The Existence of Positive Solutions for Third-Order p-Laplacian m-Point Boundary Value Problems with Sign Changing Nonlinearity on Time Scales

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doi:10.1155/2009/169321

Research Article

The Existence of Positive Solutions for Third-Order p-Laplacian m-Point Boundary Value Problems with Sign Changing Nonlinearity on Time Scales

Fuyi Xu and Zhaowei Meng

School of Science, Shandong University of Technology, Zibo, Shandong 255049, China

Correspondence should be addressed to Fuyi Xu,xfy [email protected] Received 25 February 2009; Revised 10 April 2009; Accepted 2 June 2009 Recommended by Alberto Cabada

We study the following third-orderp-Laplacianm-point boundary value problems on time scales φpu^Δ∇^∇ atft, ut 0, t ∈ 0, T _T^κ, u0 _m−2

i1 biuξi, u^ΔT 0, φpu^Δ∇0 _m−2

i1 ciφpu^Δ∇ξi, whereφpsis p-Laplacian operator, that is,φps |s|^p−2s, p > 1,φ⁻¹_p φq,1/p1/q 1, 0 < ξ1 < · · · < ξm−2 < ρT. We obtain the existence of positive solutions by using fixed-point theorem in cones. In particular, the nonlinear termft, uis allowed to change sign. The conclusions in this paper essentially extend and improve the known results.

Copyrightq2009 F. Xu and Z. Meng. 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

The theory of time scales was initiated by Hilger 1 as a mean of unifying and extending theories from diﬀerential and diﬀerence equations. The study of time scales has lead to several important applications in the study of insect population models, neural networks, heat transfer, and epidemic models, see, for example 2–6 . Recently, the boundary value problems withp-Laplacian operator have also been discussed extensively in literature; for example, see 7–18 . However, to the best of our knowledge, there are not many results concerning the higher-orderp-Laplacian mutilpoint boundary value problem on time scales.

A time scaleT is a nonempty closed subset ofR. We make the blanket assumption that 0, T are points inT. By an interval0, T_T, we always mean the intersection of the real interval 0, Twith the given time scale; that is0, T∩T.

In 19 , Anderson considered the following third-order nonlinear boundary value problemBVP:

xt ft, xt, t1≤t≤t3,

xt1 xt2 0, γxt3 δxt3 0. 1.1

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author studied the existence of solutions for the nonlinear boundary value problem by using Krasnoselskii’s fixed point theorem and Leggett and Williams fixed point theorem, respectively.

In9,10 , He considered the existence of positive solutions of thep-Laplacian dynamic equations on time scales

φpu^Δ_∇

atfut 0, t∈0, T _T, 1.2

satisfying the boundary conditions

u0−B0

u^Δ

η

0, u^ΔT 0, 1.3

or

u^Δ0 0, uT−B1

u^Δ

η

0, 1.4

whereη∈0, ρT. He obtained the existence of at least double and triple positive solutions of the problems by using a new double fixed point theorem and triple fixed point theorem, respectively.

In18 , Zhou and Ma firstly studied the existence and iteration of positive solutions for the following third-order generalized right-focal boundary value problem withp-Laplacian operator

φp

u

t qtft, ut, 0≤t≤1, u0 ^m

i1

αiuξi, un 0, u1 ⁿ

i1

βiuθi. 1.5

They established a corresponding iterative scheme for the problem by using the monotone iterative technique.

All the above works were done under the assumption that the nonlinear term is nonnegative. The key conditions used in the above papers ensure that positive solution is concave down. If the nonlinearity is negative somewhere, then the solution needs no longer to be concave down. As a result, it is diﬃcult to find positive solutions of thep- Laplacian equation when the nonlinearity changes sign. In particular, little work has been done on the existence of positive solutions for higher orderp-Laplacianm-point boundary value problems with nonlinearity f being nonnegative on time scales. Therefore, it is a natural problem to consider the existence of positive solution for higher orderp-Laplacian equations with sign changing nonlinearity on time scales. This paper attempts to fill this gap in literature.

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In this paper, by using diﬀerent method, we are concerned with the existence of positive solutions for the following third-order p-Laplacian m-point boundary value problems on time scales:

φpu^Δ∇_∇

atft, ut 0, t∈0, T _Tκ, u0 ^m−2

i1

biuξi, u^ΔT 0, φp

u^Δ∇0 ^m−2

i1

ciφp

u^Δ∇ξi ,

1.6

whereφpsisp-Laplacian operator, that is,φps |s|^p−2s,p > 1,φ⁻¹_p φq, 1/p1/q 1, andbi,ci,a,fsatisfy

H1 bi, ci∈0,∞, 0< ξ1<· · ·< ξ_m−2< ρT, 0<_m−2

i1 bi<1, 0<_m−2

i1 ci<1;

H2f : 0, T _Tκ ×0,∞ → −∞,∞is continuous,a ∈ Cld0, T _Tκ,0,∞, and there existst0∈0, T_T^κ such thatat0>0.

2. Preliminaries and Lemmas

For convenience, we list the following definitions which can be found in1–5 .

Definition 2.1. A time scaleT is a nonempty closed subset of real numbersR. Fort <supT and r > inf T, define the forward jump operatorσand backward jump operatorρ, respectively, by

σt inf{τ∈T|τ > t} ∈T,

ρr sup{τ∈T|τ < r} ∈T 2.1

for allt, r ∈T. Ifσt> t,tis said to be right scattered, ifρr< r,ris said to be left scattered;

ifσt t,tis said to be right dense, and ifρr r,ris said to be left dense. IfT has a right scattered minimumm, defineTk T− {m}; otherwise setTk T. If T has a left scattered maximumM, defineT^kT− {M}; otherwise setT^kT.

Definition 2.2. Forf :T → Randt∈T^k, the delta derivative offat the pointtis defined to be the numberf^Δt provided that it exists, with the property that for each >0, there is a neighborhoodUoftsuch that

fσt−fs−f^Δtσt−s≤|σt−s| 2.2 for alls∈U.

Forf:T → Randt∈Tk, the nabla derivative offatt, denoted byf^∇t provided it existswith the property that for each >0, there is a neighborhoodUoftsuch that

f ρt

−fs−f^∇t

ρt−s≤ρt−s 2.3

for alls∈U.

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Definition 2.3. A functionfis left-dense continuousi.e.,ld-continuous, iffis continuous at each left-dense point inT and its right-sided limit exists at each right-dense point in T.

Definition 2.4. Ifφ^Δt ft, then we define the delta integral by

b a

ftΔtφb−φa. 2.4

IfF^∇t ft, then we define the nabla integral by

b a

ft∇tFb−Fa. 2.5

Lemma 2.5. If conditionH1holds, then forh∈Cld0, T _Tκ, the boundary value problem (BVP)

u^Δ∇ht 0, t∈0, T _Tκ, u0 ^m−2

i1

biuξi, u^ΔT 0

2.6

has the unique solution

ut ^t

0

T−shs∇s _m−2

i1 bi

_ξ_i

0T−shs∇s 1−_m−2

i1 bi

. 2.7

Proof. By caculating, we can easily get2.7. So we omit it.

Lemma 2.6. If conditionH1holds, then forh∈Cld0, T _T^κ, the boundary value problem (BVP)

φp

u^Δ∇_∇

ht 0, t∈0, T _T^κ, u0 ^m−2

i1

biuξi, u^ΔT 0, φp

u^Δ∇0 ^m−2

i1

ciφp

u^Δ∇ξi 2.8

has the unique solution

ut ^t

0

T−sφq s 0

hr∇rC

∇s _m−2

i1 bi

_ξ_i

0T−sφq

_s

0hr∇rC

∇s 1−_m−2

i1 bi

, 2.9

whereC_m−2

i1 ci

_ξ_i

0hr∇r/1−_m−2

i1 ci.

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Proof. Integrating both sides of equation in2.8on0, t , we have

φp

u^Δ∇t φp

u^Δ∇0

− ^t

0

hr∇r. 2.10

So,

φp

u^Δ∇ξi φp

u^Δ∇0

− ^ξⁱ

0

hr∇r. 2.11

By boundary value conditionφpu^Δ∇0 _m−2

i1 ciφpu^Δ∇ξi, we have

φp

u^Δ∇0 −

_m−2

i1 ci

_ξ_i

0hr∇r

1−_m−2

i1 ci

. 2.12

By2.10and2.12we know

u^Δ∇t −φq

⎛

⎝ _m−2

i1 ci

_ξ_i

0hr∇r 1−_m−2

i1 ci

^t

0

hr∇r

⎞

⎠. 2.13

This together withLemma 2.5implies that

ut ^t

0

T−sφq s 0

hr∇rC

∇s _m−2

i1 bi

_ξ_i

0T−sφq

_s

0hr∇rC

∇s 1−_m−2

i1 bi

, 2.14

whereC_m−2

i1 ci

_ξ_i

0hr∇r/1−_m−2

i1 ci. The proof is complete.

Lemma 2.7. Let conditionH1holds Ifh∈Cld0, T _T^κ andht≥0, then the unique solutionut of 2.8satisfies

ut≥0, t∈0, T _Tκ. 2.15

Proof. Byu^Δ∇t −φq_m−2

i1 ci

_ξ_i

0hr∇r/1−_m−2

i1 ci

_t

0hr∇r ≤ 0, we can know that the graph ofutis concave down on0, T_Tκ, and u^Δt is nonincreasing on0, T _Tκ. This together with the assumption that the boundary conditionu^ΔT 0 implies thatu^Δt ≥0 fort∈0, T _Tκ. This implies that

t∈0,T min_Tκut u0. 2.16

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So we only proveu0≥0.By conditionH1we have

u0 _m−2

i1 bi

_ξ_i

0T−sφq

_s

0hr∇rC

∇s 1−_m−2

i1 bi

≥0. 2.17

The proof is completed.

Lemma 2.8. Let conditionH1hold. If h ∈ Cld0, T _T^κ and ht ≥ 0, then the unique positive solutionutof (BVP)2.8satisfies

t∈0,T infTκut≥σ1u, 2.18

whereσ1_m−2

i1 biξi/T−_m−2

i1 biT−ξi,usup_t∈0,T

Tκ|ut|.

Proof. Byu^Δ∇t −φq_m−2

i1 ci

_ξ_i

0hr∇r/1−_m−2

i1 ci

_t

0hr∇r ≤ 0, we can know that the graph ofutis concave down on0, T_Tκ, and u^Δt is nonincreasing on0, T _Tκ. This together with the assumption that the boundary conditionu^ΔT 0 implies thatu^Δt ≥0 fort∈0, T _T^κ. This implies that

uuT, min

t∈0,T _Tκut u0. 2.19

For alli∈ {1,2, . . . , m−2}, we have from the concavity ofuthat uξi−u0

ξi ≥ uT−u0

T , 2.20

that is,

uξi−u0 ξi

Tu0≥ ξi

TuT. 2.21

This together with the boundary conditionu0 _m−2

i1 biuξiimplies that

t∈0,TminTκut≥

_m−2

i1 biξi

T−_m−2

i1 biT−ξiuT. 2.22

This completes the proof.

LetECld0, T _T^κbe endowed with the orderingx≤yifxt≤ytfor allt∈0, T _T^κ, and u max_t∈0,T_Tκ|ut|is defined as usual by maximum norm. Clearly, it follows that E,uis a Banach space.

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For the convenience, let

ψs φq

⎛

⎝ ^s

0

ar∇r

_m−2

i1 ci

_ξ_i

0ar∇r 1−_m−2

i1 ci

⎞

⎠. 2.23

We define two cones by

P {u:u∈E, ut≥0, t∈0, T_T^κ}, K

u:u∈E, utis concave, nonincreasing and nonnegative on0, T_T^κ

t∈0,TminTκ

ut≥σu

,

2.24

whereσσ1σ2,σ1is defined inLemma 2.8and

σ2

_m−2

i1 bi

_ξ_i

0ψs∇s

1−_m−2

i1 bi

_T

0T−sψT∇s_m−2

i1 bi

_ξ_i

0ψT∇s/

1−_m−2

i1 bi

. 2.25

Define the operatorsF :P → EandS:K → Eby setting

Fut ^t

0

T−sφq s 0

arfr, ur∇rA

∇s

_m−2

i1 bi

_ξ_i

0T−sφq

_s

0arfr, ur∇rA

∇s 1−_m−2

i1 bi

,

2.26

whereA_m−2

i1 ci

_ξ_i

0arfr, ur∇r/1−_m−2

i1 ci,

Sut ^t

0

T−sϕs∇s _m−2

i1 bi

_ξ_i

0ϕs∇s

1−_m−2

i1 bi

, 2.27

whereϕs φq_s

0arfr, ur∇rA, A_m−2

i1 ci

_ξ_i

0arfr, ur∇r/1−_m−2

i1 ci, and ft, ut max{ft, ut,0}. Obviously,uis a solution of the BVP1.6if and only ifuis a fixed point of operatorF.

Lemma 2.9. S:K → Kis completely continuous.

Proof. It is easy to see thatSK⊂Kbyf ≥0 andLemma 2.8. By Arzela-Ascoli theorem and Lebesgue dominated convergence theorem, we can easily prove that operatorSis completely continuous.

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Lemma 2.10see20,21 . LetKbe a cone in a Banach spaceX. LetDbe an open bounded subset ofXwith DK D∩K /∅ andDK/K. Assume thatA : DK → Kis a compact map such that x /Axforx∈∂DK. Then the following results hold.

1IfAx ≤ x,x∈∂DK, theniKA, DK 1.

2If there existsx0 ∈K\ {0}such thatx /Axλx0 for allx ∈∂DK and allλ >0, then iKA, DK 0.

3LetUbe open inXsuch thatU⊂DK. IfiKA, DK 1 andiKA, UK 0, thenAhas a fixed point inDK\UK. The same result holds ifiKA, DK 0 andiKA, UK 1, where iKA, DKdenotes fixed point index.

We define Kρ

ut∈K:u< ρ

, Ωρ

ut∈K: min

t∈0,T Tκut< σρ

. 2.28

Lemma 2.11see20 . Ωρdefined above has the following properties:

aKσρ⊂Ωρ⊂Kρ; b Ωρis open relative to K;

cu∈∂Ωρif and only if min_t∈0,T _Tκut σρ;

difu∈∂Ωρ, thenσρ≤ut≤ρfort∈0, T _T^κ.

For the convenience, we introduce the following notations:

1

m ^T

0

T−sψT∇s _m−2

i1 bi

_ξ_i

0ψT∇s

1−_m−2

i1 bi

, 1

M

_m−2

i1 bi

_ξ_i

0ψs∇s

1−_m−2

i1 bi

. 2.29

Remark 2.12. ByH1we can know that 0< m, M <∞,MσMσ1σ2mσ1< m.

Lemma 2.13. Iffsatisfies the following condition : ft, u≤φp

mρ

, t, u∈0, T _Tκ× 0, ρ

, u /Su, u∈∂Kρ, 2.30

then

iK

S, Kρ

1. 2.31

Proof. Foru∈∂Kρ, then from2.30we have

s 0

arfr, ur∇rA ^s

0

arfr, ur∇r _m−2

i1 ci

_ξ_i

0arfr, ur∇r 1−_m−2

i1 ci

≤φp

mρ⎛

⎝ ^T

0

ar∇r

_m−2

i1 ci

_ξ_i

0ar∇r

1−_m−2

i1 ci

⎞

⎠.

2.32

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So that

ϕs φq s 0

arfr, ur∇rA

≤mρψT. 2.33

Therefore,

Sut≤ ^T

0

T−sϕs∇s _m−2

i1 bi

_ξ_i

0ϕs∇s

1−_m−2

i1 bi

≤mρ

⎛

⎝ ^T

0

T−sψT∇s

_m−2

i1 bi

_ξ_i

0ψT∇s

1−_m−2

i1 bi

⎞

⎠ρ.

2.34

This implies thatSu ≤ uforu∈∂Kρ. Hence byLemma 2.101it follows thatiKS, Kρ

1.

Lemma 2.14. Iffsatisfies the following condition:

ft, u≥φp

Mσρ

, t, u∈0, T _Tκ× σρ, ρ

, u /Su, u∈∂Ωρ, 2.35

then

iK

S,Ωρ

0. 2.36

Proof. Letet≡1 fort∈0, T _T^κ. Thene∈∂K1. We claim that

u /Suλe, u∈∂Ωρ, λ >0. 2.37

In fact, if not, there existu0∈∂Ωρandλ0>0 such thatu0Su0λ0e. Byft, u0≥φpMσρ, we have

s 0

arfr, u0r∇rA ^s

0

arfr, u0r∇r _m−2

i1 ci

_ξ_i

0arfr, u0r∇r 1−_m−2

i1 ci

≥φp

Mσρ⎛

⎝ ^s

0

ar∇r

_m−2

i1 ci

_ξ_i

0ar∇r

1−_m−2

i1 ci

⎞

⎠.

2.38

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So that

ϕs φq s 0

arfr, u0r∇rA

≥Mσρφq

⎛

⎝ ^s

0

ar∇r _m−2

i1 ci

_ξ_i

0ar∇r

1−_m−2

i1 ci

⎞

⎠

Mσρψs.

2.39

Fort∈0, T _T^κ, then

u0t Su0t λ0et

≥Su00 λ0

_m−2

i1 bi

_ξ_i

0ϕs∇s

1−_m−2

i1 bi

λ0

≥ Mσρ 1−_m−2

i1 bi m−2

i1

bi ξi

0

ψs∇sλ0

σρλ0.

2.40

This together withLemma 2.11cimplies that

σρ≥σρλ0, 2.41

a contradiction. Hence byLemma 2.102it follows thatiKS,Ωρ 0.

3. Main Results

We now give our results on the existence of positive solutions of BVP1.6.

Theorem 3.1. Suppose that conditionsH1andH2hold, and assume that one of the following conditions holds.

H3There existρ1, ρ2∈0,∞withρ1< σρ2such that ift, u≤φpmρ1,t, u∈0, T _Tκ×0, ρ1 ;

iift, u ≥ 0,t, u ∈ 0, T _T^κ ×σρ1, ρ2 , moreover ft, u ≥ φpMσρ2,t, u ∈ 0, T _T^κ ×σρ2, ρ2 .

H4There existρ1, ρ2∈0,∞withρ1< ρ2such that ift, u≤φpmρ2,t, u∈0, T _T^κ×0, ρ2 ; iift, u≥φpMσρ1,t, u∈0, T _T^κ×σ²ρ1, ρ2 . Then, the BVP1.6has at least one positive solution.

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Proof. Assume thatH3holds, we show thatShas a fixed pointu1inΩρ2\Kρ1. Byft, u≤ φpmρ1andLemma 2.13, we have that

iK

S, Kρ1

1. 3.1

Byft, u≥φpMσρ2andLemma 2.14, we have that iK

S,Ωρ2

0. 3.2

ByLemma 2.11aandρ1 < σρ2, we haveKρ1 ⊂Kσρ2 ⊂Ωρ2. It follows fromLemma 2.103 thatShas a fixed pointu1inΩρ2\Kρ1. Clearly,

u1> ρ1, min

t∈0,T _Tκu1t≥σu1> σρ1, 3.3

which implies thatσρ1≤u1t≤ρ2,t∈0, T _T^κ. By conditionH3ii, we haveft, u1t≥0, t∈0, T _Tκ, that is,ft, u1t ft, u1t. Hence,

Fu1Su1. 3.4

This means thatu1is a fixed point of operatorF.

When conditionH4holds, byft, u≤φpmρ2andLemma 2.13, we have that iK

S, Kρ2

1. 3.5

Byft, u≥φpMσρ1andLemma 2.14, we have that iK

S,Ωρ1

0. 3.6

ByLemma 2.11aandρ1 < ρ2, we haveKσρ1 ⊂ Ωρ1 ⊂ Kρ2. It follows fromLemma 2.103 thatShas a fixed pointu2inKρ2\Ωρ1. Obviously,

u2> σρ1, min

t∈0,T Tκ

u2t≥σu2> σ²ρ1, 3.7

which implies thatσ²ρ1≤u2t≤ρ2,t∈0, T _Tκ. By conditionH4ii, we haveft, u2t≥ 0,t∈0, T _T^κ, that is,ft, u2t ft, u2t. Hence,

Fu2Su2. 3.8

This means thatu2 is a fixed point of operator F. Therefore, the BVP1.6has at least one positive solution.

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Theorem 3.2. Assume that conditionsH1andH2hold, and suppose that one of the following conditions holds.

H5There existρ1, ρ2, andρ3∈0,∞withρ1< σρ2, andρ2< ρ3such that ift, u≤φpmρ1,t, u∈0, T _Tκ×0, ρ1 ;

iift, u ≥ 0,t, u ∈ 0, T _Tκ ×σρ1, ρ3 , moreover ft, u ≥ φpMσρ2,t, u ∈ 0, T _T^κ ×σρ2, ρ2 ,u /Su,∀u∈∂Ωρ2;

iiift, u≤φpmρ3,t, u∈0, T _Tκ×0, ρ3 .

H6There existρ1, ρ2, andρ3∈0,∞withρ1< ρ2< σρ3such that ift, u≥φpMσρ1,t, u∈0, T _T^κ×σ²ρ1, ρ2 ;

iift, u≤φpm1ρ2,t, u∈0, T _Tκ×0, ρ2 ,u /Su,∀u∈∂Kρ2;

iiift, u ≥ 0,t, u ∈ 0, T _T^κ ×σρ2, ρ3 , moreover,ft, u ≥ φpMσρ3,t, u ∈ 0, T _T^κ ×σρ3, ρ3 .

Then, the BVP1.6has at least two positive solutions.

Proof. Assume that conditionH5holds, we show thatShas a fixed pointu1either in∂Kρ1

or inΩρ2\Kρ1. Ifu /Suforu∈∂Kρ1∪∂Kρ3. by Lemmas2.13and2.14, we have that iK

S, Kρ1

1, iK

S, Kρ3

1, iK

S,Ωρ2

0.

3.9

ByLemma 2.11aandρ1 < σρ2, we haveKρ1 ⊂Kσρ2 ⊂Ωρ2. It follows fromLemma 2.103 thatShas a fixed pointu1inΩρ2\Kρ1. Similarly,Shas a fixed pointu2inKρ3\Ωρ2. Clearly,

u1> ρ1, min

t∈0,T _Tκu1t≥σu1> σρ1, 3.10

which implies thatσρ1≤u1t≤ρ2,t∈0, T _Tκ. By conditionH5ii, we haveft, u1t≥0, t∈0, T _Tκ, that is,ft, u1t ft, u1t. Hence,

Fu1Su1. 3.11

This means thatu1is a fixed point of operatorF. On the other hand, fromu2 ∈Kρ3\Ωρ2, ρ2<

ρ3andLemma 2.11a, we haveKσρ2 ⊂Ωρ2⊂Kρ3. Clearly, u2> σρ2, min

t∈0,T _Tκu2t≥σu2> σ²ρ2, 3.12 which implies thatσ²ρ2 ≤ u2t ≤ ρ3, t ∈ 0, T _T^κ. Byρ1 < σρ2 and conditionH5ii, we haveft, u2t≥0, t∈0, T _T^κ, that is,ft, u2t ft, u2t. Hence,

Fu2Su2. 3.13

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This means thatu2is a fixed point of operatorF. Then, the BVP1.6has at least two positive solutions.

When conditionH6holds, the proof is similar to the above, and so we omit it here.

4. An Example

In the section, we present some simple examples to explain our results.

Example 4.1. LetT 0,1/2

{1},T 1. Consider the following three-point boundary value problem withp-Laplacian

φpu^Δ∇_∇

atft, u 0, 0< t <1,

u0 1

3u 1

2

, u^Δ1 0, φp

u^Δ∇0 1

4φp

u^Δ∇

1 2

,

4.1

whereat≡1,b11/3,c11/4,ξ11/2,pq2.

By computing, we can knowσ1 1/5,σ2 1/7,M 48/5,m 48/35. Obviously, σσ1σ21/35,Mσ48/5×1/35<48/35m.

Letρ1 1,ρ2 78, thenσρ1 < ρ1 < σρ2 < ρ2. We define a sign changing nonlinearity as follows:

ft, u

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩ 48t³

35

u− 1 35

3

, 0< t <1, u∈

0, 1 35

, 48t³

35 sin 35

34 π

2u− 1 34

π 2

, 0< t <1, u∈ 1 35,1

, 48t³

35 2−u 3744

175 u−1, 0< t <1, u∈1,2 , 3744

175 t³u−2², 0< t <1, u∈2,78 , 3744

175 t³78−2²1 u−78 , 0< t <1, u∈78,∞ .

4.2

Then, by the definition offwe have

ift, u≤φpmρ1 48/35,t, u∈0,1 ×0, ρ1 ;

iift, u ≥ 0, t, u ∈ 0,1 ×σρ1, ρ2 , moreoverft, u ≥ φpMσρ2 3744/175, t, u∈0,1 ×σρ2, ρ2 .

So condition H3 holds, and by Theorem 3.1, BVP 4.1 has at least one positive solution.

(14)

Acknowledgment

This project was supported by the National Natural Science Foundation of China10471075, 10771117.

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