Báo cáo hóa học: "Research Article Existence of Solutions for a Class of Weighted p t -Laplacian System Multipoint Boundary Value Problems" doc
Bạn đang xem bản rút gọn của tài liệu. Xem và tải ngay bản đầy đủ của tài liệu tại đây (549.07 KB, 18 trang )
Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2008, Article ID 791762, 18 pages
doi:10.1155/2008/791762
Research Article
Existence of Solutions for a Class of
Weighted pt-Laplacian System Multipoint
Boundary Value Problems
Qihu Zhang,
1, 2, 3
Zheimei Qiu,
2
and Xiaopin Liu
2
1
Department of Mathematics and Information Science, Zhengzhou University of Light Industry,
Zhengzhou, Henan 450002, China
2
School of Mathematical Science, Xuzhou Normal University, Xuzhou, Jiangsu 221116, China
3
College of Mathematics and Information Science, Shaanxi Normal University, Xi’an,
Shaanxi 710062, China
Correspondence should be addressed to Zheimei Qiu,
Received 12 June 2008; Accepted 22 October 2008
Recommended by Alberto Cabada
This paper investigates the existence of solutions for weighted pt-Laplacian system multipoint
boundary value problems. When the nonlinearity term ft, ·, · satisfies sub-p
−
−1 growth condition
or general growth condition, we give the existence of solutions via Leray-Schauder degree.
Copyright q 2008 Qihu Zhang et al. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
1. Introduction
In this paper, we consider the existence of solutions for the following weighted pt-Laplacian
system:
−Δ
pt,wt
u δf
t, u,
wt
1/pt−1
u
0, t ∈ 0, 1, 1.1
with the following multipoint boundary value condition:
u0
m−2
i1
β
i
u
η
i
e
0
,u1
m−2
i1
α
i
u
ξ
i
e
1
, 1.2
where p ∈ C0, 1, R and pt > 1, −Δ
pt,wt
u −wt|u
|
pt−2
u
is called the weighted
pt-Laplacian; w ∈ C0, 1, R satisfies 0 <wt, for all t ∈ 0, 1,andwt
−1/pt−1
∈
L
1
0, 1;0<η
1
< ··· <η
m−2
< 1, 0 <ξ
1
< ··· <ξ
m−2
< 1; α
i
≥ 0, β
i
≥ 0 i 1, ,m− 2,and
0 <
m−2
i1
α
i
< 1, 0 <
m−2
i1
β
i
< 1; e
0
, e
1
∈ R
N
; δ is a positive parameter.
2 Journal of Inequalities and Applications
The study of differential equations and variational problems with variable exponent
growth conditions is a new and interesting topic. Many results have been obtained on these
problems, for example, 1–14. We refer to 2, 15, 16 the applied background on these
problems. If wt ≡ 1andpt ≡ p a constant, −Δ
pt,wt
is the well-known p-Laplacian.
If pt is a general function, −Δ
pt,wt
represents a nonhomogeneity and possesses more
nonlinearity, thus −Δ
pt,wt
is more complicated than −Δ
p
. We have the following examples.
1 If Ω ⊂ R
N
is a bounded domain, the Rayleigh quotient
λ
px
inf
u∈W
1,px
0
Ω\{0}
Ω
1/px|∇u|
px
dx
Ω
1/px|u|
px
dx
1.3
is zero in general, and only under some special conditions λ
px
> 0 see 6,but
the fact that λ
p
> 0 is very important in the study of p-Laplacian problems.
2 If wt ≡ 1andpt ≡ p a constant and −Δ
p
u>0, then u is concave, this property
is used extensively in the study of one-dimensional p-Laplacian problems, but it is
invalid for −Δ
pt,1
. It is another difference on −Δ
p
and −Δ
pt,1
.
3 On the existence of solutions of the following typical −Δ
px,1
problem:
−
u
px−2
u
|u|
qx−2
u C, x ∈ Ω ⊂ R
N
,
u 0on∂Ω,
1.4
because of the nonhomogeneity of −Δ
px,1
, if max
x∈Ω
qx < min
x∈Ω
px, then
the corresponding functional is coercive; if max
x∈Ω
px < min
x∈Ω
qx, then
the corresponding functional satisfies Palais-Smale condition see 4, 7, 12.If
min
x∈Ω
px ≤ qx ≤ max
x∈Ω
px, we can see that the corresponding functional
is neither coercive nor satisfying Palais-Smale conditions, the results on this case
are rare.
There are many results on the existence of solutions for p-Laplacian equation with
multipoint boundary value conditions see 17–20. On the existence of solutions for px-
Laplacian systems boundary value problems, we refer to 5, 7, 10, 11.Butresultsonthe
existence of solutions for weighted pt-Laplacian systems with multipoint boundary value
conditions are rare. In this paper, when pt is a general function, we investigate the existence
of solutions for weighted pt-Laplacian systems with multipoint boundary value conditions.
Moreover, the case of min
t∈0,1
pt ≤ qt ≤ max
t∈0,1
pt has been discussed.
Let N ≥ 1andI 0, 1, the function f f
1
, ,f
N
: I × R
N
× R
N
→ R
N
is assumed
to be Caratheodory, by this we mean the following:
i for almost every t ∈ I the function ft, ·, · is continuous;
ii for each x, y ∈ R
N
× R
N
the function f·,x,y is measurable on I;
iii for each R>0 there is a β
R
∈ L
1
I,R, such that for almost every t ∈ I and every
x, y ∈ R
N
× R
N
with |x|≤R, |y|≤R, one has
ft, x, y
≤ β
R
t. 1.5
Qihu Zhang et al. 3
Throughout the paper, we denote
w0
u
p0−2
u
0 lim
r → 0
wr
u
pr−2
u
r,
w1
u
p0−2
u
1 lim
r → 1
−
wr
u
pr−2
u
r.
1.6
The inner product in R
N
will be denoted by ·, ·, |·| will denote the absolute value
and the Euclidean norm on R
N
. For N ≥ 1, we set C CI, R
N
, C
1
{u ∈ C | u
∈
C0, 1, R
N
, lim
t → 0
wt|u
|
pt−2
u
t, and lim
t → 1
−
wt|u
|
pt−2
u
t exist}. For any ut
u
1
t, ,u
N
t, we denote |u
i
|
0
sup
t∈0,1
|u
i
t|, u
0
N
i1
|u
i
|
2
0
1/2
, and u
1
u
0
wt
1/pt−1
u
0
. Spaces C and C
1
will be equipped with the norm ·
0
and ·
1
,
respectively. Then C, ·
0
and C
1
, ·
1
are Banach spaces.
We say a function u : I → R
N
is a solution of 1.1 if u ∈ C
1
with wt|u
|
pt−2
u
absolutely continuous on 0, 1, which satisfies 1.1 a.e. on I.
In this paper, we always use C
i
to denote positive constants, if it cannot lead to
confusion. Denote
z
−
min
t∈I
zt, z
max
t∈I
zt, for any z ∈ CI,R. 1.7
We say f satisfies sub- p
−
− 1 growth condition, if f satisfies
lim
|u||v|→∞
ft, u, v
|u| |v|
qt−1
0, for t ∈ I uniformly, 1.8
where qt ∈ CI,R and 1 <q
−
≤ q
<p
−
. We say that f satisfies general growth condition, if
we do not know whether f satisfies sub-p
−
− 1 growth condition or not.
We will discuss the existence of solutions of 1.1-1.2 in the following two cases:
i f satisfies sub-p
−
− 1 growth condition;
ii f satisfies general growth condition.
This paper is divided into four sections. In the second section, we will do some
preparation. In the third section, we will discuss the existence of solutions of 1.1-1.2, when
f satisfies sub-p
−
− 1 growth condition. Finally, in Section 4, we will discuss the existence of
solutions of 1.1-1.2, when f satisfies general growth condition.
2. Preliminary
For any t, x ∈ I × R
N
, denote ϕt, x|x|
pt−2
x. Obviously, ϕ has the following properties.
Lemma 2.1 see 4. ϕ is a continuous function and satisfies the following:
i for any t ∈ 0, 1,ϕt, · is strictly monotone, that is,
ϕ
t, x
1
− ϕ
t, x
2
,x
1
− x
2
> 0, for any x
1
,x
2
∈ R
N
,x
1
/
x
2
; 2.1
4 Journal of Inequalities and Applications
ii there exists a function ρ : 0, ∞ → 0, ∞, ρs → ∞ as s → ∞, such that
ϕt, x,x
≥ ρ
|x|
|x|, ∀x ∈ R
N
. 2.2
It is well known that ϕt, · is a homeomorphism from R
N
to R
N
for any fixed t ∈ 0, 1. For
any t ∈ I, denote by ϕ
−1
t, · the inverse operator of ϕt, ·,then
ϕ
−1
t, x|x|
2−pt/pt−1
x, for x ∈ R
N
\{0}, ϕ
−1
t, 00. 2.3
It is clear that ϕ
−1
t, · is continuous and sends bounded sets to bounded sets. Let us
now consider the following problem with boundary value condition 1.2:
wtϕ
t, u
t
gt, 2.4
where g ∈ L
1
.Ifu is a solution of 2.4 with 1.2, by integrating 2.4 from 0 to t,wefindthat
wtϕ
t, u
t
w0ϕ
0,u
0
t
0
gs ds. 2.5
Denote a w0ϕ0,u
0.Itiseasytoseethata is dependent on gt. Define operator
F : L
1
→ C as Fgt
t
0
gs ds. By solving for u
in 2.5 and integrating, we find
utu0F
ϕ
−1
t,
wt
−1
a Fg
t. 2.6
From u0
m−2
i1
β
i
uη
i
e
0
, we have
u0
m−2
i1
β
i
η
i
0
ϕ
−1
t,
wt
−1
a Fgt
dt e
0
1 −
m−2
i1
β
i
. 2.7
From u1
m−2
i1
α
i
uξ
i
e
1
,weobtain
u0
m−2
i1
α
i
ξ
i
0
ϕ
−1
t,
wt
−1
a Fgt
dt −
1
0
ϕ
−1
t,
wt
−1
a Fgt
dt e
1
1 −
m−2
i1
α
i
.
2.8
From 2.7 and 2.8, we have
m−2
i1
β
i
η
i
0
ϕ
−1
t,
wt
−1
a Fgt
dt e
0
1 −
m−2
i1
β
i
m−2
i1
α
i
ξ
i
0
ϕ
−1
t,
wt
−1
a Fgt
dt −
1
0
ϕ
−1
t,
wt
−1
a Fgt
dt e
1
1 −
m−2
i1
α
i
.
2.9
Qihu Zhang et al. 5
For fixed h ∈ C, we denote
Λ
h
a
m−2
i1
β
i
η
i
0
ϕ
−1
t,
wt
−1
a ht
dt e
0
1 −
m−2
i1
β
i
−
m−2
i1
α
i
ξ
i
0
ϕ
−1
t,
wt
−1
a ht
dt −
1
0
ϕ
−1
t,
wt
−1
a ht
dt e
1
1 −
m−2
i1
α
i
.
2.10
Throughout the paper, we denote E
1
0
wt
−1/pt−1
dt.
Lemma 2.2. The function Λ
h
· has the following properties:
i for any fixed h ∈ C, the equation
Λ
h
a0 2.11
has a unique solution ah ∈ R
N
;
ii the function a : C → R
N
, defined in (i), is continuous and sends bounded sets to bounded
sets. Moreover,
ah
≤ 3N
E 1
1 −
m−2
i1
β
i
E
E 1
1 −
m−2
i1
α
i
E
1
p
−1
·
h
0
2N
p
e
0
e
1
p
#
−1
,
2.12
where the notation M
p
#
−1
means
M
p
#
−1
⎧
⎨
⎩
M
p
−1
,M>1
M
p
−
−1
,M≤ 1.
2.13
Proof. i It is easy to see that
Λ
h
a
m−2
i1
β
i
η
i
0
ϕ
−1
t,
wt
−1
a ht
dt e
0
1 −
m−2
i1
β
i
m−2
i1
α
i
1
ξ
i
ϕ
−1
t,
wt
−1
a ht
dt − e
1
1 −
m−2
i1
α
i
1
0
ϕ
−1
t,
wt
−1
a ht
dt.
2.14
6 Journal of Inequalities and Applications
From Lemma 2.1, it is immediate that
Λ
h
a
1
− Λ
h
a
2
,a
1
− a
2
> 0, for a
1
/
a
2
, 2.15
and hence, if 2.11 has a solution, then it is unique.
Let
t
0
3N
E 1
1 −
m−2
i1
β
i
E
E 1
1 −
m−2
i1
α
i
E
1
p
−1
·
h
0
2N
p
e
0
e
1
p
#
−1
.
2.16
If |a|≥t
0
, since wt
−1/pt−1
∈ L
1
0, 1 and h ∈ C, it is easy to see that there exists an
i ∈{1, ,N} such that the ith component a
i
of a satisfies
a
i
≥ 3
E 1
1 −
m−2
i1
β
i
E
E 1
1 −
m−2
i1
α
i
E
1
p
−1
·
h
0
2N
p
e
0
e
1
p
#
−1
.
2.17
Thus a
i
h
i
t keeps sign on I and
a
i
h
i
t
≥
a
i
−h
0
≥ 2
E 1
1 −
m−2
i1
β
i
E
E 1
1 −
m−2
i1
α
i
E
1
p
−1
·
h
0
2N
p
e
0
e
1
p
#
−1
, ∀t ∈ I,
2.18
then
a
i
h
i
t
1/pt−1
≥ 2
1/p
−1
E 1
1 −
m−2
i1
β
i
E
E 1
1 −
m−2
i1
α
i
E
1
e
0
e
1
, ∀t ∈ I.
2.19
Thus, when |a| is large enough, the ith component Λ
i
h
a of Λ
h
a is nonzero, then we
have
Λ
h
a
/
0. 2.20
Let us consider the equation
λΛ
h
a1 − λa 0,λ∈ 0, 1. 2.21
Qihu Zhang et al. 7
It is easy to see that all the solutions of 2.21 belong to bt
0
{x ∈ R
N
||x| <t
0
}. So,
we have
d
B
Λ
h
a,b
t
0
, 0
d
B
I,b
t
0
, 0
/
0, 2.22
it means the existence of solutions of Λ
h
a0.
In this way, we define a function ah : C0, 1 → R
N
, which satisfies
Λ
h
ah
0. 2.23
ii By the proof of i, we also obtain that a sends bounded sets to bounded sets, and
ah
≤ 3N
E 1
m−2
i1
β
i
E
E 1
m−2
i1
α
i
E
1
p
−1
·
h
0
e
0
e
1
p
#
−1
. 2.24
It only remains to prove the continuity of a.Let{u
n
} be a convergent sequence in C
and u
n
→ u as n → ∞. Since {au
n
} is a bounded sequence, then it contains a convergent
subsequence {au
n
j
}.Letau
n
j
→ a
0
as j → ∞. Since Λ
u
n
j
au
n
j
0, letting j → ∞,
we have Λ
u
a
0
0. From i,wegeta
0
au, it means that a is continuous. T his completes
the proof.
Now, we define a : L
1
→ R
N
as
aua
Fu
. 2.25
It is clear that a· is continuous and sends bounded sets of L
1
to bounded sets of R
N
,
and hence it is a complete continuous mapping.
If u is a solution of 2.4 with 1.2, then
utu0F
ϕ
−1
t,
wt
−1
agFgt
t, ∀t ∈ 0, 1. 2.26
The boundary condition 1.2 implies that
u0
m−2
i1
β
i
η
i
0
ϕ
−1
t,
wt
−1
agFgt
dt e
0
1 −
m−2
i1
β
i
. 2.27
We denote that
K
1
ht :
K
1
◦ h
tF
ϕ
−1
t,
wt
−1
ahFh
t, ∀t ∈ 0, 1. 2.28
Lemma 2.3. The operator K
1
is continuous and sends equi-integrable sets in L
1
to relatively compact
sets in C
1
.
8 Journal of Inequalities and Applications
Proof. It is easy to check that K
1
ht ∈ C
1
. Since wt
−1/pt−1
∈ L
1
and
K
1
h
tϕ
−1
t,
wt
−1
ahFh
, ∀t ∈ 0, 1, 2.29
it is easy to check that K
1
is a continuous operator from L
1
to C
1
.
Let now U be an equi-integrable set in L
1
, then there exists ρ
∗
∈ L
1
, such that
ut
≤ ρ
∗
t a.e. in I, for any u ∈ U. 2.30
We want to show that
K
1
U ⊂ C
1
is a compact set.
Let {u
n
} be a sequence in K
1
U, then there exists a sequence {h
n
}∈U such that
u
n
K
1
h
n
. For any t
1
,t
2
∈ I, we have
F
h
n
t
1
− F
h
n
t
2
t
1
0
h
n
t dt −
t
2
0
h
n
t dt
t
2
t
1
h
n
t dt
≤
t
2
t
1
ρ
∗
t dt
.
2.31
Hence the sequence {Fh
n
} is uniformly bounded and equicontinuous. By Ascoli-
Arzela theorem, there exists a subsequence of {Fh
n
} which we rename the same
convergent in C. According to the bounded continuous operator a, we can choose a
subsequence of {ah
n
Fh
n
} which we still denote {ah
n
Fh
n
} which is convergent
in C, then wtϕt, K
1
h
n
t ah
n
Fh
n
is convergent in C.
Since
K
1
h
n
tF
ϕ
−1
t,
wt
−1
a
h
n
F
h
n
t, ∀t ∈ 0, 1, 2.32
according to the continuity of ϕ
−1
and the integrability of wt
−1/pt−1
in L
1
, we can see that
K
1
h
n
is convergent in C.Thus{u
n
} is convergent in C
1
. This completes the proof.
Let us define P : C
1
→ C
1
as
Ph
m−2
i1
β
i
K
1
◦ h
η
i
e
0
1 −
m−2
i1
β
i
. 2.33
It is easy to see that P is compact continuous.
We denote N
f
u : 0, 1 × C
1
→ L
1
the Nemytski operator associated to f defined by
N
f
utf
t, ut,
wt
1/pt−1
u
t
, a.e. on I. 2.34
Lemma 2.4. u is a solution of 1.1-1.2 if and only if u is a solution of the following abstract
equation:
u P
δN
f
u
K
1
δN
f
u
. 2.35
Qihu Zhang et al. 9
Proof. If u is a solution of 1.1-1.2, by integrating 1.1 from 0 to t,wefindthat
wtϕ
t, u
t
a
δN
f
u
F
δN
f
u
t. 2.36
From 2.36, we have
utu0F
ϕ
−1
r,
wr
−1
a
δN
f
u
F
δN
f
u
t,
u0
m−2
i1
β
i
u0F
ϕ
−1
r,
wr
−1
a
δN
f
u
F
δN
f
u
η
i
e
0
,
2.37
then we have
u0
m−2
i1
β
i
F
ϕ
−1
r,
wr
−1
a
δN
f
u
F
δN
f
u
η
i
e
0
1 −
m−2
i1
β
i
m−2
i1
β
i
K
1
δN
f
u
η
i
e
0
1 −
m−2
i1
β
i
P
δN
f
u
.
2.38
So we have
u P
δN
f
u
K
1
δN
f
u
. 2.39
Conversely, if u is a solution of 2.35,itiseasytoseethat
u0P
δN
f
u
m−2
i1
β
i
K
1
δN
f
u
η
i
e
0
1 −
m−2
i1
β
i
,
u0
m−2
i1
β
i
u0K
1
δN
f
u
η
i
e
0
m−2
i1
β
i
u
η
i
e
0
,
u1P
δN
f
u
K
1
δN
f
u
1.
2.40
By the condition of the mapping a,
u0
m−2
i1
β
i
K
1
δN
f
u
η
i
e
0
1 −
m−2
i1
β
i
m−2
i1
α
i
K
1
δN
f
u
ξ
i
− K
1
δN
f
u
1e
1
1 −
m−2
i1
α
i
,
2.41
then we have
u1
m−2
i1
α
i
K
1
δN
f
u
ξ
i
− K
1
δN
f
u
1e
1
1 −
m−2
i1
α
i
K
1
δN
f
u
1, 2.42
10 Journal of Inequalities and Applications
thus
u1
m−2
i1
α
i
u1 − K
1
δN
f
u
1K
1
δN
f
u
ξ
i
e
1
m−2
i1
α
i
P
δN
f
u
K
1
δN
f
u
ξ
i
e
1
m−2
i1
α
i
u
ξ
i
e
1
,
2.43
from 2.40 and 2.43,weobtain1.2.
From 2.35, we have
u
tϕ
−1
t,
wt
−1
a F
δN
f
u
,
wtϕ
t, u
δN
f
ut.
2.44
Hence u is a solution of 1.1-1.2. This completes the proof.
Lemma 2.5. If u is a solution of 1.1-1.2, then for any j 1, ,N,there exists a ς
j
∈ 0, 1, such
that
u
j
ς
j
≤ C
∗
:
e
0
1 −
m−2
i1
β
i
e
1
1 −
m−2
i1
α
i
. 2.45
Proof. For any j 1, ,N,if there exists ς
j
∈ 0, 1 such that u
j
ς
j
0, then 2.45 is valid.
If it is false, then u
j
is strictly monotone.
i If u
j
is strictly decreasing in 0, 1, then
u
j
0 >u
j
ξ
i
>u
j
1,u
j
0 >u
j
η
i
>u
j
1,i 1, ,m− 2. 2.46
Thus
u
j
0
m−2
i1
β
i
u
j
η
i
e
j
0
<
m−2
i1
β
i
u
j
0e
j
0
,
u
j
1
m−2
i1
α
i
u
j
ξ
i
e
j
1
>
m−2
i1
α
i
u
j
1e
j
1
,
2.47
it means that
e
j
0
1 −
m−2
i1
β
i
>u
j
0 >u
j
1 >
e
j
1
1 −
m−2
i1
α
i
, 2.48
Qihu Zhang et al. 11
then there exists a t
j
∈ 0, 1 such that
0 >
u
j
t
j
u
j
1 − u
j
0
1 − 0
> −
e
j
1
1 −
m−2
i1
α
i
e
j
0
1 −
m−2
i1
β
i
. 2.49
ii If u
j
is strictly increasing in 0, 1, then
u
j
0 <u
j
ξ
i
<u
j
1,u
j
0 <u
j
η
i
<u
j
1,i 1, ,m− 2. 2.50
Thus
u
j
0
m−2
i1
β
i
u
j
η
i
e
j
0
>
m−2
i1
β
i
u
j
0e
j
0
,
u
j
1
m−2
i1
α
i
u
j
ξ
i
e
j
1
<
m−2
i1
α
i
u
j
1e
j
1
,
2.51
it means that
e
j
0
1 −
m−2
i1
β
i
<u
j
0 <u
j
1 <
e
j
1
1 −
m−2
i1
α
i
, 2.52
then there exists a r
j
∈ 0, 1 such that
0 <
u
j
r
j
u
j
1 − u
j
0
1 − 0
<
e
j
1
1 −
m−2
i1
α
i
e
j
0
1 −
m−2
i1
β
i
. 2.53
Combining 2.49 and 2.53, then we obtain 2.45.
This completes the proof.
3. f satisfies sub-p
−
− 1 growth condition
In this section, we will apply Leray-Schauder’s degree to deal with the existence of solutions
for 1.1-1.2, when f satisfies sub-p
−
− 1 growth condition.
Theorem 3.1. If f satisfies sub-p
−
− 1 growth condition, then for any fixed parameter δ, problem
1.1-1.2 has at least one solution.
Proof. Denote Ψ
f
u, λ : PλδN
f
u K
1
λδN
f
u, where N
f
u is defined in 2.34.We
know that 1.1-1.2 has the same solution of
u Ψ
f
u, λ, 3.1
when λ 1.
12 Journal of Inequalities and Applications
It is easy to see that the operator P is compact continuous. According to Lemmas 2.2
and 2.3, then we can see that Ψ
f
·,λ is compact continuous from C
1
to C
1
for any λ ∈ 0, 1.
We claim that all the solutions of 3.1 are uniformly bounded for λ ∈ 0, 1.Infact,if
it is false, we can find a sequence of solutions {u
n
,λ
n
} for 3.1 such that u
n
1
→ ∞ as
n → ∞,andu
n
1
> 1 for any n 1, 2,
Let t
n
∈ 0, 1 such that
1
2
sup
t∈0,1
wt
u
n
t
pt−1
≤ w
t
n
u
n
t
n
pt
n
−1
, n 1, 2, 3.2
For any fixed n 1, 2, ,there exists an i
n
∈{1, ,N} such that
u
i
n
n
t
n
≥
1
N
u
n
t
n
. 3.3
Thus, {u
i
n
n
} becomes a sequence with respect to n.
Since u
n
,λ
n
are solutions of 3.1, according to Lemma 2.5, for any n 1, 2, ,there
exists ξ
i
n
n
∈ 0, 1 such that |u
i
n
n
ξ
i
n
n
|≤C
∗
, then
wt
u
n
pt−2
u
i
n
n
tw
ξ
i
n
n
u
n
ξ
i
n
n
pξ
i
n
n
−2
u
i
n
n
ξ
i
n
n
t
ξ
i
n
n
u
n
qr−1
1
λ
n
δf
i
n
r, u
n
,
wr
1/pr−1
u
n
u
n
qr−1
1
dr, ∀t ∈ 0, 1.
3.4
For any t ∈ 0, 1, we have
wt
u
n
pt−2
u
i
n
n
t
≤ w
ξ
i
n
n
u
n
ξ
i
n
n
pξ
i
n
n
−2
u
i
n
n
ξ
i
n
n
t
ξ
i
n
n
u
n
qr−1
1
λ
n
δf
i
n
r, u
n
,
wr
1/pr−1
u
n
u
n
qr−1
1
dr
.
3.5
Without loss of generality, we assume that |u
n
ξ
i
n
n
| > 0.
1
◦
If pξ
i
n
n
− 2 ≤ 0, then
w
ξ
i
n
n
|u
n
ξ
i
n
n
|
pξ
i
n
n
−2
|
u
i
n
n
ξ
i
n
n
| w
ξ
i
n
n
|
u
i
n
n
ξ
i
n
n
|
|u
n
ξ
i
n
n
|
2−pξ
i
n
n
≤ w
ξ
i
n
n
|
u
i
n
n
ξ
i
n
n
|
|
u
i
n
n
ξ
i
n
n
|
2−pξ
i
n
n
w
ξ
i
n
n
|
u
i
n
n
ξ
i
n
n
|
pξ
i
n
n
−1
≤ w
ξ
i
n
n
C
pξ
i
n
n
−1
∗
,
3.6
where C
∗
is defined in 2.45.
Qihu Zhang et al. 13
Combining 3.2, 3.3, 3.5,and3.6, we have
1
2N
sup
t∈0,1
wt
u
n
pt−1
≤
1
N
w
t
n
u
n
t
n
pt
n
−1
≤ w
t
n
u
n
t
n
pt
n
−2
u
i
n
n
t
n
≤ w
ξ
i
n
n
C
pξ
i
n
n
−1
∗
t
n
ξ
i
n
n
u
n
qr−1
1
λ
n
δf
i
n
r, u
n
,
wr
1/pr−1
u
n
u
n
qr−1
1
dr
≤ C
1
C
2
u
n
q
−1
1
.
3.7
Then we have
wt
u
n
t
pt−1
≤ 2N
C
1
C
2
u
n
q
−1
1
, ∀t ∈ 0, 1. 3.8
Denoting α q
− 1/p
−
− 1, we have
sup
t∈0,1
wt
1/pt−1
u
n
t
≤ C
3
u
n
α
1
. 3.9
Thus
wt
1/pt−1
u
n
t
0
≤ NC
3
u
n
α
1
. 3.10
2
◦
If pξ
i
n
n
− 2 > 0, since |u
i
n
n
ξ
i
n
n
|≤C
∗
, we have
w
ξ
i
n
n
u
n
ξ
i
n
n
pξ
i
n
n
−2
u
i
n
n
ξ
i
n
n
w
ξ
i
n
n
1/pξ
i
n
n
−1
u
i
n
n
ξ
i
n
n
w
ξ
i
n
n
u
n
ξ
i
n
n
pξ
i
n
n
−1
pξ
i
n
n
−2/pξ
i
n
n
−1
≤ C
4
sup
t∈0,1
wt
u
n
t
pt−1
1
, where
p
− 2
p
− 1
.
3.11
According to 3.2, 3.3, 3.5,and3.11, we have
1
2N
sup
t∈0,1
wt
u
n
t
pt−1
≤ w
t
n
u
n
t
n
pt
n
−2
u
i
n
n
t
n
≤ C
4
sup
t∈0,1
wt
u
n
t
pt−1
ε
1
C
2
u
n
q
−1
1
.
3.12
14 Journal of Inequalities and Applications
Since <1 is a positive constant, 3.12 means that
sup
t∈0,1
wt
u
n
t
pt−1
≤ C
5
u
n
q
−1
1
. 3.13
Thus
wt
1/pt−1
u
n
t
0
≤ NC
6
u
n
α
1
. 3.14
Summarizing this argument, we have
wt
1/pt−1
u
n
t
0
≤ C
7
u
n
α
1
. 3.15
Since |ah|≤C
8
Fh
0
|e
0
| |e
1
|
p
#
−1
, then we have
a
δN
f
≤ C
8
F
N
f
0
e
0
e
1
p
#
−1
≤ C
9
u
q
−1
1
1.
3.16
Thus
a
δN
f
u
n
F
δN
f
u
n
≤
a
δN
f
u
n
F
δN
f
u
n
≤ C
10
u
n
q
−1
1
.
3.17
Combining 2.38 and 3.17, we have
u
n
0
≤ C
11
u
n
α
1
, where α
q
− 1
p
−
− 1
. 3.18
For any j 1, ,N,since
u
j
n
t
u
j
n
0
t
0
u
j
n
r dr
≤
u
j
n
0
t
0
wr
−1/pr−1
sup
t∈0,1
wt
1/pt−1
u
j
n
t
dr
≤
u
n
α
1
C
11
C
7
E
,
3.19
we have
u
j
n
0
≤ C
12
u
n
α
1
,j 1, ,N; n 1, 2, 3.20
Thus
u
n
0
≤ NC
12
u
n
α
1
,n 1, 2, 3.21
Combining 3.15 and 3.21, then we obtain that {u
n
1
} is bounded.
Qihu Zhang et al. 15
Thus, there exists a large enough R
0
> 0 such that all the solutions of 3.1 belong to
BR
0
{u ∈ C
1
|u
1
<R
0
}, then the Leray-Schauder degree d
LS
I − Ψ
f
·,λ,BR
0
, 0 is
well defined for λ ∈ 0, 1,and
d
LS
I − Ψ
f
·, 1,B
R
0
, 0
d
LS
I − Ψ
f
·, 0,B
R
0
, 0
. 3.22
Let
u
0
m−2
i1
β
i
η
i
0
ϕ
−1
t,
wt
−1
a0
dt e
0
1 −
m−2
i1
β
i
r
0
ϕ
−1
t,
wt
−1
a0
dt, 3.23
where a0 is defined in 2.25,thus u
0
is the unique solution of u Ψ
f
u, 0.
It is easy to see that u is a solution of u Ψ
f
u, 0 if and only if u is a solution of the
following:
I
⎧
⎪
⎪
⎨
⎪
⎪
⎩
−Δ
pt,wt
u 0, t ∈ 0, 1,
u0
m−2
i1
β
i
u
η
i
e
0
,u1
m−2
i1
α
i
u
ξ
i
e
1
.
3.24
Obviously, system I possesses only one solution u
0
. Since u
0
∈ BR
0
, thus the Leray-
Schauder degree
d
LS
I − Ψ
f
·, 1,B
R
0
, 0
d
LS
I − Ψ
f
·, 0,B
R
0
, 0
1
/
0, 3.25
therefore, we obtain that 1.1-1.2 has at least one solution. This completes the proof.
4. f satisfies general growth condition
In the following, we will deal with the existence of solutions for pt-Laplacian ordinary
system, when f satisfies general growth condition.
Denote
Ω
ε
u ∈ C
1
| max
1≤i≤N
u
i
0
wt
1/pt−1
u
i
0
<ε
, θ
ε
2 1/E
. 4.1
Assumption 4.1. Let positive constant ε satisfy u
0
∈ Ω
ε
, |P0| <θ,and|a0| < 1/N2E
1min
t∈I
|θ/E|
pt−1
, where u
0
is defined in 3.23, a· is defined in 2.25.
It is easy to see that Ω
ε
is an open bounded domain in C
1
.
Theorem 4.2. Assume that Assumption 4.1 is satisfied. If positive parameter δ is small enough, then
the problem 1.1-1.2 has at least one solution on
Ω
ε
.
16 Journal of Inequalities and Applications
Proof. Denote Ψ
f
u, λPλδN
f
uK
1
λδN
f
u. According to Lemma 2.4, u is a solution
of
−Δ
pt,wt
u λδf
t, u,
wt
1/pt−1
u
0,t∈ 0, 1, 4.2
with 1.2 if and only if u is a solution of the following abstract equation:
u Ψ
f
u, λ. 4.3
From Lemmas 2.2 and 2.3, then we can see that Ψ
f
·,λ is compact continuous from C
1
to C
1
for any λ ∈ 0, 1. According to Leray-Schauder degree theory, we only need to prove
that
1
◦
u Ψ
f
u, λ has no solution on ∂Ω
ε
for any λ ∈ 0, 1,
2
◦
d
LS
I − Ψ
f
·, 0, Ω
ε
, 0
/
0.
Then we can conclude that the system 1.1-1.2 has a solution on
Ω
ε
.
1
◦
If it is false, then there exists a λ ∈ 0, 1 and u ∈ ∂Ω
ε
is a solution of 4.2 with
1.2.Thusu, λ satisfies
wtϕ
t, u
t
a
λδN
f
λδF
N
f
t. 4.4
Since u ∈ ∂Ω
ε
, then there exists an i such that |u
i
|
0
|wt
1/pt−1
u
i
|
0
ε.
i Suppose that |u
i
|
0
≥ 2θ, then |wt
1/pt−1
u
i
|
0
≤ ε − 2θ θ/E. On the other
hand, for any t, t
∈ I, we have
u
i
t − u
i
t
t
t
u
i
r dr
≤
1
0
wr
−1/pr−1
wr
1/pr−1
u
i
r
dr ≤ θ.
4.5
This implies that |u
i
t|≥θ for each t ∈ I.
Notice that u ∈
Ω
ε
, then |ft, u,wt
1/pt−1
u
|≤β
Nε
t,holding|FN
f
|≤
1
0
β
Nε
t dt. Since P· is continuous, when 0 <δis small enough, from Assumption 4.1,we
have
u0
P
λδN
f
u
<θ. 4.6
It is a contradiction to |u
i
t|≥θ for each t ∈ I.
ii Suppose that |u
i
|
0
< 2θ, then θ/E < |wr
1/pr−1
u
i
|
0
≤ ε. This implies that
|wt
2
1/pt
2
−1
u
i
t
2
|≥θ/E for some t
2
∈ I. Since u ∈ Ω
ε
,itiseasytoseethat
w
t
2
1/pt
2
−1
u
i
t
2
≥
θ
E
Nε
N2E 1
≥
w
t
2
1/pt
2
−1
u
t
2
N2E 1
. 4.7
Qihu Zhang et al. 17
Combining 4.4 and 4.7, we have
|θ/E|
pt
2
−1
N2E 1
≤
1
N2E 1
w
t
2
u
i
t
2
pt
2
−1
≤
1
N2E 1
w
t
2
u
t
2
pt
2
−1
≤ w
t
2
u
t
2
pt
2
−1
u
i
t
2
≤
a
λδN
f
λ
δF
N
f
t
.
4.8
Since u ∈
Ω
ε
and f is Caratheodory, it is easy to see that |ft, u, wt
1/pt−1
u
|≤
β
Nε
t,thus|δFN
f
|≤δ
1
0
β
Nε
t dt. According to Lemma 2.2, a· is continuous, we have
a
λδN
f
−→
a0
as δ −→ 0. 4.9
Thus, when 0 <δis small enough, from Assumption 4.1, we can conclude that
|θ/E|
pt
2
−1
N2E 1
≤
a
λδN
f
λ
δF
N
f
t
<
1
N2E 1
min
t∈I
θ
E
pt−1
. 4.10
It is a contradiction. Summarizing this argument, for each λ ∈ 0, 1, the problem 4.2
with 1.2 has no solution on ∂Ω
ε
when positive parameter δ is small enough.
2
◦
According to Assumption 4.1, u
0
∈ Ω
ε
, where u
0
is defined in 3.23,thusu
0
is
the unique solution of u Ψ
f
u, 0, then the Leray-Schauder degree
d
LS
I − Ψ
f
·, 0, Ω
ε
, 0
d
LS
I − Ψ
f
·, 1, Ω
ε
, 0
1
/
0. 4.11
This completes the proof.
Similar to the proof of Theorem 4.2, we have Theorem 4.3.
Theorem 4.3. Assume ft, x, yσt|x|
q
1
t−2
x μt|y|
q
2
t−2
y,whereq
1
,q
2
,σ,μ ∈ CI,R
satisfy max
t∈I
pt <q
1
t,q
2
t, ∀t ∈ I.Ifδ 1 and |e
0
|, |e
1
| are small enough, then the problem
1.1-1.2 possesses at least one solution.
On the typical case, we have Corollary 4.4.
Corollary 4.4. Assume that ft, x, yσt|x|
q
1
t−2
x μt|y|
q
2
t−2
y,whereq
1
,q
2
,σ,μ∈ CI,R
satisfy min
t∈I
pt ≤ q
1
t, q
2
t ≤ max
t∈I
pt. On the conditions of Theorem 4.2, the problem 1.1-
1.2 possesses at least one solution.
18 Journal of Inequalities and Applications
Acknowledgments
This work is supported by the National Science Foundation of China 10701066 and
10671084, China Postdoctoral Science Foundation 20070421107, the Natural Science Foun-
dation of Henan Education Committee 2008-755-65, and the Natural Science Foundation of
Jiangsu Education Committee 08KJD110007.
References
1 E. Acerbi and G. Mingione, “Regularity results for a class of functionals with non-standard growth,”
Archive for Rational Mechanics and Analysis , vol. 156, no. 2, pp. 121–140, 2001.
2 Y. Chen, S. Levine, and M. Rao, “Variable exponent, linear growth functionals in image restoration,”
SIAM Journal on Applied Mathematics, vol. 66, no. 4, pp. 1383–1406, 2006.
3 L. Diening, “Maximal function on generalized Lebesgue spaces L
p·
,” Mathematical Inequalities &
Applications, vol. 7, no. 2, pp. 245–253, 2004.
4 X L. Fan and Q H. Zhang, “Existence of solutions for px-Laplacian Dirichlet problem,” Nonlinear
Analysis: Theory, Methods & Applications, vol. 52, no. 8, pp. 1843–1852, 2003.
5 X L. Fan, H Q. Wu, and F Z. Wang, “Hartman-type results for pt-Laplacian systems,” Nonlinear
Analysis: Theory, Methods & Applications, vol. 52, no. 2, pp. 585–594, 2003.
6 X L. Fan, Q. Zhang, and D. Zhao, “Eigenvalues of px-Laplacian Dirichlet problem,” Journal of
Mathematical Analysis and Applications, vol. 302, no. 2, pp. 306–317, 2005.
7 A. El Hamidi, “Existence results to elliptic systems with nonstandard growth conditions,” Journal of
Mathematical Analysis and Applications, vol. 300, no. 1, pp. 30–42, 2004.
8 H. Hudzik, “On generalized Orlicz-Sobolev space,” Functiones et Approximatio Commentarii Mathe-
matici, vol. 4, pp. 37–51, 1976.
9 O. Kov
´
a
ˇ
cik and J. R
´
akosn
´
ık, “On spaces L
px
Ω and W
k,px
Ω,” Czechoslovak Mathematical Journal,
vol. 41, no. 4, pp. 592–618, 1991.
10 Q. Zhang, “Existence of positive solutions for elliptic systems with nonstandard px-growth
conditions via sub-supersolution method,” Nonlinear Analysis: Theory, Methods & Applications, vol.
67, no. 4, pp. 1055–1067, 2007.
11 Q. Zhang, “Existence of positive solutions for a class of px-Laplacian systems,” Journal of
Mathematical Analysis and Applications, vol. 333, no. 2, pp. 591–603, 2007.
12 Q. Zhang, “Existence of radial solutions for px-Laplacian equations in
R
N
,” Journal of Mathematical
Analysis and Applications, vol. 315, no. 2, pp. 506–516, 2006.
13 Q. Zhang, “Boundary blow-up solutions to px-Laplacian equations with exponential nonlineari-
ties,” Journal of Inequalities and Applications, vol. 2008, Article ID 279306, 8 pages, 2008.
14 Q. Zhang, X. Liu, and Z. Qiu, “The method of subsuper solutions for weighted pr-Laplacian
equation boundary value problems,” Journal of Inequalities and Applications, vol. 2008, Article ID
621621, 18 pages, 2008.
15 M. R
˚
u
ˇ
zi
ˇ
cka, Electrorheological Fluids: Modeling and Mathematical Theory, vol. 1748 of Lecture Notes in
Mathematics, Springer, Berlin, Germany, 2000.
16 V. V. Zhikov, “Averaging of functionals of the calculus of variations and elasticity theory,” Mathematics
of the USSR—Izvestija, vol. 29, no. 1, pp. 33–66, 1987.
17 Y. Guo, W. Shan, and W. Ge, “Positive solutions for second-order m-point boundary value problems,”
Journal of Computational and Applied Mathematics, vol. 151, no. 2, pp. 415–424, 2003.
18 Y. Wang and W. Ge, “Existence of multiple positive solutions for multipoint boundary value problems
with a one-dimensional p-Laplacian,” Nonlinear Analysis: Theory, Methods & Applications, vol. 67, no.
2, pp. 476–485, 2007.
19 Y. Wang and W. Ge, “Multiple positive solutions for multipoint boundary value problems with one-
dimensional p-Laplacian,” Journal of Mathematical Analysis and Applications, vol. 327, no. 2, pp. 1381–
1395, 2007.
20 Y. Wang and W. Ge, “Positive solutions for multipoint boundary value problems with a one-
dimensional p-Laplacian,” Nonlinear Analysis: Theory, Methods & Applications, vol. 66, no. 6, pp. 1246–
1256, 2007.
