Three Solutions for a Discrete Nonlinear Neumann Problem Involving the p-Laplacian

doi:10.1155/2010/862016

Research Article

Three Solutions for a Discrete Nonlinear Neumann Problem Involving the p-Laplacian

Pasquale Candito

and Giuseppina D’Agu`ı

1DIMET University of Reggio Calabria, Via Graziella (Feo Di Vito), 89100 Reggio Calabria, Italy

2Department of Mathematics of Messina, DIMET University of Reggio Calabria, 89100 Reggio Calabria, Italy

Correspondence should be addressed to Giuseppina D’Agu`ı,[email protected] Received 26 October 2010; Accepted 20 December 2010

Academic Editor: E. Thandapani

Copyrightq2010 P. Candito and G. D’Agu`ı. 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.

We investigate the existence of at least three solutions for a discrete nonlinear Neumann boundary value problem involving thep-Laplacian. Our approach is based on three critical points theorems.

1. Introduction

In these last years, the study of discrete problems subject to various boundary value con- ditions has been widely approached by using different abstract methods as fixed point theorems, lower and upper solutions, and Brower degreesee, e.g.,1–3and the reference given therein. Recently, also the critical point theory has aroused the attention of many authors in the study of these problems4–12.

The main aim of this paper is to investigate different sets of assumptions which guarantee the existence and multiplicity of solutions for the following nonlinear Neumann boundary value problem

−Δ

φpΔuk−1

qkφpuk λfk, uk, k∈1, N,

Δu0 ΔuN 0, P_λ^f whereN is a fixed positive integer,1, Nis the discrete interval{1, . . . , N},qk > 0 for all k∈1, N,λis a positive real parameter,Δuk: uk1−uk,k 0,1, . . . , N1, is the forward difference operator,φps : |s|^p−2s, 1 < p < ∞, andf : 1, N×^Ê → ^Ê is a continuous function.

In particular, for every λ lying in a suitable interval of parameters, at least three solutions are obtained under mutually independent conditions. First, we require that the primitiveF off isp-sublinear at infinity and satisfies appropriate local growth condition Theorem 3.1. Next, we obtain at least three positive solutions uniformly bounded with respect toλ, under a suitable sign hypothesis onf, an appropriate growth conditions onFin a bounded interval, and without assuming asymptotic condition at infinity onfTheorem 3.4, Corollary 3.6. Moreover, the existence of at least two nontrivial solutions for problemP_λ^fis obtained assuming thatFisp-sublinear at zero andp-superlinear at infinityTheorem 3.5.

It is worth noticing that it is the first time that this type of results are obtained for discrete problem with Neumann boundary conditions; instead of Dirichlet problem, similar results have been already given in 6, 9, 13. Moreover, in 14, the existence of multiple solutions to problem P_λ^f is obtained assuming different hypotheses with respect to our assumptionsseeRemark 3.7.

Investigation on the relation between continuous and discrete problems are available in the papers15,16. General references on difference equations and their applications in different fields of research are given in17,18. While for an overview on variational methods, we refer the reader to the comprehensive monograph19.

2. Critical Point Theorems and Variational Framework

LetX be a real Banach space, letΦ,Ψ:X → ^Êbe two functions of classC¹ onX, and letλ be a positive real parameter. In order to study problemP_λ^f, our main tools are critical points theorems for functional of typeΦ−λΨwhich insure the existence at least three critical points for everyλbelonging to well-defined open intervals. These theorems have been obtained, respectively, in6,20,21.

Theorem 2.1see11, Theorem 2.6. LetX be a reflexive real Banach space,Φ : X → ^Êbe a coercive, continuously Gˆateaux differentiable and sequentially weakly lower semicontinuous func- tional whose Gˆateaux derivative admits a continuous inverse onX^∗,Ψ:X → ^Êbe a continuously Gˆateaux differentiable functional whose Gˆateaux derivative is compact such that

Φ0 Ψ0 0. 2.1

Assume that there existr >0 andv∈X, withr <Φvsuch that a1sup_Φu≤rΨu/r <Ψv/Φv,

a2for eachλ∈Λr : Φv/Ψv, r/sup_Φu≤rΨuthe functionalΦ−λΨis coercive.

Then, for eachλ∈Λr, the functionalΦ−λΨhas at least three distinct critical points inX.

Theorem 2.2see7, Corollary 3.1. LetX be a reflexive real Banach space,Φ : X → ^Ê be a convex, coercive, and continuously Gˆateaux differentiable functional whose Gˆateaux derivative admits a continuous inverse onX^∗, and letΨ:X → ^Êbe a continuously Gˆateaux differentiable functional whose Gˆateaux derivative is compact such that

infX Φ Φ0 Ψ0 0. 2.2

Assume that there exist two positive constantsr1,r2andv∈X, with 2r1<Φv< r2/2 such that b1sup_Φu≤r

1Ψu/r1<2/3Ψv/Φv, b2sup_Φu≤r

2Ψu/r2<1/3Ψv/Φv,

b3for eachλ∈Λ : 3/2Φv/Ψv, min{r1/sup_Φu≤r1Ψv, r2/2sup_Φu≤r2Ψu}

and for everyu1, u2 ∈ X, which are local minima for the functional Φ−λΨ such that Ψu1≥0 andΨu2≥0, and one has inf_t∈0,1Ψtu1 1−tu2≥0.

Then, for each λ ∈ Λ, the functional Φ− λΨ admits at least three critical points which lie in Φ⁻¹− ∞, r2.

Finally, for allr >infXΦ, we put

ϕr inf

u∈Φ⁻¹−∞,r

sup_u∈Φ−1−∞,rΨu

−Ψu

r−Φu ,

λ^∗: 1

inf_{r>infXΦ}ϕr,

2.3

where we read 1/0 : ∞if this case occurs.

Theorem 2.3see8, Theorem 2.3. LetXbe a finite dimensional real Banach space. Assume that for eachλ∈0, λ^∗one has

elim _{u → ∞}Φ−λΨ −∞.

Then, for eachλ∈0, λ^∗, the functionalΦ−λΨadmits at least three distinct critical points.

Remark 2.4. It is worth noticing that whenever X is a finite dimensional Banach space, a careful reading of the proofs of Theorems2.1and2.2shows that regarding to the regularity of the derivative ofΦandΨ, it is enough to require only thatΦandΨare two continuous functionals onX^∗.

Now, consider theN-dimensional normed spaceW {u :0, N1 → ^Ê : Δu0

ΔuN 0}endowed with the norm

u : _N1

k 1

|Δuk−1|^pN

k 1

qk|uk|^p 1/p

, ∀u∈W. 2.4

In the sequel, we will use the following inequality:

k∈0,N1max |uk| ≤ u

q^1/p, ∀u∈W whereq: min

k∈1,Nqk. 2.5

Moreover, put

Φu: u ^p

p , Ψu: ^N

k 1

Fk, uk, ∀u∈W, 2.6

whereFk, t: ₀^tfk, ξdξfor everyk, t∈1, N×^Ê. It is easy to show thatΦandΨare twoC¹-functionals onW.

Next lemma describes the variational structure of problem P_λ^f, for the reader convenience we give a sketch of the proof, see also14,

Lemma 2.5. W, · is a Banach space. Letu∈W,ube a solution of problemP_λ^fif and only ifu is a critical point of the functionalΦ−λΨ.

Proof. Bearing in mind both that a finite dimensional normed space is a Banach space and the following partial sum:

−^N

k 1

Δ

φpΔuk−1 vk

N1

k 1

φpΔuk−1

Δvk−1, 2.7

for everyuandv∈W, standard variational arguments complete the proof.

Finally, we point out the following strong maximum principle for problemP_λ^f. Lemma 2.6. Fixu∈Wsuch that

−Δ

φpΔu_k−1

qk|uk|^p−2uk≥0 ∀k∈1, N. 2.8 Then, eitheru >0 in1, N, oru≡0.

Proof. Letj∈1, Nbe such thatuj min_k∈1,Nuk. An immediate computation gives Δuj≥0, Δu_j−1≤0. 2.9

From this, by2.8, we obtain

qjuj^p−2uj≥Δuj^p−2Δuj−Δuj−1^p−2Δuj−1≥0, 2.10 souj ≥ 0, that isu≥ 0. Moreover, assuming thatuj 0, from the preciding inequality and nonnegativity ofuj−1, uj1, one has

0≤Δuj^p−2 uj1

Δu_j−1^p−2 uj−1

≤0, 2.11

souj−1 uj1 0. Thus, repeating these arguments, the conclusion follows at once.

3. Main Results

For each positive constantscandd, we write

Ac: _N

k 1max_|t|≤cFk, t

c^p , Bd:

_N

k 1Fk, d

d^p , Q: ^N

k 1

qk. 3.1

Now, we give the main results.

Theorem 3.1. Assume that there exist three positive constantsc,d, andswithc < d, ands < psuch that

i1Ac<q/QBd,

i2max_k∈1,Nlim sup_{|t| →∞}Fk, t/|t|^s<∞.

Then, for every

λ∈ Q

p 1 Bd,q

p 1 Ac

, 3.2

problemP_λ^fadmits at least three solutions.

Proof. We applyTheorem 2.1, by puttingΦand Ψdefined as in2.6on the spaceW. An easy computation ensures the regularity assumptions required onΦandΨ; seeRemark 2.4.

Therefore, it remains to verify assumptionsa1anda2. To this hand, we put r q

pc^p, 3.3

and we pickv∈W, defined by putting

vk d for everyk∈1, N. 3.4

Clearly, sincec < d, one hasr <Φv Q/pd^p, and in addition, by2.5, we have sup_u∈Φ−1−∞,rΨu

r ≤ sup _u

∞≤cΨ q/p

c^p ≤ p

qAc. 3.5

On the other hand, we compute

Ψv Φv

p

QBd. 3.6

Therefore, byi1, combining3.5and3.6, it is clear thata1holds. Moreover, one has Q

p 1 Bd,q

p 1 Ac

⊂Λr. 3.7

Now, fixλas in the conclusion; first, we observe that for every 1≤s≤p, one has N

k 1

|uk|^s≤Nq^−s/p u ^s, ∀u∈W. 3.8

Next, byi2, there exist two positive constantsM1andM2such that

Fk, ξ≤M1|ξ|^sM2, ∀k, ξ∈1, N×^Ê. 3.9

Hence, for everyu∈W, we get

Φu−λΨu≥ u ^p p −λM1

N 1

|uk|^s−λNM2

≥ u ^p

p −λM1 N

q^s/p u ^s−λNM2.

3.10

At this point, sinces < p, it is clear that the functionalΦ−λΨturns out to be coercive.

Remark 3.2. We note that hypothesisi2can be replaced with the following:

i₂max_k∈1,Nlim sup_{|t| →∞}Fk, t/|t|^p< Ac/N.

Arguing as before, there exist two constantL1 < Ac/NandL2such that

Fk, ξ≤L1|ξ|^pL2, ∀k, ξ∈1, N×^Ê. 3.11

Hence, for everyu∈W, it easy to see that Φu−λΨu≥ u ^p

p −q p

1 AcL1N

q u ^p−λNL2≥ 1 p

1− NL1

Ac

u ^p−λNL2, 3.12

with1−NL1/Ac>0.

Remark 3.3. It is worth noticing that a careful reading of the proof of Theorem 3.1 shows that, provided thatAc 0 and under the only conditioni2, problemP_λ^fadmits at least one solution for everyλ >0 and at least three solutions for everyλ ∈Q/p1/Bd,∞, whenever there existsd >0 for whichBd>0.

Theorem 3.4. Let f be a continuous function in1, N×0,∞such thatfk,0/0 for some k ∈1, N. Assume that there exist three positive constantsc1,d, andc2 with2q/Q^1/pc1 < d <

1/2q/Q^1/pc2 such that

j1fk, ξ≥0 for eachk, ξ∈1, N×0, c2, j2max{Bc1,2Bc2}<2/3q/QBd.

Then, for eachλ∈3/2Q/p1/Bd,q/pmin{1/Bc1,1/2Bc2}, problemP_λ^fadmits at least three positive solutionsuⁱ,i 1,2,3, such that

uⁱ_k< c2, 3.13

for allk∈1, N,i 1,2,3.

Proof. Consider the auxiliary problem

−Δ

φpΔuk−1

qkφpuk λfk, u k, k∈1, N, Δu0 ΔuN 0,

P_λ^f

wheref:1, N×^Ê → ^Êis a continuous function defined putting

fk, ξ

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

fk,0, if ξ <0, fk, ξ, if 0≤ξ≤c2, fk, c2, if ξ > c2.

3.14

From j1, owing to Lemma 2.6, any solution of problem P_λ^f is positive. In addition, if it satisfies also the condition 0 ≤ uk ≤ c2, and for every k ∈ 1, N, clearly it turns to be also a positive solution of P_λ^f. Therefore, for our goal, it is enough to show that our conclusion holds for P_λ^f. In this connection, our aim is to apply Theorem 2.2. Fix λ in 3/2Q/p1/Bd,q/pmin{1/Bc1,1/2Bc2} and let Φ, Ψ and W as before.

Now, take

r1

q

pc^p₁, r2

q

pc^p₂. 3.15

From2.5, arguing as before, we obtain

k∈1,Tmax|uk| ≤c1, 3.16

for allu∈Wsuch that u ≤pr1^1/p, and

k∈1,Tmax|uk| ≤c2, 3.17

for allu∈Wsuch that u ≤pr2^1/p. Therefore, one has

sup_u∈Φ−1−∞,r1Ψu r1

sup _{u <pr1}^1/p_N

k 1Fk, uk

r1 ≤

_N

k 1Fk, c1 r1

p

qBc1, 3.18

as well as

sup_u∈Φ−1−∞,r2Ψu

r2 ≤ p

qBc2. 3.19

On the other hand, pick v ∈ W, defined as in 3.4, bearing in mind 3.6, and from 2q/Q^1/pc1 < d < 1/2q/Q^1/pc2, we obtain 2r1 < Φv < c2/2 Moreover, taking into account3.18,3.19, fromj1, assumptionsb1andb2follow. Further, again from3.18, 3.19, and3.6, one has that

λ∈ 3

2 Q

p 1 Bd,q

pmin 1

Bc1, 1 2Bc2

⊂Λ. 3.20

Now, letu1andu2be two local minima forΦ−λΨsuch thatΨu1≥0 andΨu2≥0. Owing to Lemmas2.5and2.6, they are two positive solutions forP_λ^fsotu¹_k 1−tu²_k ≥0, for all k ∈1, Nand for allt∈ 0,1. Hence, since one hasΨtu¹ 1−tu² ≥0 for allt ∈0,1, b3is verified.

Therefore, the functionalΦ−λΨadmits at least three critical pointsuⁱ,i 1,2,3, which are three positive solutions ofP_λ^f. Finally, from2.5, fori 1,2,3, one has

k∈1,Nmax

uⁱ_k≤c2, 3.21

and the proof is completed.

Theorem 3.5. Letf : 1, N×^Ê → ^Ê be a continuous function such thatfk,0/0 for some k ∈ 1, N. Assume that there exist four constantsM1,M2,s, and α, with M1 > 0, s > pand 0≤α < ssuch that

lFk, ξ≥M1|ξ|^s−M2|ξ|^α, for allk, ξ∈1, N×^Ê. Then, for eachλ∈0, λ^∗, where

λ^∗: q p

1

sup_c>0Ac, 3.22

problemP_λ^fadmits at least three nontrivial solutions.

Proof. Our aim is to applyTheorem 2.3withΦandΨas above. Fixλ∈0, λ, and there isc >0 such thatλ <q/p1/Ac. Settingr q/pc^pand arguing as in the proof ofTheorem 3.1, one has

1

λ^∗ ≤ϕr≤ sup_u∈Φ−1−∞,rΨu

r ≤ p

qAc< 1

λ, 3.23

that isλ < λ^∗. Moreover, denote

q max

k∈1,Nqk, 3.24

it is a simple matter to show that for eachu∈W, one has N

k 1

|uk|^s≥ u ^s N12^pq_s/p

N^s−p/p,

N k 1

|uk|^α≤Nq^−α/p u ^α. 3.25

Hence, froml, for eachu∈W, we get

Φu−λΨu≤ u ^p

p − λM1

N12^pq_s/p

N^s−p/p u ^sλM2Nq^−α/p u ^α. 3.26 Therefore, since s > p ands > α, condition e is verified. Hence, from Theorem 2.3, the functional Φ− λΨ admits three critical points, which are three solutions for P_λ^f. Since fk,0/0 for somek∈1, N, they are nontrivial solutions, and the conclusion is proved.

Corollary 3.6. Letf : 1, N×^Ê → ^Ê be a continuous function such that fk,0/0 for some k ∈1, N. Assume that there exist four constantsM1,M2,c, andαwithM1 >0 and 0≤ α < p such that

l1Ac< qM1/N12^pq,

l2Fk, ξ≥M1|ξ|^p−M2|ξ|^α, for allk, ξ∈1, N×^Ê. Then, for every

λ∈

N12^pq pM1 ,q

p 1 Ac

, 3.27

problemP_λ^fadmits at least three solutions.

Proof. Our claim is to prove that conditioneofTheorem 2.3holds for everyλ∈N12^p q/pM1,q/p1/Ac⊂0, λ^∗. Indeed, froml1, arguing as in3.23, one has thatλ < λ^∗. Moreover, byl2, from3.26withs p, for everyu∈W, we have

Φu−λΨu≤ u ^p

p − λM1

N12^pq u ^pλM2Nq^−α/p u ^α

≤ 1

p− λM1

N12^pq

u ^pλM2Nq^−α/p u ^α,

3.28

where1/p−λM1/N12^pq<0, which implies conditione.

Remark 3.7. In14, by Mountain Pass Theorem, the authors established the existence of at least one solution for problemP_λ^frequiring the following conditions:

θ1fk, t ◦|t|^p−1fort → 0 uniformly ink∈1, N,

θ2there exist two positive constantsρandswiths > psuch that

0< sFk, t≤tfk, t, 3.29

for every|t|> ρandk, ξ∈1, N×^Ê.

Moreover, they remember that the above conditions imply, respectively, the following:

θ3Fk, t ◦|t|^pfort → 0 uniformly ink∈1, N, θ4there exist two positive constantsM1andM2such that

Fk, ξ≥M1|ξ|^s−M2, ∀k, ξ∈1, N×^Ê. 3.30 Next result shows that under more general conditions thanθ3andθ4, problemP₁^f has at least two nontrivial solutions.

Theorem 3.8. Assume that (l2) holds and in addition θ5max_k∈1,Nlim sup_{|t| →0}Fk, t/|t|^p<∞.

Then, problem (P₁^f) has at least two nontrivial solutions.

Proof. We claim that the functionalΦ−Ψadmits a local minimum at zero and a local nonzero maximum. To this end, we observe that byθ5, there existM >0 andρ >0 such that

Fk, t≤M1|t|^p, for every|t| ≤ρ, k∈1, N. 3.31 Hence, bearing in mindLemma 2.5and3.25, withs p, for everyu∈Wwith u ≤ρ√^p

q, we get

Φu−Ψu≥ 1

p−MN q

u ^p

p ≥0 Φ0−Ψ0, 3.32

that is, 0 is a local minimum. Moreover, by l2, by now, it is evident that the functional Φ−Ψis anticoercive inW. Hence, by the regularity ofΦ−Ψ, there existsu∈Wwhich is a global maximum for the functional. Therefore, since it is not restrictive to suppose thatu /0 otherwise, there are infinitely many critical points, our conclusion follows: if dimX ≥ 2, from Corollary 2.11 of22which ensures a third critical point different from 0 anduand by standards arguments if dimX 1.

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