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Volume 2008, Article ID 791762, 18 pagesdoi:10.1155/2008/791762 Research Article Existence of Solutions for a Class of Boundary Value Problems Qihu Zhang, 1, 2, 3 Zheimei Qiu, 2 and Xiao

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Volume 2008, Article ID 791762, 18 pages

doi:10.1155/2008/791762

Research Article

Existence of Solutions for a Class of

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

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

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:

−Δp t,wt u δft, u,

w t1/pt−1u

 0, t ∈ 0, 1, 1.1 with the following multipoint boundary value condition:

u0 m−2

i1

β i u

η i

 e0 , u1 m−2

i1

α i u

ξ i

 e1 , 1.2

where p ∈ C0, 1, R and pt > 1, −Δ p t,wt u −wt|u|p t−2 uis called the weighted

p t-Laplacian; w ∈ C0, 1, R satisfies 0 < wt, for all t ∈ 0, 1, and wt −1/pt−1 ∈

L10, 1; 0 < η1 < · · · < η m−2 < 1, 0 < ξ1 < · · · < 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; e0, e1∈ RN ; δ is a positive parameter.

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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 ≡ 1 and pt ≡ p a constant, −Δ p t,wt is the well-known p-Laplacian.

If pt is a general function, −Δ p t,wt represents a nonhomogeneity and possesses more nonlinearity, thus−Δp t,wtis more complicated than−Δp We have the following examples

1 If Ω ⊂ RNis a bounded domain, the Rayleigh quotient

λ p x inf

u ∈W01,pxΩ\{0}

Ω1/px|∇u| p x dx

Ω1/px|u| p x dx 1.3

is zero in general, and only under some special conditions λ p x > 0see 6, but

the fact that λ p > 0 is very important in the study of p-Laplacian problems.

2 If wt ≡ 1 and pt ≡ 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−Δp t,1 It is another diﬀerence on −Δpand−Δp t,1

3 On the existence of solutions of the following typical −Δp x,1problem:

−up x−2 u

 |u| q x−2 u C, x ∈ Ω ⊂ R N ,

because of the nonhomogeneity of −Δp x,1, if maxx∈Ωq x < min x∈Ωp x, then

the corresponding functional is coercive; if maxx∈Ωp x < min x∈Ωq x, then

the corresponding functional satisfies Palais-Smale condition see 4, 7, 12 If minx∈Ωp x ≤ qx ≤ max x∈Ωp x, 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 conditionssee 17–20 On the existence of solutions for

px-Laplacian systems boundary value problems, we refer to 5,7,10, 11 But results on the

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 mint ∈0,1 p t ≤ qt ≤ max t ∈0,1 p t has been discussed.

Let N ≥ 1 and I 0, 1, the function f f1, , f N : I × R N× RN → RNis 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× RN the function f ·, x, y is measurable on I;

iii for each R > 0 there is a β R ∈ L1I, R, such that for almost every t ∈ I and every

x, y ∈ R N× RNwith|x| ≤ R, |y| ≤ R, one has

f t, x, y ≤ β R t. 1.5

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Throughout the paper, we denote

w0up0−2u0 lim

r→ 0 w rup r−2 ur,

w1up0−2u1 lim

r→ 1 −w rup r−2 ur. 1.6

The inner product inRN will be denoted by

and the Euclidean norm on RN For N ≥ 1, we set C CI, R N , C1  {u ∈ C | u ∈

C 0, 1, R N, limt→ 0 w t|u|p t−2 ut, and lim t→ 1 −w t|u|p t−2 ut exist} For any ut 

u1t, , u N t, we denote |u i|0 supt ∈0,1 |u i t|, u0 N

i1|u i|2

01/2 , and u1

u0  wt 1/pt−1 u0 Spaces C and C1 will be equipped with the norm·0 and ·1, respectively ThenC, ·0 and C1,·1 are Banach spaces

We say a function u : I → RN is a solution of1.1 if u ∈ C1 with wt|u|p t−2 u

absolutely continuous on0, 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 z t, z max

t ∈I z t, for any z ∈ CI, R. 1.7

We say f satisfies sub-p−− 1 growth condition, if f satisfies

lim

|u||v| → ∞

f t, u, v

|u| |v|q t−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 of1.1-1.2 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 of1.1-1.2, when

f satisfies sub-p−− 1 growth condition Finally, inSection 4, we will discuss the existence of solutions of1.1-1.2, when f satisfies general growth condition.

2 Preliminary

For anyt, x ∈ I × R N , denote ϕt, x |x| p t−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, x1

− ϕt, x2

, x1− x2 > 0, for any x1, x2∈ RN , x1/ x2; 2.1

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ii there exists a function ρ : 0, ∞ → 0, ∞, ρs → ∞ as s → ∞, such that

ϕ t, x, x ≥ ρ|x||x|, ∀x ∈ R N 2.2

It is well known that ϕ t, · is a homeomorphism from R N toRN for any fixed t ∈ 0, 1 For

any t ∈ I, denote by ϕ−1t, · the inverse operator of ϕt, ·, then

ϕ−1t, x |x| 2−pt/pt−1 x, for x∈ RN \ {0}, ϕ−1t, 0 0. 2.3

It is clear that ϕ−1t, · is continuous and sends bounded sets to bounded sets Let us

now consider the following problem with boundary value condition1.2:

w tϕt, ut gt, 2.4

where g ∈ L1 If u is a solution of2.4 with 1.2, by integrating 2.4 from 0 to t, we find that

w tϕt, ut w0ϕ0, u0 t

0

g s ds. 2.5

Denote a w0ϕ0, u0 It is easy to see that a is dependent on gt Define operator

F : L1 → C as Fgt t

0g s ds By solving for uin2.5 and integrating, we find

u t u0 Fϕ−1

t,

w t−1a Fg t. 2.6

From u0 m−2

i1 β i u η i e0, we have

u0

m−2

i1 β i

η i

0ϕ−1

t,

w t−1a Fgt dt e0

1−m−2

i1 β i . 2.7

From u1 m−2

i1 α i u ξ i e1, we obtain

u0

m−2

i1 α i

ξ i

0ϕ−1

t,

w t−1a Fgt dt−1

0ϕ−1

t,

1−m−2

2.8 From2.7 and 2.8, we have

m−2

i1 β i

η i

0ϕ−1

t,

1−m−2

i1 β i

m−2

i1 α i

ξ i

0ϕ−1

t,

w t−1a Fgt dt−1

0ϕ−1

t,

1−m−2

2.9

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For fixed h ∈ C, we denote

Λh a 

m−2

i1 β iη i

0ϕ−1

t,

w t−1a ht dt e0

1−m−2

i1 β i

−

m−2

i1 α iξ i

0ϕ−1

t,

w t−1a ht dt−1

0ϕ−1

t,

1−m−2

2.10

Throughout the paper, we denote E1

0wt −1/pt−1 dt.

Lemma 2.2 The function Λ h · has the following properties:

i for any fixed h ∈ C, the equation

Λh a 0 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

·h0 2N pe0 e1p# −1 ,

2.12

where the notation M p# −1means

M p#−1

⎧

⎨

⎩

M p−1, M > 1

M p−−1, M ≤ 1. 2.13

Proof. i It is easy to see that

Λh a 

m−2

i1 β i

η i

0ϕ−1

t,

1−m−2

i1 β i

m−2

i1 α i

1

ξ i ϕ−1

t,

w t−1a ht dt − e1

1−m−2

i1 α i

1 0

ϕ−1

t,

w t−1a ht dt.

2.14

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FromLemma 2.1, it is immediate that

Λh

a1

− Λh

a2

, a1− a2 > 0, for a1/ a2 , 2.15

and hence, if2.11 has a solution, then it is unique

Let

t0 3N

E 1

1−m−2

i1 β i

E E 1

1−m−2

i1 α i

E 1

p−1

·h0 2N pe0 e1p# −1 .

2.16

If|a| ≥ t0 , since wt −1/pt−1 ∈ L10, 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

·h0 2N pe0 e1p# −1 .

2.17

Thusa i h i t keeps sign on I and

a i h i t ≥ a i − h0

≥ 2

E 1

1−m−2

i1 β i

E E 1

1−m−2

i1 α i

E 1

p −1

·h0 2N pe0 e1p# −1 , ∀t ∈ I,

2.18 then

a i h i t1/pt−1≥ 21/p −1 E 1

1−m−2

i1 β i

E E 1

1−m−2

i1 α i

E 1e0 e1, ∀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

Λh a / 0. 2.20 Let us consider the equation

λΛh a 1 − λa 0, λ ∈ 0, 1. 2.21

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It is easy to see that all the solutions of2.21 belong to bt0 {x ∈ R N | |x| < t0} So,

we have

d B

Λh a, bt0

, 0 d B

I, b

t0

, 0 / 0, 2.22

it means the existence of solutions ofΛh a 0.

In this way, we define a functionah : C0, 1 → R N, which satisfies

Λh

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 EE 1 m−2

i1 α i E 1

p−1

·h0e0 e1p# −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 } Let au n j → a0 as j → ∞ Since Λu nj au n j 0, letting j → ∞,

we haveΛu a0 0 From i, we get a0 au, it means that a is continuous This completes

the proof

Now, we define a : L1 → RNas

a u aF u. 2.25

It is clear that a· is continuous and sends bounded sets of L1to bounded sets ofRN, and hence it is a complete continuous mapping

If u is a solution of2.4 with 1.2, then

u t u0 Fϕ−1

t,

w t−1a g Fgt t, ∀t ∈ 0, 1. 2.26 The boundary condition1.2 implies that

u0

m−2

i1 β iη i

0ϕ−1

t,

w t−1a g Fgt dt e0

1−m−2

i1 β i . 2.27

We denote that

K1ht :K1◦ ht Fϕ−1

t,

w t−1a h Fh t, ∀t ∈ 0, 1. 2.28

Lemma 2.3 The operator K1is continuous and sends equi-integrable sets in L1to relatively compact sets in C1.

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Proof It is easy to check that K1ht ∈ C 1 Sincewt −1/pt−1 ∈ L1and

K1ht ϕ−1

t,

w t−1a h Fh , ∀t ∈ 0, 1, 2.29

it is easy to check that K1is a continuous operator from L1to C1

Let now U be an equi-integrable set in L1, then there exists ρ∗∈ L1, such that

u t ≤ ρ∗t a.e in I, for any u ∈ U. 2.30

We want to show that K1U ⊂ C1is a compact set

Let {u n } be a sequence in K1U, then there exists a sequence {h n } ∈ U such that

u n K1h n For any t1 , t2∈ I, we have

F

h n

t1

− Fh n

t2 t1

0

h n t dt − t2

0

h n t dt

t2

t1

h n t dt

≤ t2

t1

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, K1h nt ah n Fh n is convergent in C.

Since

K1

h n

t Fϕ−1

t,

w t−1a

h n

 Fh n t, ∀t ∈ 0, 1, 2.32

according to the continuity of ϕ−1and the integrability of wt −1/pt−1 in L1, we can see that

K1hn is convergent in C Thus {u n } is convergent in C1 This completes the proof

Let us define P : C1 → C1as

P h 

m−2

i1 β i

K1◦ hη i

 e0

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 × C1 → L1the Nemytski operator associated to f defined by

N f ut ft, u t,w t1/pt−1 ut, 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 PδN f u K1δN f u. 2.35

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Proof If u is a solution of1.1-1.2, by integrating 1.1 from 0 to t, we find that

w tϕt, ut aδN f u FδN f ut. 2.36 From2.36, we have

u t u0 Fϕ−1

r,

w r−1a

δN f u FδN f u t,

u0 m−2

i1

β i

u 0 Fϕ−1

r,

w r−1a

δN f u FδN f u η i e0, 2.37

then we have

u0

m−2

i1 β i F

ϕ−1

r,

w r−1a

δN f u FδN f u η i

 e0

1−m−2

i1 β i

m−2

i1 β i K1

δN f uη i

 e0

1−m−2

i1 β i PδN f u.

2.38

So we have

u PδN f u K1δN f u. 2.39

Conversely, if u is a solution of2.35, it is easy to see that

u 0 PδN f u

m−2

i1 β i K1

δN f uη i

 e0

1−m−2

i1 β i

,

u0 m−2

i1

β i

u 0 K1δN f uη i e0m−2

i1

β i u

η i

 e0 ,

u 1 PδN f u K1δN f u1.

2.40

By the condition of the mapping a,

u0

m−2

i1 β i K1

δN f uη i

 e0

1−m−2

i1 β i

m−2

i1 α i K1

δN f uξ i

− K1δN f u1 e1

1−m−2

2.41 then we have

u1

m−2

i1 α i K1

δN f uξ i

− K1δN f u1 e1

1−m−2

i1 α i K1δN f u1, 2.42

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u1 m−2

i1

α i

u 1 − K1δN f u1 K1δN f uξ i e1

m−2

i1

α i

P

δN f u K1δN f uξ i e1m−2

i1

α i u

ξ i

 e1,

2.43

from2.40 and 2.43, we obtain 1.2

From2.35, we have

ut ϕ−1

t,

w t−1a FδN f u ,

w tϕt, u

 δN f ut. 2.44 Hence u is a solution of1.1-1.2 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∗:

e0

1−m−2

i1 β i e1

1−m−2

i1 α i

. 2.45

Proof For any j 1, , N, if there exists ς j ∈ 0, 1 such that u jς j 0, then 2.45 is valid

If it is false, then u jis strictly monotone

i If u jis strictly decreasing in0, 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 j0 m−2

i1

β i u j

η i

 e j

0<

m−2

i1

β i u j 0 e j

0,

u j1 m−2

i1

α i u j

ξ i

 e j

1>

m−2

i1

α i u j 1 e j

1,

2.47

it means that

e j0

1−m−2

i1 β i > u

j 0 > u j 1 > e

j

1

1−m−2

i1 α i, 2.48

w t −1a h t dt.

2.14 Trang 6

FromLemma...

λΛh a 1 − λ a 0, λ ∈ 0, 1. 2.21 Trang 7

It is easy to see that all the solutions. ..

1−m−2

2.9 Trang 5

For fixed h ∈ C, we denote

Λh

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