Three Solutions to Dirichlet Boundary Value Problems for Laplacian Difference Equations

doi:10.1155/2008/345916

Research Article

Three Solutions to Dirichlet Boundary Value

Problems for

p

-Laplacian Difference Equations

Liqun Jiang1, 2 _{and Zhan Zhou}2, 3

1_{Department of Mathematics and Computer Science, Jishou University, Jishou, Hunan 416000, China} 2_{Department of Applied Mathematics, Hunan University, Changsha, Hunan 410082, China}

3_{Department of Applied Mathematics, Guangzhou University, Guangzhou 510006, Guangdong, China}

Correspondence should be addressed to Liqun Jiang,liqunjianghn@yahoo.com

Received 2 March 2007; Revised 16 July 2007; Accepted 15 October 2007

Recommended by Svatoslav Stanek

We deal with Dirichlet boundary value problems forp-Laplacian difference equations depending on a parameterλ. Under some assumptions, we verify the existence of at least three solutions when

λlies in two exactly determined open intervals respectively. Moreover, the norms of these solutions are uniformly bounded in respect toλbelonging to one of the two open intervals.

Copyrightq2008 L. Jiang and Z. Zhou. 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

Let R, Z,N be all real numbers, integers, and positive integers, respectively. Denote Za {a, a1, . . .}andZa, b {a, a1, . . . , b}witha < bfor anya, b∈Z.

In this paper, we consider the following discrete Dirichlet boundary value problems:

Δφ_pΔxk−1 λfk, xk 0, k∈Z1, T,

x0 0xT 1, 1.1

whereTis a positive integer,p >1 is a constant,Δis the forward difference operator defined by

Δxk xk1−xk,φ_psis ap-Laplacian operator, that is,φ_ps |s|p−2_{s,fk,·∈CR,R}

for anyk∈Z1, T.

In the last years, a few authors have gradually paid more attentions to applying critical point theory to deal with problems for nonlinear second discrete systems; we refer to 5–9. But all these systems do not concern with thep-Laplacian. For the reader’s convenience, we recall the definition of the weak closure.

Suppose thatE⊂X. We denoteEwas the weak closure ofE, that is,x∈Ewif there exists a sequence{xn} ⊂Esuch thatΛxn→Λxfor everyΛ∈X∗.

Very recently, based on a new variational principle of Ricceri 10, the following three critical points was established by Bonanno 11.

Theorem 1.1 see 11, Theorem 2.1. Let X be a separable and reflexive real Banach space.

Φ : X→R a nonnegative continuously Gˆateaux differentiable and sequentially weakly lower semicontinuous functional whose Gˆateaux derivative admits a continuous inverse on X∗. J : X→R a continuously Gˆateaux differentiable functional whose Gˆateaux derivative is compact. Assume that there existsx0∈Xsuch thatΦx0 Jx0 0and that

ilimx→∞Φx−λJx ∞for allλ∈ 0,∞ ; Further, assume that there arer >0, x1∈Xsuch that

iir <Φx1;

iiisup_x∈Φ−1_−∞,r wJx<r/r Φx1Jx1. Then, for each

λ∈Λ1

Φx1

Jx1 −sup_x∈Φ−1_−∞,r wJx

, r

sup_x∈Φ−1_−∞,r wJx

, 1.2

the equation

Φ_{x−λJx 0 1.3}

has at least three solutions inXand, moreover, for eachh >1, there exists an open interval

Λ2 ⊆

0, hr

rJx1/Φx1−sup_x∈Φ−1_−∞,r wJx

1.4

and a positive real numberσ such that, for eachλ ∈Λ2,1.3has at least three solutions inXwhose

norms are less thanσ.

Here, our principle aim is by employingTheorem 1.1to establish the existence of at least three solutions for thep-Laplacian discrete boundary value problem1.1.

The paper is organized as follows. The next section is devoted to give some basic defi-nitions. InSection 3, under suitable hypotheses, we prove that the problem1.1possesses at least three solutions whenλlies in exactly determined two open intervals, respectively; more-over, all these solutions are uniformly bounded with respect toλbelonging to one of the two open intervals. At last, a consequence is presented.

2. Preliminaries

The classHof the functionsx:Z0, T1→Rsuch thatx0 xT1 0 is aT-dimensional Hilbert space with inner product

x, z T

k1

We denote the induced norm by

Furthermore, for any constantp >1, we define other norms

xp

SinceHis a finite dimensional space, there exist constantsc2p≥c1p>0 such that

c1pxp≤ xP≤c2pxp. 2.4

The following two functionals will be used later:

Φx 1

Remark 2.1. Obviously, for anyx, z∈H, corresponds to a solution of the problem1.1. Thus, we reduce the existence of a solution for the problem1.1to the existence of a critical point ofΦ−λJ onH.

The following estimate will play a key role in the proof of our main results.

Lemma 2.2. For anyx∈Handp >1, the relation

thus,

Combining the above inequality with the estimate2.13, we have

|xτ|<T−τ11/q

Now, we claim that the inequality

T−τ11/q

holds, which leads to the required inequality2.10. In fact, we define a continuous function υ:0, T 1 →Rby

This implies the assertion2.18.Lemma 2.2is proved.

3. Main results

First, we present our main results as follows.

and for eachh >1,

the problem1.1admits at least three solutions inHand, moreover, for eachh >1, there exist an open intervalΛ2 ⊆ 0, aand a positive real numberσ such that, for eachλ∈Λ2, the problem1.1admits

at least three solutions inHwhose norms inHare less thanσ.

Remark 3.2. By the conditionA1, we have

Proof ofTheorem 3.1. Let X be the Hilbert space H. Thanks to Remark 2.1, we can apply

Taking into account the fact thatα < p, we obtain, for allλ∈ 0,∞ ,

lim

x→∞Φx−λJx ∞. 3.8

The conditioniofTheorem 1.1is satisfied. Now, we let

So, the assumptioniiofTheorem 1.1is obtained. Next, we verify that the assumption

iiiofTheorem 1.1holds. FromLemma 2.2, the estimateΦx≤rimplies that

On the other hand, we get

Therefore, it follows from the assumptionA1that

sup x∈Φ−1_−∞,r w

Jx≤ r r Φx1

Jx1, 3.16

that is, the conditioniiiofTheorem 1.1is satisfied. Note that

Φx1

Jx1−sup_x∈Φ−1_−∞,r wJx

≤ 2dp

pT_k1Fk, d−Tmaxk,ξ∈Z1,T×−c,cFk, ξ

1

ϕ₂,

r

sup_x∈Φ−1_−∞,r wJx

≥ 2cp

pT 1p−1Tmaxk,ξ∈Z1,T×−c,cFk, ξ 1

ϕ₁.

3.17

By a simple computation, it follows from the condition A1 that ϕ₂ > ϕ₁. Applying Theorem 1.1, for eachλ∈Λ11/ϕ₂,1/ϕ₁ , the problem1.1admits at least three solutions in

H.

For eachh >1, we easily see that

hr

rJx1/Φx1−sup_x∈Φ−1_−∞,r wJx

≤ h2cdp

2p−1_pcpT

k1Fk, d −TT1p−1pdpmaxk,ξ∈Z1,T×−c,cFk, ξ

a.

3.18

Taking the conditionA1into account, it forces thata > 0. Then fromTheorem 1.1, for each

h > 1, there exist an open interval Λ2 ⊆ 0, aand a positive real number σ, such that, for

λ ∈Λ2, the problem1.1admits at least three solutions inHwhose norms inHare less than

σ. The proof ofTheorem 3.1is complete.

As a special case of the problem1.1, we consider the following systems:

Δφ_pΔxk−1 λwkgxk 0, k∈Z1, T,

x0 0xT1, 3.19

wherew:Z1, T→Randg∈CR,Rare nonnegative. Define

Wk k

t1

wt, Gξ

ξ

0

gsds. 3.20

Corollary 3.3. Letw:Z1, T→Randg ∈CR,Rbe two nonnegative functions. Assume that there exist four positive constantsc, d, η, αwithc <T1/2p−1/pdandα < psuch that

A₁maxk∈Z1, Twk<2cpWT/T 2cp2T1p−1dpGd/Gc;

A₂Gξ≤η1|ξ|αfor anyξ∈R.

Furthermore, put

ϕ₁ pT1

p−1_TGcmax

k∈Z1,Twk

2cp ,

ϕ₂ p WTGd−TGcmaxk∈Z1,Twk

2dp ,

3.21

and for eachh >1,

a 2cd

p_h

2p−1_pcp_WTGd−pdp_TT1p−1_Gcmax

k∈Z1,Twk

. 3.22

Then, for each

λ∈Λ1

1 ϕ₂,

1 ϕ₁

, 3.23

the problem 3.19admits at least three solutions in H and, moreover, for eachh > 1, there exist an open intervalΛ2 ⊆ 0, aand a positive real numberσ such that, for eachλ ∈Λ2, the problem3.19

admits at least three solutions inHwhose norms inH are less thanσ.

Proof. Note that from factfk, s wkgsfor anyk ∈Z1, T×R, we have

max

k,ξ∈Z1,T×−c, cFk, ξ Gck∈maxZ1,Twk. 3.24

On the other hand, we takeμη maxk∈Z1,Twk. Obviously, all assumptions ofTheorem 3.1

are satisfied.

To the end of this paper, we give an example to illustrate our main results.

Example 3.4. We consider1.1withfk, s kgs, T 15, p 3, where

gs

es_{, s≤4d,}

se4d_{−4d, s >4d.} 3.25

We have thatWk 1/2kk1and

Gξ

⎧ ⎨ ⎩

eξ−1, ξ≤4d,

1 2ξ

2 _e4d_{−4dξ 1−4de}4d_8d2_{−1, ξ >4d.} 3.26

Acknowledgments

This work is supported by the National Natural Science Foundation of Chinano. 10571032 and Doctor Scientific Research Fund of Jishou universityno. jsdxskyzz200704.

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