1. Introduction
The study of diferential equations and variational problems with variable exponent has been a new and interesting topic. It arises from nonlinear elasticity theory, electrorheological fluids, etc., (see123). Many results have been obtained on this kind of problems, for example 1345678. In 7, Fan gives the regularity of weak solutions for differential equations with variable exponent. On the existence of solutions for elliptic systems with variable exponent, we refer to 89. In this article, we mainly consider the existence of positive weak solutions for the system
(
P
)
-
Δ
p
(
x
)
u
=
λ
p
(
x
)
[
g
(
x
)
a
(
u
)
+
f
(
v
)
]
in
Ω
,
-
Δ
p
(
x
)
v
=
λ
p
(
x
)
[
g
(
x
)
b
(
v
)
+
h
(
u
)
]
in
Ω
,
u
=
v
=
0
on
∂
Ω
,
where Ω ⊂ ℝN is a bounded domain with C2 boundary ∂Ω, 1 < p(x) ∈ C1 (Ω̄)
is a function. The operator Δp(x)u = div (|∇u|p(x)-2∇u)
is called p(x)-Laplacian. Especially, if p(x) ≡ p (a constant), (P) is the well-known p-Laplacian system. There are many articles on the existence of solutions for p-Laplacian elliptic systems, for example 510. Owing to the nonhomogeneity of p(x)-Laplacian problems are more complicated than those of p-Laplacian, many results and methods for p-Laplacian are invalid for p(x)-Laplacian; for example, if Ω is bounded, then the Rayleigh quotient
λ
p
(
x
)
=
inf
u
∈
W
0
1
,
p
(
x
)
(
Ω
)
\
{
0
}
∫
Ω
1
p
(
x
)
|
∇
u
|
p
(
x
)
d
x
∫
Ω
1
p
(
x
)
|
u
|
p
(
x
)
d
x
is zero in general, and only under some special conditions λp(x) > 0 (see 11), and maybe the first eigenvalue and the first eigenfunction of p(x)-Laplacian do not exist, but the fact that the first eigenvalue λp > 0 and the existence of the first eigenfunction are very important in the study of p-Laplacian problems. There are more difficulties in discussing the existence of solutions of variable exponent problems.
Hai and Shivaji 10, consider the existence of positive weak solutions for the following p-Laplacian problems
(
I
)
-
Δ
p
u
=
λ
f
(
v
)
in
Ω
,
-
Δ
p
v
=
λ
g
(
u
)
in
Ω
,
u
=
v
=
0
on
∂
Ω
the first eigenfunction is used to construct the subsolution of p-Laplacian problems success-fully. On the condition that λ is large enough and
lim
u
→
+
∞
f
M
(
g
(
u
)
)
1
(
p
-
1
)
u
p
-
1
=
0
,
for every
M
>
0
,
the authors give the existence of positive solutions for problem (I).
Chen 5, considers the existence and nonexistence of positive weak solution to the following quasilinear elliptic system:
(
II
)
-
Δ
p
u
=
λ
f
(
u
,
v
)
=
λ
u
α
v
γ
in
Ω
,
-
Δ
q
v
=
λ
g
(
u
,
v
)
=
λ
u
δ
v
β
in
Ω
,
u
=
v
=
0
on
∂
Ω
the first eigenfunction is used to construct the subsolution of problem(II), the main results are as following
(i) If α, β ≥ 0, γ, δ > 0, θ = (p - 1 - α)(q - 1 - β) - γδ > 0, then problem (II) has a positive weak solution for each λ > 0;
(ii) If θ = 0 and pγ = q(p - 1 - α), then there exists λ0 > 0 such that for 0 < λ < λ0, then problem (II) has no nontrivial nonnegative weak solution.
On the p(x)-Laplacian problems, maybe the first eigenvalue and the first eigenfunction of p(x)-Laplacian do not exist. Even if the first eigenfunction of p(x)-Laplacian exist, because of the nonhomogeneity of p(x)-Laplacian, the first eigenfunction cannot be used to construct the subsolution of p(x)-Laplacian problems. Zhang 12 investigated the existence of positive solutions of the system
-
Δ
p
(
x
)
u
=
λ
p
(
x
)
f
(
v
)
in
Ω
,
-
Δ
p
(
x
)
v
=
λ
p
(
x
)
g
(
u
)
in
Ω
,
u
=
v
=
0
on
∂
Ω
,
In this article, we consider the existence of positive solutions of the system
-
Δ
p
(
x
)
u
=
λ
p
(
x
)
F
(
x
,
u
,
v
)
in
Ω
,
-
Δ
p
(
x
)
v
=
λ
p
(
x
)
G
(
x
,
u
,
v
)
in
Ω
,
u
=
v
=
0
on
∂
Ω
,
where p(x) ∈ C1 (Ω̄)
is a function, F(x, u, v) = [g(x)a(u) + f(v)], G(x, u, v) = [g(x)b(v) +h(u)], λ is a positive parameter and Ω ⊂ ℝN is a bounded domain.
To study p(x)-Laplacian problems, we need some theory on the spaces Lp(x)(Ω), W1,p(x)(Ω) and properties of p(x)-Laplacian which we will use later (see 613). If Ω ⊂ ℝN is an open domain, write
C
+
(
Ω
)
=
{
h
:
h
∈
C
(
Ω
)
,
h
(
x
)
>
1
f
o
r
x
∈
Ω
}
,
h
+
=
sup
x
∈
Ω
h
(
x
)
,
h
-
=
inf
x
∈
Ω
h
(
x
)
,
for any
h
∈
C
(
Ω
)
.
Throughout the article, we will assume that:
(H1) Ω ⊂ ℝN is an open bounded domain with C2 boundary ∂Ω.
(H2) p(x) ∈ C1 (Ω̄)
and 1 < p- ≤ p+.
(H3) a, b ∈ C1([0, ∞)) are nonnegative, nondecreasing functions such that
lim
u
→
+
∞
a
(
u
)
u
p
-
-
1
=
0
,
lim
u
→
+
∞
b
(
u
)
u
p
-
-
1
=
0
.
(H4) f, h : [0, +∞) → R are C1, monotone functions, limu→+∞ f(u) = +∞, limu→+∞ h(u) = +∞, and
lim
u
→
+
∞
f
M
(
h
(
u
)
)
1
(
p
-
-
1
)
u
p
-
-
1
=
0
,
∀
M
>
0
.
(H5) g : [0, +∞) → (0, +∞) is a continuous function such that
L
1
=
min
x
∈
Ω
̄
g
(
x
)
, and
L
2
=
max
x
∈
Ω
̄
g
(
x
)
.
Denote
L
p
(
x
)
(
Ω
)
=
u
|
u
is a measurable real - valued function
,
∫
Ω
|
u
(
x
)
|
p
(
x
)
d
x
<
∞
.
We introduce the norm on Lp(x)(Ω) by
|
u
|
p
(
x
)
=
inf
λ
>
0
:
∫
Ω
u
(
x
)
λ
p
(
x
)
d
x
≤
1
,
and (Lp(x)(Ω), |.|p(x)) becomes a Banach space, we call it generalized Lebesgue space. The space (Lp(x)(Ω), |.|p(x)) is a separable, reflexive, and uniform convex Banach space (see [6, Theorems 1.10 and 1.14]).
The space W1,p(x)(Ω) is defined by W1,p(x)(Ω) = {u ∈ Lp(x) : | ∇u| ∈ Lp(x)}, and it is equipped with the norm
u
=
|
u
|
p
(
x
)
+
|
∇
u
|
p
(
x
)
,
∀
u
∈
W
1
,
p
(
x
)
(
Ω
)
.
We denote by W01,p(x)(Ω) is the closure of
C
0
∞
Ω
in W1,p(x)(Ω). W1,p(x)(Ω) and
W
0
1
,
p
(
x
)
(
Ω
)
are separable, reflexive, and uniform convex Banach space (see [6, Theorem 2.1] We define
(
L
(
u
)
,
v
)
=
∫
Ω
|
∇
u
|
p
(
x
)
-
2
∇
u
∇
v
d
x
,
∀
v
,
u
∈
W
0
1
,
p
(
x
)
(
Ω
)
,
then L
:
W
0
1
,
p
(
x
)
(
Ω
)
→
(
W
0
1
,
p
(
x
)
(
Ω
)
)
*
is a continuous, bounded, and strictly monotone operator, and it is a homeomorphism (see [14, Theorem 3.1]).
If u
,
v
∈
W
0
1
,
p
(
x
)
(
Ω
)
,
(
u
,
v
)
is called a weak solution of (P) if it satisfies
∫
Ω
|
∇
u
|
p
(
x
)
-
2
∇
u
⋅
∇
q
d
x
=
∫
Ω
λ
p
(
x
)
F
(
x
,
u
,
v
)
q
d
x
,
∀
q
∈
W
0
1
,
p
(
x
)
(
Ω
)
,
∫
Ω
|
∇
v
|
p
(
x
)
-
2
∇
v
⋅
∇
q
d
x
=
∫
Ω
λ
p
(
x
)
G
(
x
,
u
,
v
)
q
d
x
,
∀
q
∈
W
0
1
,
p
(
x
)
(
Ω
)
.
Define A
:
W
1
,
p
(
x
)
(
Ω
)
→
(
W
0
1
,
p
(
x
)
(
Ω
)
)
*
as
<
A
u
,
φ
>
=
∫
Ω
(
|
∇
u
|
p
(
x
)
-
2
∇
u
∇
φ
+
l
(
x
,
u
)
φ
)
d
x
,
∀
u
∈
W
1
,
p
(
x
)
(
Ω
)
,
∀
φ
∈
W
0
1
,
p
(
x
)
(
Ω
)
,
where l(x, u) is continuous on
Ω
̄
×
ℝ
, and l(x, .) is increasing. It is easy to check that A is a continuous bounded mapping. Copying the proof of 15, we have the following lemma.
Lemma 1.1. (Comparison Principle). Let u, v ∈ W1,p(x)(Ω) satisfying Au - Av ≥ 0 in
(
W
0
1
,
p
(
x
)
(
Ω
)
)
*
,
φ
(
x
)
=
min
{
u
(
x
)
-
v
(
x
)
,
0
}
. If φ
(
x
)
∈
W
0
1
,
p
(
x
)
(
Ω
)
(i.e., u ≥ v on ∂Ω ), then u ≥ v a.e. in Ω.
Here and hereafter, we will use the notation d(x, ∂Ω) to denote the distance of x ∈ Ω to the boundary of Ω.
Denote d(x) = d(x, ∂Ω) and ∂
Ω
ϵ
=
{
x
∈
Ω
|
d
(
x
,
∂
Ω
)
<
ϵ
}
. Since ∂Ω is C2 regularly, then there exists a constant δ ∈ (0, 1) such that d
(
x
)
∈
C
2
(
∂
Ω
3
δ
¯
)
, and |∇d(x)| ≡ 1.
Denote
v
1
(
x
)
=
γ
d
(
x
)
,
d
(
x
)
<
δ
,
γ
δ
+
∫
δ
d
(
x
)
γ
2
δ
-
t
δ
2
p
-
-
1
(
L
1
+
1
)
2
p
-
-
1
d
t
,
δ
≤
d
(
x
)
<
2
δ
,
γ
δ
+
∫
δ
2
δ
γ
2
δ
-
t
δ
2
p
-
-
1
(
L
1
+
1
)
2
p
-
-
1
d
t
,
2
δ
≤
d
(
x
)
.
Obviously, 0
≤
v
1
(
x
)
∈
C
1
(
Ω
̄
)
. Considering
-
Δ
p
(
x
)
w
(
x
)
=
η
in
Ω
,
w
=
0
on
∂
Ω
,
we have the following result
Lemma 1.2. (see 16). If positive parameter η is large enough and w is the unique solution of (1), then we have
(i) For any θ∈ (0, 1) there exists a positive constant C1 such that
C
1
η
1
p
+
-
1
+
θ
≤
max
x
∈
Ω
̄
w
(
x
)
;
(ii) There exists a positive constant C2 such that
max
x
∈
Ω
̄
w
(
x
)
≤
C
2
η
1
p
-
-
1
.
2. Existence results
In the following, when there be no misunderstanding, we always use Ci to denote positive constants.
Theorem 2.1. On the conditions of (H1) - (H5), then (P) has a positive solution when λ is large enough.
Proof. We shall establish Theorem 2.1 by constructing a positive subsolution (Φ1, Φ2) and supersolution (z1, z2) of (P), such that Φ1 ≤ z1 and Φ2 ≤ z2. That is (Φ1, Φ2) and (z1, z2) satisfies
∫
Ω
|
∇
Φ
1
|
p
(
x
)
-
2
∇
Φ
1
⋅
∇
q
d
x
≤
∫
Ω
λ
p
(
x
)
g
(
x
)
a
(
Φ
1
)
q
d
x
+
∫
Ω
λ
p
(
x
)
f
(
Φ
2
)
q
d
x
,
∫
Ω
|
∇
Φ
2
|
p
(
x
)
-
2
∇
Φ
2
⋅
∇
q
d
x
≤
∫
Ω
λ
p
(
x
)
g
(
x
)
b
(
Φ
2
)
q
d
x
+
∫
Ω
λ
p
(
x
)
h
(
Φ
1
)
q
d
x
,
and
∫
Ω
|
∇
z
1
|
p
(
x
)
-
2
∇
z
1
⋅
∇
q
d
x
≥
∫
Ω
λ
p
(
x
)
g
(
x
)
a
(
z
1
)
q
d
x
+
∫
Ω
λ
p
(
x
)
f
(
z
2
)
q
d
x
,
∫
Ω
|
∇
z
2
|
p
(
x
)
-
2
∇
z
2
⋅
∇
q
d
x
≥
∫
Ω
λ
p
(
x
)
g
(
x
)
b
(
z
2
)
q
d
x
+
∫
Ω
λ
p
(
x
)
h
(
z
1
)
q
d
x
,
for all q
∈
W
0
1
,
p
(
x
)
(
Ω
)
with q ≥ 0. According to the sub-supersolution method for p(x)-Laplacian equations (see 16), then (P) has a positive solution.
Step 1. We construct a subsolution of (P).
Let σ ∈ (0, δ) is small enough. Denote
ϕ
(
x
)
=
e
k
d
(
x
)
-
1
,
d
(
x
)
<
σ
,
e
k
σ
-
1
+
∫
σ
d
(
x
)
k
e
k
σ
2
δ
-
t
2
δ
-
σ
2
p
-
-
1
d
t
,
σ
≤
d
(
x
)
<
2
δ
,
e
k
σ
-
1
+
∫
σ
2
δ
k
e
k
σ
2
δ
-
t
2
δ
-
σ
2
p
-
-
1
d
t
,
2
δ
≤
d
(
x
)
.
It is easy to see that ϕ
∈
C
1
(
Ω
̄
)
. Denote
α
=
min
inf
p
(
x
)
-
1
4
(
sup
|
∇
p
(
x
)
|
+
1
)
,
1
,
ζ
=
min
{
a
(
0
)
L
1
+
f
(
0
)
,
b
(
0
)
L
1
+
h
(
0
)
,
-
1
}
.
By computation
-
Δ
p
(
x
)
ϕ
=
-
k
(
k
e
k
d
(
x
)
)
p
(
x
)
-
1
(
p
(
x
)
-
1
)
+
(
d
(
x
)
+
ln
k
k
)
∇
p
∇
d
+
▵
d
k
,
d
(
x
)
<
σ
,
1
2
δ
-
σ
2
(
p
(
x
)
-
1
)
p
-
-
1
-
2
δ
-
d
2
δ
-
σ
ln
k
e
k
σ
2
δ
-
d
2
δ
-
σ
2
p
-
-
1
∇
p
∇
d
+
Δ
d
×
(
k
e
k
σ
)
p
(
x
)
-
1
2
δ
-
d
2
δ
-
σ
2
(
p
(
x
)
-
1
)
p
-
-
1
-
1
(
L
1
+
1
)
,
σ
<
d
(
x
)
<
2
δ
,
0
,
2
δ
<
d
(
x
)
.
From (H3) and (H4), there exists a positive constant M > 1 such that
f
(
M
-
1
)
≥
1
,
h
(
M
-
1
)
≥
1
.
Let σ
=
1
k
ln
M
, then
σ
k
=
ln
M
.
If k is sufficiently large, from (2), we have
-
Δ
p
(
x
)
ϕ
≤
-
k
p
(
x
)
α
,
d
(
x
)
<
σ
.
Let -λζ = kα, then
k
p
(
x
)
α
≥
-
λ
p
(
x
)
ζ
,
from (3), then we have
-
Δ
p
(
x
)
ϕ
≤
λ
p
(
x
)
(
a
(
0
)
L
1
+
f
(
0
)
)
≤
λ
p
(
x
)
(
g
(
x
)
a
(
ϕ
)
+
f
(
ϕ
)
)
,
d
(
x
)
<
σ
.
Since d
(
x
)
∈
C
2
(
∂
Ω
3
δ
¯
)
, then there exists a positive constant C3 such that
-
Δ
p
(
x
)
ϕ
≤
(
k
e
k
σ
)
p
(
x
)
-
1
2
δ
-
d
2
δ
-
σ
2
(
p
(
x
)
-
1
)
p
-
-
1
-
1
.
2
(
p
(
x
)
-
1
)
(
2
δ
-
σ
)
(
p
-
-
1
)
-
2
δ
-
d
2
δ
-
σ
ln
k
e
k
σ
2
δ
-
d
2
δ
-
σ
2
p
-
-
1
∇
p
∇
d
+
Δ
d
≤
C
3
(
k
e
k
σ
)
p
(
x
)
-
1
In
k
,
σ
<
d
(
x
)
<
2
δ
.
If k is sufficiently large, let -λζ = kα, we have
C
3
(
k
e
k
σ
)
p
(
x
)
-
1
ln
k
=
C
3
(
k
M
)
p
(
x
)
-
1
ln
k
≤
λ
p
(
x
)
,
then
-
Δ
p
(
x
)
ϕ
≤
λ
p
(
x
)
(
L
1
+
1
)
,
σ
<
d
(
x
)
<
2
δ
.
Since ϕ (x) ≥ 0 and a, f are monotone, when λ is large enough, then we have
-
Δ
p
(
x
)
ϕ
≤
λ
p
(
x
)
(
g
(
x
)
a
(
ϕ
)
+
f
(
ϕ
)
)
,
σ
<
d
(
x
)
<
2
δ
.
Obviously
-
Δ
p
(
x
)
ϕ
=
0
≤
λ
p
(
x
)
(
L
1
+
1
)
≤
λ
p
(
x
)
(
g
(
x
)
a
(
ϕ
)
+
f
(
ϕ
)
)
,
2
δ
<
d
(
x
)
.
Combining (4), (5), and (6), we can conclude that
-
Δ
p
(
x
)
ϕ
≤
λ
p
(
x
)
(
g
(
x
)
a
(
ϕ
)
+
f
(
ϕ
)
)
,
a
.
e
.
on
Ω
.
Similarly
-
Δ
p
(
x
)
ϕ
≤
λ
p
(
x
)
(
g
(
x
)
b
(
ϕ
)
+
h
(
ϕ
)
)
,
a
.
e
.
on
Ω
.
From (7) and (8), we can see that (ϕ1, ϕ2) = (ϕ, ϕ) is a subsolution of (P).
Step 2. We construct a supersolution of (P).
We consider
-
Δ
p
(
x
)
z
1
=
λ
p
+
μ
(
L
2
+
1
)
in
Ω
,
-
Δ
p
(
x
)
z
2
=
λ
p
+
(
L
2
+
1
)
h
(
β
(
λ
p
+
(
L
2
+
1
)
μ
)
)
in
Ω
,
z
1
=
z
2
=
0
on
∂
Ω
,
where β
=
β
(
λ
p
+
(
L
2
+
1
)
μ
)
=
max
x
∈
Ω
̄
z
1
(
x
)
. We shall prove that (z1, z2) is a supersolution for (p).
For q
∈
W
0
1
,
p
(
x
)
(
Ω
)
with q ≥ 0, it is easy to see that
∫
Ω
|
∇
z
2
|
p
(
x
)
-
2
∇
z
2
⋅
∇
q
d
x
=
∫
Ω
λ
p
+
(
L
2
+
1
)
h
(
β
(
λ
p
+
(
L
2
+
1
)
μ
)
)
q
d
x
≥
∫
Ω
λ
p
+
L
2
h
(
β
(
λ
p
+
(
L
2
+
1
)
μ
)
)
q
d
x
+
∫
Ω
λ
p
+
h
(
z
1
)
q
d
x
.
Since
lim
u
→
+
∞
f
M
(
h
(
u
)
)
1
(
p
-
-
1
)
u
p
-
-
1
=
0
,when μ is sufficiently large, combining Lemma 1.2 and (H3), then we have
h
(
β
(
λ
p
+
(
L
2
+
1
)
μ
)
)
≥
b
C
2
λ
p
+
(
L
2
+
1
)
h
(
β
(
λ
p
+
(
L
2
+
1
)
μ
)
)
1
p
-
-
1
≥
b
(
z
2
)
Hence
∫
Ω
|
∇
z
2
|
p
(
x
)
-
2
∇
z
2
⋅
∇
q
d
x
≥
∫
Ω
λ
p
+
g
(
x
)
b
(
z
2
)
q
d
x
+
∫
Ω
λ
p
+
h
(
z
1
)
q
d
x
.
Also
∫
Ω
|
∇
z
1
|
p
(
x
)
-
2
∇
z
1
⋅
∇
q
d
x
=
∫
Ω
λ
p
+
(
L
2
+
1
)
μ
q
d
x
By (H3), (H4), when μ is sufficiently large, combining Lemma 1.2 and (H3), we have
(
L
2
+
1
)
μ
≥
1
λ
p
+
1
C
2
β
(
λ
p
+
(
L
2
+
1
)
μ
)
p
-
-
1
≥
L
2
a
(
β
(
λ
p
+
(
L
2
+
1
)
μ
)
)
+
f
C
2
λ
p
+
(
L
2
+
1
)
h
(
β
(
λ
p
+
(
L
2
+
1
)
μ
)
)
1
p
-
-
1
.
Then
∫
Ω
|
∇
z
1
|
p
(
x
)
-
2
∇
z
1
⋅
∇
q
d
x
≥
∫
Ω
λ
p
+
g
(
x
)
a
(
z
1
)
q
d
x
+
∫
Ω
λ
p
+
f
(
z
2
)
q
d
x
.
According to (11) and (12), we can conclude that (z1, z2) is a supersolution for (P).
It only remains to prove that ϕ1 ≤ z1 and ϕ2 ≤ z2.
In the definition of v1(x), let γ
=
δ
2
(
max
x
∈
Ω
̄
ϕ
(
x
)
+
max
x
∈
Ω
̄
|
∇
ϕ
(
x
)
|
)
. We claim that
ϕ
(
x
)
≤
v
1
(
x
)
,
∀
x
∈
Ω
.
From the definition of v1, it is easy to see that
ϕ
(
x
)
≤
2
max
x
∈
Ω
̄
ϕ
(
x
)
≤
v
1
(
x
)
,
when
d
(
x
)
=
δ
,
and
ϕ
(
x
)
≤
2
max
x
∈
Ω
̄
ϕ
(
x
)
≤
v
1
(
x
)
,
when
d
(
x
)
≥
δ
.
It only remains to prove that
ϕ
(
x
)
≤
v
1
(
x
)
,
when
d
(
x
)
<
δ
.
Since
v
1
-
ϕ
∈
C
1
(
∂
Ω
δ
¯
)
,
then there exists a point
x
0
∈
∂
Ω
δ
¯
such that
v
1
(
x
0
)
-
ϕ
(
x
0
)
=
min
x
0
∈
∂
Ω
δ
¯
[
v
1
(
x
)
-
ϕ
(
x
)
]
.
If v1(x0) - ϕ(x0) < 0, it is easy to see that 0 < d(x0) < δ, and then
∇
v
1
(
x
0
)
-
∇
ϕ
(
x
0
)
=
0
.
From the definition of v1, we have
|
∇
v
1
(
x
0
)
|
=
γ
=
2
δ
max
x
∈
Ω
̄
ϕ
(
x
)
+
max
x
∈
Ω
̄
|
∇
ϕ
(
x
)
|
>
|
∇
ϕ
(
x
0
)
|
.
It is a contradiction to ∇v1(x0) - ∇ϕ(x0) = 0. Thus (13) is valid.
Obviously, there exists a positive constant C3 such that
γ
≤
C
3
λ
.
Since d
(
x
)
∈
C
2
(
∂
Ω
3
δ
¯
)
, according to the proof of Lemma 1.2, then there exists a positive constant C4 such that
-
Δ
p
(
x
)
v
1
(
x
)
≤
C
*
γ
p
(
x
)
-
1
+
θ
≤
C
4
λ
p
(
x
)
-
1
+
θ
,
a
.
e
.
in
Ω
,
where
θ
∈
(
0
,
1
)
.
When η
≥
λ
p
+
is large enough, we have
-
Δ
p
(
x
)
v
1
(
x
)
≤
η
.
According to the comparison principle, we have
v
1
(
x
)
≤
w
(
x
)
,
∀
x
∈
Ω
.
From (13) and (14), when η
≥
λ
p
+
and λ ≥ 1 is sufficiently large, we have
ϕ
(
x
)
≤
v
1
(
x
)
≤
w
(
x
)
,
∀
x
∈
Ω
.
According to the comparison principle, when μ is large enough, we have
v
1
(
x
)
≤
w
(
x
)
≤
z
1
(
x
)
,
∀
x
∈
Ω
.
Combining the definition of v1(x) and (15), it is easy to see that
ϕ
1
(
x
)
=
ϕ
(
x
)
≤
v
1
(
x
)
≤
w
(
x
)
≤
z
1
(
x
)
,
∀
x
∈
Ω
.
When μ ≥ 1 and λ is large enough, from Lemma 1.2, we can see that β
(
λ
p
+
(
L
2
+
1
)
μ
)
is large enough, then
λ
p
+
(
L
2
+
1
)
h
(
β
(
λ
p
+
(
L
2
+
1
)
μ
)
)
is large enough. Similarly, we have ϕ2 ≤ z2. This completes the proof. □
3. Asymptotic behavior of positive solutions
In this section, when parameter λ → +∞, we will discuss the asymptotic behavior of maximum of solutions about parameter λ, and the asymptotic behavior of solutions near boundary about parameter λ.
Theorem 3.1. On the conditions of (H1)-(H5), if (u, v) is a solution of (P) which has been given in Theorem 2.1, then
(i) There exist positive constants C1 and C2 such that
C
1
λ
≤
max
x
∈
Ω
̄
u
(
x
)
≤
C
2
λ
p
+
(
L
2
+
1
)
μ
1
p
-
-
1
C
1
λ
≤
max
x
∈
Ω
̄
v
(
x
)
≤
C
2
λ
p
+
(
L
2
+
1
)
h
C
2
(
λ
p
+
(
L
2
+
1
)
μ
)
1
p
-
-
1
1
p
-
-
1
(ii) for any θ ∈ (0, 1), there exist positive constants C3 and C4 such that
C
3
λ
d
(
x
)
≤
u
(
x
)
≤
C
4
(
λ
p
+
(
L
2
+
1
)
μ
)
1
/
(
p
-
-
1
)
(
d
(
x
)
)
θ
,
a
s
d
(
x
)
→
0
,
C
3
λ
d
(
x
)
≤
v
(
x
)
≤
C
4
λ
p
+
(
L
2
+
1
)
h
C
2
(
λ
p
+
(
L
2
+
1
)
μ
)
1
p
-
-
1
1
p
-
-
1
(
d
(
x
)
)
θ
,
a
s
d
(
x
)
→
0
where μ satisfies (10).
Proof. (i) Obviously, when 2δ ≤ d(x), we have
u
(
x
)
,
v
(
x
)
≥
ϕ
(
x
)
=
e
k
σ
-
1
+
∫
σ
2
δ
k
e
k
σ
2
δ
-
t
2
δ
-
σ
2
p
-
-
1
d
t
≥
-
λ
ζ
α
∫
σ
2
δ
M
2
δ
-
t
2
δ
-
σ
2
p
-
-
1
d
t
,
then there exists a positive constant C1 such that
C
1
λ
≤
max
x
∈
Ω
̄
u
(
x
)
and
C
1
λ
≤
max
x
∈
Ω
̄
v
(
x
)
.
It is easy to see
u
(
x
)
≤
z
1
(
x
)
≤
max
x
∈
Ω
̄
z
1
(
x
)
≤
C
2
(
λ
p
+
(
L
2
+
1
)
μ
)
1
p
-
-
1
,
then
max
x
∈
Ω
̄
u
(
x
)
≤
C
2
λ
p
+
(
L
2
+
1
)
μ
1
p
-
-
1
.
Similarly
max
x
∈
Ω
̄
v
(
x
)
≤
C
2
λ
p
+
(
L
2
+
1
)
h
C
2
λ
p
+
(
L
2
+
1
)
μ
1
p
-
-
1
1
p
-
-
1
Thus (16) and (17) are valid.
(ii) Denote
v
3
(
x
)
=
α
(
d
(
x
)
)
θ
,
d
(
x
)
≤
ρ
,
where θ ∈ (0, 1) is a positive constant, ρ ∈ (0, δ) is small enough.
Obviously, v3(x) ∈ C1(Ωρ), By computation
-
Δ
p
(
x
)
v
3
(
x
)
=
-
(
α
θ
)
p
(
x
)
-
1
(
θ
-
1
)
(
p
(
x
)
-
1
)
(
d
(
x
)
)
(
θ
-
1
)
(
p
(
x
)
-
1
)
-
1
(
1
+
Π
(
x
)
)
,
d
(
x
)
<
ρ
,
where
Π
x
=
d
∇
p
∇
d
In
α
θ
θ
-
1
p
x
-
1
+
d
∇
p
∇
d
In
d
p
x
-
1
+
d
Δ
d
θ
-
1
p
x
-
1
.
Let α
=
1
ρ
C
2
λ
p
+
L
2
+
1
μ
1
/
p
-
-
1
,
where ρ > 0 is small enough, it is easy to see that
α
p
x
-
1
≥
λ
p
+
μ
L
2
+
1
and
Π
x
≤
1
2
.
where ρ > 0 is small enough, then we have
-
Δ
p
x
v
3
x
≥
λ
p
+
μ
L
2
+
1
.
Obviously v3(x) ≥ z1(x) on ∂Ωρ. According to the comparison principle, we have v3(x) ≥ z1 (x) on Ωρ. Thus
u
x
≤
C
4
λ
p
+
L
2
+
1
μ
1
/
p
-
-
1
d
x
θ
,
a
s
d
x
→
0
.
Let α
=
1
ρ
C
2
λ
p
+
L
2
+
1
h
C
2
λ
p
+
L
2
+
1
μ
1
p
-
-
1
1
p
-
-
1
,
when ρ > 0 is small enough, it is easy to see that
α
p
x
-
1
≥
λ
p
+
L
2
+
1
h
C
2
λ
p
+
L
2
+
1
μ
1
p
-
-
1
.
Similarly, when ρ > 0 is small enough, we have
v
x
≤
C
4
λ
p
+
L
2
+
1
h
C
2
λ
p
+
L
2
+
1
μ
1
p
-
-
1
1
p
-
-
1
d
x
θ
a
s
d
x
→
0
Obviously, when d(x) < σ, we have
u
x
,
v
x
≥
ϕ
x
=
e
k
d
x
-
1
≥
C
3
λ
d
x
.
Thus (18) and (19) are valid. This completes the proof. □
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
All authors typed, read and approved the final manuscript.