-Laplacian operators.
We study the following nonlinear system
−∆p,P u = a(x)|u|p−2 u + b(x)|u|α |v|β v + f
in Ω,
−∆Q,q v = c(x)|u|α |v|β u + d(x)|v|q−2 v + g
in Ω,
(1.4)
u = v = 0 on ∂Ω,
where Ω is a bounded subset of Rn with a smooth boundary ∂Ω, ∆p,P with p > 1,
p 6= 2 and P (x) a weight function, denotes the degenerate p-Laplacian defined by
∆p,P u = div[P (x)|∇u|p−2 ∇u], α, β ≥ 0, f, g are given functions and a(x), b(x), c(x)
and d(x) are bounded variable coefficients. We consider here a generalization for
the p-Laplacian to the degenerated p-Laplacian. We obtain necessary and sufficient
conditions on the variable coefficients for having the maximum principle for system
(1.4) and then we prove the existence of positive solutions for this system by using
an approximation method.
This paper is organized as follows: In section 2, we give some assumptions on
the coefficients a(x), b(x), c(x) and d(x), and on the functions f, g to insure the
existence of solution for system (1.4) in a suitable weighted Sobolev space. We
also introduce some technical results and some notations, which are established
in [1, 2, 3, 10]. Section 3 is devoted to the maximum principle of system (1.4).
Finally, in section 4, we prove the existence of solutions for system (1.4) using an
approximation method already used in [4].
2. Technical Results
Now, we introduce some technical results [10] concerning the degenerated homogeneous eigenvalue problem
−∆H,P u = div[H(x)|∇u|p−2 ∇u] = λG(x)|u|p−2 u
in Ω,
u = 0 on ∂Ω,
(2.1)
where H(x) and G(x) are measurable functions satisfying
ν(x)
< H(x) < c1 ν(x),
(2.2)
c1
for a.e. x ∈ Ω with some constant c1 ≥ 1, where ν(x) is a weight function, i.e., a
function which is measurable and positive a.e. in Ω, satisfying the conditions
ν ∈ L1Loc (Ω),
1
ν − p−1 ∈ L1Loc (Ω),
ν −s ∈ L1 (Ω),
(2.3)
EJDE-2007/66
MAXIMUM PRINCIPLE
3
with
s∈(
N
1
, ∞) ∩ [
, ∞),
p
p−1
(2.4)
k
G(x) ∈ L k−p (Ω),
for some constant k satisfying p < k <
p∗s .
p∗s ,
where
p∗s
(2.5)
=
N Ps
N −Ps
with Ps =
ps
s+1
<p<
Lemma 2.1. There exists the least (i.e. the first or principal) eigenvalue λ =
λG (p, Ω) > 0 and at least one corresponding eigenfunction u = uG ≥ 0 a.e. in Ω of
the eigenvalue problem (2.1).
Theorem 2.2. Let H(x) satisfy (2.2) and G(x) satisfy (2.5), then (2.1) admits a
positive principal eigenvalue λG (p). Moreover, it is characterized by
Z
Z
p
G(x)|u| ≤
H(x)|∇u|p .
(2.6)
λG (p)
Ω
Ω
Now, let us introduce the weighted Sobolev space W 1,p (ν, Ω) which is the set of
all real valued functions u defined in Ω for which (see [3, 10])
Z
hZ
i1/p
p
kukW 1,p (ν,Ω) =
|u| +
ν(x)|∇u|p
< ∞.
(2.7)
Ω
Ω
Since we are dealing with the Dirichlet problem, we introduce also the space
W01,p (ν, Ω) as the closure of C0∞ (Ω) in W 1,p (ν, Ω) with respect to the norm
hZ
i1/p
< ∞,
(2.8)
kukW 1,p (ν,Ω) =
ν(x)|∇u|p
0
Ω
which is equivalent to the norm given by (2.7). Both spaces W 1,p (ν, Ω) and
W01,p (ν, Ω) are well defined reflexive Banach Spaces. The space W01,p (ν, Ω) is compactly imbedding into the space Lp (Ω), under the conditions given by (2.3) and
(2.4), i.e.
W01,p (ν, Ω) ,→,→ Lp (Ω),
(2.9)
which means that
Z
Z
p
|u| ≤ c2
ν(x)|∇u|p , i.e., kukLp (Ω) ≤ c kukW 1,p (ν,Ω) .
(2.10)
Ω
0
Ω
3. maximum principle
In this paper, we assume that
α, β ≥ 0;
∗
∗
f ∈ Lp (Ω),
and
p, q > 1,
g ∈ Lq (Ω),
α+1 β+1
+
= 1,
p
q
1
1
1
1
+ ∗ = 1,
+ ∗ = 1.
p p
q
q
1
(P (x))− p−1 ∈ L1Loc (Ω), (P (x))−s ∈ L1 (Ω)
N
1
with s ∈ ( , ∞) ∩ [
, ∞),
p
p−1
P (x) ∈ L1Loc (Ω),
Q(x) ∈ L1Loc (Ω),
(3.1)
1
(Q(x))− q−1 ∈ L1Loc (Ω), (Q(x))−t ∈ L1 (Ω)
N
1
with t ∈ ( , ∞) ∩ [
, ∞),
q
q−1
(3.2)
4
S. A. KHAFAGY, H. M. SERAG,
EJDE-2007/66
We also assume that the variable coefficients a(x), b(x), c(x), and d(x) are bounded
smooth positive functions such that
k
a(x) ∈ L k−p (Ω) ∩ Lp (Ω),
with p < k < p∗s ,
l
d(x) ∈ L l−q (Ω) ∩ Lq (Ω),
(3.3)
with q < l < qt∗
and
b(x) < (a(x))
α+1
p
(d(x))
β+1
q
,
c(x) < (a(x))
α+1
p
(d(x))
β+1
q
.
(3.4)
We say that system (1.4) satisfies the maximum principle if f ≥ 0, g ≥ 0 implies
u ≥ 0, v ≥ 0 for any solution (u, v) for system (1.4)
Theorem 3.1. Assume that (3.1)–(3.4) are satisfied. Then, the maximum principle
holds for system (1.4) if
λa (p) > 1,
(λa (p) − 1)
α+1
p
λd (q) > 1,
(λd (q) − 1)
β+1
q
− 1 > 0.
(3.5)
(3.6)
Conversely, if the maximum principle holds, then (3.5) and (3.7) are satisfied,
where
β+1
α+1
b(x) α+1
c(x) β+1
) p inf (
) q ,
(3.7)
(λa (p) − 1) p (λd (q) − 1) q > Θ inf (
x∈Ω a(x)
x∈Ω d(x)
where
Θ=
p
α+1 β+1
p
q
p
α+1 β+1
p
q
φ
inf Ω ( ψ
q)
φ
supΩ ( ψ
q)
< 1,
and φ (respectively ψ) is the positive eigenfunction associated to λa (p) (respectively
λd (q)) normalized by kφk∞ = kψk∞ = 1.
Proof. The condition is necessary: If λa (p) ≤ 1, then the functions f := a(x)(1 −
λa (p))φp−1 , g := 0 are nonnegative, nevertheless (−φ, 0) satisfies (1.4), which contradicts the maximum principle.
Similarly, if λd (q) ≤ 1, then the functions f := 0, g := d(x)(1 − λd (q))ψ q−1 are
nonnegative, nevertheless (0, −ψ) satisfies (1.4), which means that the maximum
principle does not hold.
Now suppose that λa (p) > 1, λd (q) > 1 and (3.7), and hence (3.6), does not
hold, i.e.
(λa (p) − 1)
α+1
p
(λd (q) − 1)
β+1
q
≤ Θ inf
x∈Ω
b(x) α+1
c(x) β+1
p
q
inf
.
x∈Ω d(x)
a(x)
Now, we want to fined a positive real number ξ such that
β+1
φp α+1
p
q
≤ ξ,
q
ψ
ψ q α+1 β+1
1
B( p ) p q ≤ ,
φ
ξ
A
A > 0,
(3.8)
B > 0.
Equation (3.8) is satisfied if
A sup
Ω
β+1
β+1
φp α+1
1
φp α+1
p
q
p
q
≤
ξ
≤
inf
.
ψq
B Ω ψq
(3.9)
EJDE-2007/66
MAXIMUM PRINCIPLE
5
Let
a(x) α+1
A = (λa (p) − 1) sup(
) p
x∈Ω b(x)
d(x) β+1
and B = (λd (q) − 1) sup(
) q ,
x∈Ω c(x)
then (3.9) becomes
a(x) α+1
φp α+1 β+1
(λa (p) − 1) sup(
) p sup q p q
ψ
Ω
x∈Ω b(x)
β+1
1
φp α+1
p
q
≤ξ≤ .
β+1 inf
q
Ω
ψ
q
(λd (q) − 1) supx∈Ω ( d(x)
)
c(x)
Then, ξ exists if
(λa (p) − 1)
≤
α+1
p
(λd (q) − 1)
β+1
q
1
(supx∈Ω ( a(x)
b(x) ))
α+1
p
(supx∈Ω ( d(x)
c(x) ))
β+1
q
inf Ω
φp
ψq
supΩ
φp
ψq
β+1
α+1
p
q
β+1
α+1
p
q
Therefore,
(λa (p) − 1)
α+1
p
(λd (q) − 1)
β+1
q
≤ Θ inf (
x∈Ω
b(x) α+1
c(x) β+1
) p inf (
) q .
x∈Ω d(x)
a(x)
So, ζ exists if (3.7), and hence (3.6), does not hold.
q α+1 β+1
p
q
with C, D > 0, then (3.8) implies
If ξ = D
p
C
β+1
a(x) α+1
(Cφ)p α+1
(λa (p) − 1) sup(
) p (
) p q
q
b(x)
(Dψ)
x∈Ω
β+1
1
(Cφ)p α+1
≤1≤ (
) p q ,
β+1
q
d(x) q
(Dψ)
(λd (q) − 1) supx∈Ω ( c(x) )
and hence, for some x ∈ Ω, we have
a(x) ( Cφ)p β+1
a(x) (Cφ)p β+1
q
q
(λa (p) − 1)(
)
)
≤
(λ
(p)
−
1)
sup
(
≤ 1,
a
b(x)
(Dψ)q
(Dψ)q
x∈Ω b(x)
c(x)
( d(x) )
(Cφ)p α+1
(Cφ)p α+1
p
1≤ (
)
≤
(
) p ,
(Dψ)q
(λd (q) − 1) (Dψ)q
(λd (q) − 1) supx∈Ω ( d(x)
)
c(x)
1
which implies
a(x)(λa (p) − 1)((Cφ)p )
β+1
q
≤ b(x)(Dψ)β+1 ,
d(x)(λd (q) − 1)((Dψ)q )
α+1
p
≤ c(x)(Cφ)α+1 .
Using (3.1), we have
a(x)(λa (p) − 1)(Cφ)p−1 ≤ b(x)(Dψ)β+1 (Cφ)α ,
d(x)(λd (q) − 1)(Dψ)q−1 ≤ c(x)(Cφ)α+1 (Dψ)β .
Then
f = −a(x)(λa (p) − 1)(Cφ)p−1 + b(x)(Dψ)β+1 (Cφ)α ≥ 0,
g = −d(x)(λd (q) − 1)(Dψ)q−1 + c(x)(Cφ)α+1 (Dψ)β ≥ 0,
6
S. A. KHAFAGY, H. M. SERAG,
EJDE-2007/66
are nonnegative functions, nevertheless (−Cφ, −Dψ) is a solution of (1.4), and the
maximum principle does not hold.
The condition is sufficient: Assume that (3.5) and (3.6) hold; if (u, v) is a solution
of (1.4) for f, g ≥ 0, we obtain by multiplying the first equation of (1.4) by u− :=
max(0, −u) and integrating over Ω
Z
P (x)|∇u− |p
Ω
Z
Z
Z
Z
=
a(x)|u− |p −
b(x)|u− |α+1 |v + |β+1 +
b(x)|u− |α+1 |v − |β+1 −
f u− ,
Ω
Ω
Ω
Ω
then
Z
P (x)|∇u− |p ≤
Ω
Z
a(x)|u− |p +
Ω
Z
b(x)|u− |α+1 |v − |β+1 .
Ω
By using (2.6) and (3.4), we have
Z
Z
− p
(λa (p) − 1)
a(x)|u | ≤
b(x)|u− |α+1 |v − |β+1
Ω
Ω
Z
β+1
α+1
≤ (a(x)|u− |p ) p (d(x)|v − |q ) q .
Ω
Applying Hölder inequality, we get
Z
hZ
i α+1
hZ
i β+1
β+1
p
q
(λa (p) − 1)
a(x)|u− |p ≤
(a(x)|u− |p )
(d(x)|v − |q ) q
,
Ω
Ω
Ω
and hence
β+1
Z
i Z
α+1
h
Z
β+1
q
q
p
(d(x)|v − |q )
−
a(x)|u− |p
≤ 0.
a(x)|u− |p
(λa (p) − 1)
Ω
Ω
Ω
Now, if
Z
a(x)|u− |p = 0,
Ω
then u− = 0, (where a(x) 6= 0 for any x), which implies that u ≥ 0. If not, we get
β+1
β+1
hZ
i α+1
hZ
i α+1
α+1
p
q
p
q
− p
p
a(x)|u |
≤
d(x)|v − |q
.
(3.10)
(λa (p) − 1)
Ω
Ω
Similarly, from the second equation of (1.4), we deduce that
β+1
β+1
hZ
i α+1
hZ
i α+1
β+1
p
q
p
q
(λd (q) − 1) q
d(x)|v − |q
≤
a(x)|u− |p
.
Ω
(3.11)
Ω
Multiplying (3.10) by (3.11), we obtain
β+1 h Z
β+1
i α+1
hZ
i α+1
β+1
α+1
p
q
p
q
− p
p
q
((λa (p)−1)
d(x)|v − |q
≤ 0.
(λd (q)−1)
−1)
a(x)|u |
Ω
Ω
Using (3.6), we have u− = v − = 0, which implies that u ≥ 0, v ≥ 0, i.e. the
maximum principle holds.
EJDE-2007/66
MAXIMUM PRINCIPLE
7
4. Existence of Positive Solutions
Now, we shall prove that the system (1.4) has a solution in the space W01,p (P, Ω)×
W01,q (Q, Ω), by an approximation method.
Following [4], for ∈ (0, 1), we introduce the system
(|u |p−2 u )
|v |β v
|u |α
+
b(x)
+ f,
(1 + |1\p u |p−1 )
(1 + |1\q v |β+1 ) (1 + |1\p u |α )
|v |β
|u |α u
(|v |q−2 v )
−∆Q,q v = c(x)
+
d(x)
+ g,
(1 + |1\q v |β ) (1 + |1\p u |α+1 )
(1 + |1\q v |q−1 )
u = v = 0 on ∂Ω.
(4.1)
Letting (ζ, η) = (u , v ), then the system above can be written in the form
−∆P,p u = a(x)
−∆P,p ζ = h(ζ, η) + f
in Ω,
−∆Q,q η = k(ζ, η) + g
in Ω,
ζ = η = 0 on ∂Ω.
where
(|ζ|p−2 ζ)
|η|β η
|ζ|α
+
b(x)
,
(1 + |1\p ζ|p−1 )
(1 + |1\q η|β+1 ) (1 + |1\p ζ|α )
|η|β
|ζ|α ζ
(|η|q−2 η)
k(ζ, η) = c(x)
+ d(x)
.
1\q
β
1\p
α+1
(1 + | η| ) (1 + | ζ|
)
(1 + |1\q η|q−1 )
h(ζ, η) = a(x)
It is easy to prove that h(ζ, η) and k(ζ, η) are bounded, since a(x), b(x), c(x) and
d(x) are also bounded. Then, there exists M > 0 such that |h(ζ, η)| ≤ M and
|k(ζ, η)| ≤ M for all ζ, η.
Lemma 4.1. System (4.1) has a solution U = (u , v ) in W01,p (P, Ω)×W01,q (Q, Ω).
Proof. We complete the proof in the following steps
(a) Construction of sub-super solutions for system (4.1) as follows; Let
ζ 0 ∈ W01,p (P, Ω) be a solution of − ∆P,p ζ 0 = M + f,
η 0 ∈ W01,q (Q, Ω) be a solution of − ∆Q,q η 0 = M + g,
ζ0 ∈ W01,p (P, Ω) be a solution of − ∆P,p ζ0 = −M + f,
(4.2)
η0 ∈ W01,q (Q, Ω) be a solution of − ∆Q,q η0 = −M + g.
Then , as in [14], we say that (ζ 0 , η 0 ) is a super solution of (4.1) and (ζ0 , η0 ) is a
sub solution of the same system, since we have
−∆P,p ζ 0 − h(ζ 0 , η) − f ≥ −∆P,p ζ 0 − (M + f ) = 0
∀η ∈ [η0 , η 0 ] in Ω,
−∆Q,q η 0 − k(ζ, η 0 ) − g ≥ −∆Q,q η 0 − (M + g) = 0
∀ζ ∈ [ζ0 , ζ 0 ] in Ω.
−∆P,p ζ0 − h(ζ0 , η) − f ≤ −∆P,p ζ0 + M − f = 0
∀η ∈ [η0 , η 0 ] in Ω,
−∆Q,q η0 − k(ζ, η0 ) − g ≤ −∆Q,q η0 + M − g = 0
∀ζ ∈ [ζ0 , ζ 0 ]in Ω,
Let us assume that K = [ζ0 , ζ 0 ] × [η0 , η 0 ].
8
S. A. KHAFAGY, H. M. SERAG,
EJDE-2007/66
(b) Definition of the operator T : We define the operator T : (ζ, η) → (w, z) by
−∆P,p w = h(ζ, η) + f
in Ω,
−∆Q,q z = k(ζ, η) + g
in Ω,
(4.3)
w = z = 0 on ∂Ω.
(c) T (K) ⊂ K: Since (ζ, η) ∈ [ζ0 , ζ 0 ] × [η0 , η 0 ], then from(4.2) and (4.3), we get
−∆P,p w + ∆P,p ζ 0 ≤ h(ζ, η) − M.
(4.4)
Multiplying this equation by (w − ζ 0 )+ = max(w − ζ 0 , 0) and integrating over Ω,
we obtain
Z
Z
0
0 +
(−∆P,p w + ∆P,p ζ )(w − ζ ) ≤ (h(ζ, η) − M )(w − ζ 0 )+ ≤ 0,
Ω
Ω
which implies
Z
P (x) |∇w|p−2 ∇w − |∇ζ 0 |p−2 ∇ζ 0 ∇(w − ζ 0 )+ ≤ 0.
Ω
It is well known by [17], that the following inequality holds
γ
γ
|x − y|p ≤ C{(|x|p−2 x − |y|p−2 y)(x − y)} 2 (|x|p + |y|p )1− 2 .
(4.5)
for all x, y ∈ RN , where γ = p if 1 < p ≤ 2 and γ = 2 if p > 2.
Applying (4.5), we obtain
Z
P (x)|∇(w − ζ 0 )|p ≤ 0
Ω
which implies that, since P (x) is a weight function, (w − ζ 0 )+ = 0, and hence
w ≤ ζ 0.
Again, as above, we can deduce that w ≥ ζ0 . So, we have ζ0 ≤ w ≤ ζ 0 .
Similarly, we can deduce that η0 ≤ z ≤ η 0 , and hence (w, z) ∈ [ζ0 , ζ 0 ] × [η0 , η 0 ].
(d) T is completely continuous: First, we prove that T is continuous; for this, we
need the following lemma.
Lemma 4.2. If (ζk , ηk ) → (ζ, η), in Lp (Ω) × Lq (Ω), then, as in [5, 16],
Z
p∗ 1/p∗
|ζk |p−2 ζk
|ζ|p−2 ζ
a(x)[
−
]
→ 0,
(1 + |1\p ζk |p−1 ) (1 + |1\p ζ|p−1 )
Ω
Z
|ηk |β ηk
|ζk |α
b(x)[
1\p
α
(1 + | ζk | ) (1 + |1\q ηk |β+1 )
Ω
p∗ 1/p∗
|ζ|α
|η|β η
]
→ 0,
−
(1 + |1\p ζ|α ) (1 + |1\q η|β+1 )
Z
|ζk |α ζk
|ηk |β
c(x)[
(1 + |1\p ζk |α+1 ) (1 + |1\q ηk |β )
Ω
q∗ 1/q∗
α
|ζ| ζ
|η|β
−
]
→ 0,
(1 + |1\p ζ|α+1 ) (1 + |1\q η|β )
Z
q∗ 1/q∗
|ηk |q−2 ηk
|η|q−2 η
d(x)[
−
]
→ 0,
(1 + |1\q ηk |q−1 ) (1 + |1\q η|q−1 )
Ω
as k → +∞.
EJDE-2007/66
MAXIMUM PRINCIPLE
9
Proof. If (ζk ) → (ζ) in Lp (Ω), then there exists a subsequence still denoted by (ζk )
itself such that ζk (x) → ζ(x) a.e. on Ω and |ζk (x)| ≤ l(x) a.e. on Ω, for all k, with
l ∈ Lp (Ω). Hence,
∗
|ζk |p−2 ζk ≤ |ζk |p−1 ≤ lp−1 ∈ Lp (Ω),
1\p
p−1
1 + | ζk |
and, since a(x) 6= 0 is bounded, we have
a(x)
|ζk (x)|p−2 ζk (x)
|ζ(x)|p−2 ζ(x)
→
a(x)
1 + |1\p ζk (x)|p−1
1 + |1\p ζ(x)|p−1
a.e. on Ω as k → +∞.
Thus from the Dominated Convergence Theorem, we obtain the first statement of
this theorem.
Prove of the second statement: If (ηk ) → (η) in Lq (Ω), then there exists a
subsequence still denoted also by (ηk ), such that ηk (x) → η(x) a.e. on Ω and
|ηk (x)| ≤ m(x) a.e. on Ω, for all k, with m ∈ Lq (Ω).
Now, from (3.1) and (3.2), we obtain
αp∗
(β + 1)p∗
+
= 1,
p
q
and hence,
Z
∗
[lα mβ+1 ]p ≤
Ω
Z
∗
∗
|l|αp |m|(β+1)p ≤
Ω
Z
|l|p
p∗ α/p Z
Ω
|m|q
p∗ (β+1)/q
< ∞.
Ω
So that
∗
|(|ζk |α |ηk |β+1 )| ≤ lα mβ+1 ∈ Lp (Ω).
Hence, since b(x) is bounded, we have
b(x)
|ζk |α
|ηk |β ηk
|ζ|α
|η|β η
→ b(x)
,
1\p
α
(1 + | ζ| ) 1 + |1\q η|β+1
1 + |1\p ζk |α 1 + |1\q ηk |β+1
a.e. on Ω as k → +∞. Thus from the Dominated Convergence Theorem, we obtain
the second statement in this theorem. Similarly we prove the third and fourth
statements.
Now, we prove the continuity of T . Assume that (ζk , ηk ) → (ζ, η), in Lp (Ω) ×
Lq (Ω), then we have from the first equation of (4.3)
− ∆P,p wk + ∆P,p w
(|ζk |p−2 ζk )
(|ζ|p−2 ζ)
= a(x)
−
1\p
p−1
1\p
p−1
(1 + | ζk | ) (1 + | ζ| )
|ζk |α
|ηk |β ηk
|ζ|α
|η|β η
+ b(x)
−
,
(1 + |1\p ζk |α ) (1 + |1\q ηk |β+1 ) (1 + |1\p ζ|α ) (1 + |1\q η|β+1 )
10
S. A. KHAFAGY, H. M. SERAG,
EJDE-2007/66
multiplying this equation by (wk − w) and integrating over Ω, we obtain
Z
P (x) |∇wk |p−2 ∇wk − |∇w|p−2 ∇w ∇(wk − w)
Ω
Z
(|ζk |p−2 ζk )
(|ζ|p−2 ζ)
−
=
a(x)
(wk − w)
1\p
p−1
1\p
p−1
(1 + | ζk | ) (1 + | ζ| )
Ω
Z
|ηk |β ηk
|ζk |α
+
b(x)
(1 + |1\p ζk |α ) (1 + |1\q ηk |β+1 )
Ω
α
|ζ|
|η|β η
−
(wk − w).
(1 + |1\p ζ|α ) (1 + |1\q η|β+1 )
Using Hölder’s inequality, we get
Z
P (x) |∇wk |p−2 ∇wk − |∇w|p−2 ∇w ∇(wk − w)
Ω
Z
1/p
Z
p∗ 1/p∗ |ζ|p−2 ζ
|ζk |p−2 ζk
p
−
]
|w
−
w|
(a(x)[
≤
k
(1 + |1\p ζk |p−1 ) (1 + |1\p ζ|p−1 )
Ω
Ω
Z
|ηk |β ηk
|ζk |α
+
b(x)[
1\p
α
(1 + | ζk | ) (1 + |1\q ηk |β+1 )
Ω
Z
1/p
p∗ 1/p∗ |ζ|α
|η|β η
p
]
|w
−
w|
.
−
k
(1 + |1\p ζ|α ) (1 + |1\q η|β+1 )
Ω
Applying (4.5) and lemma 4.2, we obtain
Z
P (x)|∇(wk − w)|p → 0
as k → +∞,
Ω
which implies that wk → w in W01,p (P, Ω). Similarly, we can deduce that zk → z
in W01,q (Q, Ω). Then, (wk , zk ) → (w, z) in W01,p (P, Ω) × W01,q (Q, Ω).
To prove that T is compact, let (ζj , ηj ) be a bounded sequence in K. Multiplying
the first equation in (4.3) by wj and integrating over Ω, we obtain
Z
P (x)|∇wj |p
Ω
Z h
Z
i
|ζj |p−2 ζj
|ηj |β ηj
|ζj |α
a(x)
=
+
b(x)
w
+
f wj
j
1 + |1\p ζj |p−1
1 + |1\p ζj |α 1 + |1\q ηj |β+1
Ω
Ω
Z
Z
Z
≤
a(x)|ζj |p−2 ζj wj +
b(x)|ζj |α |ηj |β ηj wj +
f wj
Ω
Ω
Ω
Z
Z
∗
1/p
∗ 1/p
≤
[a(x)|ζj |p−1 ]p
(wj )p
Ω
Ω
Z
∗ Z
1/p Z
1/p∗ Z
1/p
∗ 1/p
α
β+1 p
p
P∗
+
[b(x)|ζj | |ηj |
]
(wj )
+
|f |
(wj )p
.
Ω
Ω
1,p
W0 (P, Ω) and
Ω
Ω
Hence, (wj ) is bounded in
it possesses a strongly convergent subsequence in Lp (Ω). The same is true for (zj ) in Lq (Ω).
Since K is a convex, bounded, closed subset of Lp (Ω) × Lq (Ω), we can apply
Schauder’s Fixed Point Theorem to obtain the existence of a fixed point for T ,
which gives the existence of solution U = (u , v ) of (4.1), and this completes the
proof.
Now, we are in a position to prove the existence of a solution for system (1.4).
EJDE-2007/66
MAXIMUM PRINCIPLE
11
Theorem 4.3. Assume that (2.2)-(2.6) and (2.10) are satisfied, then system (1.4)
admits a solution (u, v) in W01,p (P, Ω) × W01,q (Q, Ω).
Proof. This proof is done in three steps:
(a) First, we proof that (1\p u , 1\q v ) is bounded in W01,p (P, Ω) × W01,q (Q, Ω).
Multiplying the first equation of ( 4.1) by (u ) and integrating over Ω , we obtain
Z
P (x)|∇(1\p u )|p
Ω
Z
Z
Z
(4.6)
1\p
p
1\p
1\p∗
≤
a(x)|( u )| +
b(x)|( u )| + |f | × |(1\p u )|.
Ω
Ω
Ω
From (2.6), we get
Z
Z
Z
1\p
1\p∗
1\p
p
|f | × |(1\p u )|.
b(x)|( u )| + a(x)|( u )| ≤
(λa (p) − 1)
Ω
Ω
Ω
Using Hölder’s inequality, we have
Z
1/p∗
(λa (p) − 1)
|(1\p u )|p
≤ c with c > 0,
Ω
which implies that (1\p u ) is bounded in Lp (Ω) and from (4.6) it is bounded in
W01,p (P, Ω). similarly (1\q v ) is bounded in W01,q (Q, Ω).
(b) (1\p u , 1\q v ) converges to (0, 0) strongly in W01,p (P, Ω) × W01,q (Q, Ω).
From (a), (1\p u , 1\q v ) converges to (u∗ , v∗ ) strongly in LP (Ω) × Lq (Ω) and
weakly in W01,p (P, Ω) × W01,q (Q, Ω).
∗
Multiplying the first equation of (4.1) by (1\p ), we get
− ∆P,p (1\p u )
= a(x)
∗
|1\p u |p−2 (1\p u )
|1\p u |α
(1\q |v |)β (1\q v )
+ b(x)
+ f 1\p ,
1\p
p−1
1\p
α
(1 + | u | )
(1 + | u | ) (1 + |1\q v |β+1 )
since the sequence (1\p u ) is bounded in W01,p (P, Ω), then, we can find subsequence
(1\p u ) such that
1\p u → u∗
weakly in W01,p (P, Ω)
and 1\p u → u∗
a.e. on Ω.
Again, using Dominated Convergence Theorem as in Lemma 4.2, we have
a(x)
|u∗ |p−2 u∗
|1\p u |p−2 (1\p u )
→
a(x)
,
(1 + |u∗ |p−1 )
(1 + |1\p u |p−1 )
∗
strongly in LP (Ω) and
b(x)
|v∗ |β v∗
|1\p u |α
(1\q |v |)β (1\q v )
|u∗ |α
,
→
b(x)
(1 + |u∗ |α ) (1 + |v∗ |β+1 )
(1 + |1\p u |α ) (1 + |1\q v |β+1 )
∗
strongly in LP (Ω). Using a classical result in [15], and passing to the limit, we
obtain
|u∗ |p−2 u∗
|u∗ |α
|v∗ |β v∗
−∆P,p (u∗ ) = a(x)
+
b(x)
.
(4.7)
(1 + |u∗ |p−1 )
(1 + |u∗ |α ) (1 + |v∗ |β+1 )
Multiplying this equation by (u−
∗ ) and integrating over Ω, then applying (2.6), we
get
Z
Z
− p
α+1 − β+1
(λa (p) − 1)
a(x)|u∗ | ≤
b(x)|u−
|v∗ |
.
∗|
Ω
Ω
12
S. A. KHAFAGY, H. M. SERAG,
EJDE-2007/66
Using (3.4) and applying Hölder’s inequality, as in the proof of theorem 3.1, we
deduce
β+1
β+1
Z
α+1
Z
α+1
α+1
p
q
p
q
− p
p
(λa (p) − 1)
a(x)|u∗ |
≤
,
(4.8)
d(x)|v∗− |q
Ω
Ω
similarly, from the second equation of (4.1), we have
β+1
β+1
Z
α+1
Z
α+1
β+1
p
q
p
q
− q
p
q
(λd (q) − 1)
d(x)|v∗ |
≤
.
a(x)|u−
∗|
Ω
(4.9)
Ω
Multiplying (4.8) by (4.9), we obtain
((λa (p) − 1)
α+1
p
(λd (q) − 1)
β+1
q
Z
β+1
Z
α+1
p
q
≤ 0.
− 1)
a(x)|u− |p
d(x)|v − |q
Ω
u−
∗
Ω
v∗−
From (3.6), we have
=
= 0, which implies that u∗ , v ∗ ≥ 0.
Now, we show that u∗ = v∗ = 0. Multiplying equation (4.7) by (u∗ ) and integrating over Ω, we get, as above
Z
β+1
Z
α+1
β+1
α+1
p
q
a(x)|u∗ |p
d(x)|v ∗ |q
≤ 0,
((λa (p) − 1) p (λd (q) − 1) q − 1)
Ω
∗
Ω
∗
which implies that u = v = 0.
(c) (u , v ) is bounded in W01,p (P, Ω) × W01,q (Q, Ω). Assume that
ku kW 1,p (P,Ω) → ∞ or
0
kv kW 1q (Q,Ω) → ∞.
0
Set
q
p
t = max(ku kW 1,p (P,Ω) , kv kW 1q (Q,Ω) ),
0
0
1\p∗
Dividing the first equation of (4.1) by (t
−∆P,p z = a(x)
z = u t−1\p
,
w = v t−1\q
.
1\q ∗
) and the second by (t
), we obtain
(|z |p−2 z )
|z |α
|w |β w
+
b(x)
(1 + |1\p u |p−1 )
(1 + |1\p u |α ) (1 + |1\q v |β+1 )
∗
+ f t−1\p
,
−∆Q,q w = d(x)
(|w |q−2 w )
|v |β
|z |α z
+ c(x)
1\q
q−1
1\p
α+1
(1 + | v | )
(1 + | u |
) (1 + |1\q v |β )
∗
+ gt−1\q
.
As in (b) above, we can prove that (z , w ) → (z, w) strongly in W01,p (P, Ω) ×
W01,q (Q, Ω), and taking the limit, as → 0, we obtain
−∆P,p z = a(x)|z|p−2 z + b(x)|w|β |z|α w,
−∆Q,q w = d(x)|w|q−2 w + c(x)|w|β |z|α z,
and hence, we deduce that w = z = 0.
Since there exists a sequence (n )n∈N such that either kzn k = 1 or kwn k = 1,
we obtain a contradiction.
Hence, (u , v ) is bounded in W01,p (P, Ω) × W01,q (Q, Ω), we can extract a subsequence denoted by (u , v ) which converges to (u0 , v0 ) strongly in LP (Ω)×Lq (Ω) and
weakly in W01,p (P, Ω)×W01,q (Q, Ω) as → 0. By using a similar procedure as above,
we can prove that (u , v ) converges strongly to (u0 , v0 ) in W01,p (P, Ω)×W01,q (Q, Ω).
EJDE-2007/66
MAXIMUM PRINCIPLE
13
Indeed, since 1\p u → 0 a.e. on Ω, we have
|u |p−2 u ≤ |u |p−1 ≤ lp−1 ∈ LP ∗ (Ω),
1 + |1\p u |p−1
with l ∈ LP (Ω) and
|u |p−2 u
→ a(x)|u0 |p−2 u0 a.e. on Ω.
1 + |1\p u |p−1
Hence, from the Dominated Convergence Theorem, we obtain
Z
−1
p∗
a(x)[(|u |p−2 u ) 1 + |1\p u |p−1
− |u0 |p−2 u0 ]
→ 0,
a(x)
as → 0.
Ω
Also, since 1\p u → 0 and 1\q v → 0 a.e. on Ω, then
−1 β
−1
|u |α 1 + |1\p u |α
|v | v 1 + |1\q v |β+1
→ |u0 |α |v0 |β v0 ,
a.e. on Ω,
and, as in the proof of Lemma 4.2, we have
∗
|v |β v
|u |α
≤ |u |α |v |β+1 ≤ lα mβ+1 ∈ LP (Ω)
1\p
α
1\q
β+1
1 + | u |
1 + | v |
with m ∈ Lq (Ω). From the Dominated Convergence Theorem, we have
Z
p∗
|u |α
|v |β v
α
β
b(x)[
−
|u
|
|v
|
v
]
→ 0,
0
0
0
(1 + |1\p u |α ) (1 + |1\q v |β+1 )
Ω
as → 0. Similarly, we have
Z
∗
|u |α u
|v |β
c(x)[
− |u0 |α u0 |v0 |β ])q → 0 as → 0,
1\p u |α+1 ) (1 + |1\q v |β )
(1
+
|
Ω
Z
∗
|v |q−2 v
− |v0 |q−2 v0 )]q → 0 as → 0.
d(x)[
1\q v |q−1 )
(1
+
|
Ω
Therefore, passing to the limit, (u , v ) → (u0 , v0 ), and hence we obtain from (4.1)
−∆P,p u0 = a(x)|u0 |p−2 u0 + b(x)|u0 |α |v0 |β v0 + f
in Ω,
−∆Q,q v0 = d(x)|v0 |q−2 v0 + c(x)|u0 |α u0 |v0 |β + g
in Ω.
Hence, (u0 , v0 ) satisfies the system (1.4).
Remark 4.4. (i) When P (x) = Q(x) = 1, p = q = 2 and α = β = 0, we obtain
some results presented in [4]. (ii) When P (x) = Q(x) = 1 and the coefficients
a(x), b(x), c(x) and d(x) are constants, we have some results presented in [5].
References
[1] Adams, R.; Sobolev Spaces, Academic Press, New York, 1975.
[2] Barles, G.; Remarks on Uniqueness Results of the First Eigenvalue of the p-Laplacian, Ann.
Fac. Sc. Toulouse, Vol. t., 9, 65-75, 1988.
[3] Benci, V. and Forturato, D.; Some Compact Embedding Theorem for Weighted Sobolev
Spaces, Bull. U. M. I., Vol. 13B, 832-843, 1976.
[4] Boccardo, L., Fleckinger, J. and de Thélin, F.; Existence of Solutions for Some Nonlinear
Cooperative Systems and Some Applications, Diff. and Int. Eqns., Vol. 7, No. 3, 689-698,
1994.
[5] Bouchekif, M., Serag, H. and de Thélin, F.; On maximum principle and Existence of Solutions
for Some Nonlinear Elliptic Systems, Rev. Mat. Apl., Vol. 16, 1-16, 1995.
[6] de Figueiredo, D. and Mitidieri, E.; A maximum principle for an Elliptic System and Applications to Semilinear Problem, S.I.A.M.J. Mat. Anal., Vol. 17, 836-849, 1986.
14
S. A. KHAFAGY, H. M. SERAG,
EJDE-2007/66
[7] de Figueriedo, D. and Mitidieri, E.; maximum principle for Cooperative Elliptic Systems,
Comptes Rendus Acad. Sc. Paris, Vol. 310, 49-52, 1990.
[8] de Figueiredo, D. and Mitidieri, E.; maximum principles for Linear Elliptic Systems,
Quaterno Matematico, Vol. 177, Dip. Sc. Mat., Univ. Trieste, 1988.
[9] Diaz, J.; Nonlinear Partial Differential Equations and free Boundaries, Pitman Publ., Program, 1985.
[10] Drabek, P., Kufner, A., and Nicolosi, F.; Quasilinear elliptic equation with degenerations and
singularities, Walter de Gruyter, Bertin, New York, 1997.
[11] Fleckinger, J. Hernándes, J. and de Thélin, F.; On maximum principle and Existence of
Positive Solutions for Cooperative Elliptic Systems, J. Diff. and Int. Eqns. Vol. 8, 69-85,
1995.
[12] Fleckinger, J. and Serag, H.; On maximum principle and Existence of Solutions for Elliptic
Systems on Rn , J. Egypt. Math. Soc., Vol. 2, 45-51, 1994.
[13] Fleckinger, J. and Serag, H.; Semilinear Cooperative Elliptic Systems on Rn , Rend. di
Mat.,Vol. Seri VII 15 Roma, 89-108, 1995.
[14] Hernández, J.; Some Existence and Stability Results for Solutions of Reaction-Diffusion
Systems with Nonlinear Boundary Conditions, P. de Mottoni, L. Salvadori (eds.), Nonlinear
Differential Equations: Invariance Stability and Bifurcation, New York, Acad. Press, 161-173,
1981.
[15] Lions, J.; Quelques Méthodes de Résolution des Problémes aux Limites Nonlinéaires, Dunod,
Paris, 1969.
[16] Serag, H. and El-Zahrani, E.; maximum principle and Existence of Positive Solution for
Nonlinear Systems on Rn , Electron. J. Diff. Eqns., Vol. 2005, No. 85, 1-12, 2005.
[17] Simon, J.; Régularité de la solution d’un probléme aux limites non linéare, Ann. Fac. Sc.
Toulous, t.3, 247-274, 1981.
Salah A. Khafagy
Mathematics Department, Faculty of Science, Al-Azhar University, Nasr City (11884),
Cairo, Egypt
E-mail address: el gharieb@hotmail.com
Hassan M. Serag
Mathematics Department, Faculty of Science, Al-Azhar University, Nasr City (11884),
Cairo, Egypt
E-mail address: serraghm@yahoo.com