Singular Boundary Value Problems for Ordinary Differential Equations

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p-LAPLACIAN PROBLEMS VIA VARIATIONAL METHODS

RAVI P. AGARWAL, KANISHKA PERERA, AND DONAL O’REGAN Received 31 March 2005

We obtain multiple positive solutions of singular discrete p-Laplacian problems using variational methods.

1. Introduction

We consider the boundary value problem

−∆ϕ_p∆u(k−1)=fk,u(k), k∈[1,n], u(k)>0, k∈[1,n],

u(0)=0=u(n+ 1),

(1.1)

wherenis an integer greater than or equal to 1, [1,n] is the discrete interval{1,. . .,n},

∆u(k)=u(k+ 1)−u(k) is the forward diﬀerence operator,ϕp(s)= |s|^p⁻²s, 1< p <∞, and we only assume that f ∈C([1,n]×(0,∞)) satisfies

a0(k)≤f(k,t)≤a1(k)t⁻^γ, (k,t)∈[1,n]× 0,t0

(1.2) for some nontrivial functionsa0,a1≥0 andγ,t0>0, so that it may be singular att=0 and may change sign.

Letλ1,ϕ1>0 be the first eigenvalue and eigenfunction of

−∆ϕp

∆u(k−1)=λϕp

u(k), k∈[1,n],

u(0)=0=u(n+ 1). (1.3)

Theorem1.1. If (1.2) holds and lim sup

t→∞

f(k,t)

t^p⁻¹ < λ1, k∈[1,n], (1.4) then (1.1) has a solution.

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Theorem1.2. If (1.2) holds and

f(k,t1)≤0, k∈[1,n], (1.5)

for somet1> t0, then (1.1) has a solutionu1< t1. If, in addition,

lim inf

t→∞

f(k,t)

t^p⁻¹ > λ1, k∈[1,n], (1.6) then there is a second solutionu2> u1.

Example 1.3. Problem (1.1) with f(k,t)=t⁻^γ+λt^β has a solution for allγ >0 and λ (resp.,λ < λ1,λ≤0) ifβ < p−1 (resp.,β=p−1,β > p−1) byTheorem 1.1.

Example 1.4. Problem (1.1) with f(k,t)=t⁻^γ+e^t−λhas two solutions for allγ >0 and suﬃciently largeλ >0 byTheorem 1.2.

Our results seem new even forp=2. Other results on discretep-Laplacian problems can be found in [1,2] in the nonsingular case and in [3,4,5,6] in the singular case.

2. Preliminaries

First we recall theweak comparison principle(see, e.g., Jiang et al. [2]).

Lemma2.1. If

−∆ϕp

∆u(k−1)≥ −∆ϕp

∆v(k−1), k∈[1,n],

u(0)≥v(0), u(n+ 1)≥v(n+ 1), (2.1)

thenu≥v.

Next we prove a local comparison result.

Lemma2.2. If

−∆ϕp

∆u(k−1)≥ −∆ϕp

∆v(k−1),

u(k)=v(k), u(k±1)≥v(k±1), (2.2)

thenu(k±1)=v(k±1).

Proof. We have

−ϕp

∆u(k)+ϕp

∆u(k−1)≥ −ϕp

∆v(k)+ϕp

∆v(k−1), (2.3)

∆u(k)≥∆v(k), ∆u(k−1)≤∆v(k−1). (2.4) Combining with the strict monotonicity ofϕpshows that

0≤ϕp

∆u(k)−ϕp

∆v(k)≤ϕp

∆u(k−1)−ϕp

∆v(k−1)≤0, (2.5)

and hence, the equalities hold in (2.4).

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The followingstrong comparison principleis now immediate.

Lemma2.3. If

−∆ϕp

∆u(k−1)≥ −∆ϕp

∆v(k−1), k∈[1,n],

u(0)≥v(0), u(n+ 1)≥v(n+ 1), (2.6)

then eitheru > vin[1,n], oru≡v. In particular, if

−∆ϕp

∆u(k−1)≥0, k∈[1,n],

u(0)≥0, u(n+ 1)≥0, (2.7)

then eitheru >0in[1,n]oru≡0.

Consider the problem

−∆ϕ_p∆u(k−1)=gk,u(k), k∈[1,n],

u(0)=0=u(n+ 1), (2.8)

whereg∈C([1,n]×R). The classWof functionsu: [0,n+ 1]→Rsuch thatu(0)=0= u(n+ 1) is ann-dimensional Banach space under the norm

u = _n+1

k=1

∆u(k−1)^p 1/ p

. (2.9)

Define

Φg(u)=

n+1

k=1

1

p∆u(k−1)^p−Gk,u(k), u∈W, (2.10) whereG(k,t)= 0^tg(k,s)ds. Then the functionalΦgisC¹with

Φg(u),v₌

n+1

k=1

ϕ_p∆u(k₋1)∆v(k₋1)−gk,u(k)v(k)

= − n k=1

∆ϕp

∆u(k−1)+gk,u(k)v(k)

(2.11)

(summing by parts), so solutions of (2.8) are precisely the critical points ofΦg. Lemma2.4. If

lim sup

|t|→∞

g(k,t)

|t|^p⁻²t< λ1, k∈[1,n], (2.12) thenΦghas a global minimizer.

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Proof. By (2.12), there is aλ∈[0,λ1) such that G(k,t)≤ λ

p^|t|^p+C, (2.13)

whereCdenotes a generic positive constant. Since λ1= min

u∈W\{0}

_n+1

k=1∆u(k−1)^p _n

k=1u(k)^p , (2.14)

then

Φg(u)≥1 p

1− λ λ1

u^p−Cu, (2.15)

soΦgis bounded from below and coercive.

Lemma2.5. If

lim inf

t→+_∞

g(k,t)

t^p⁻¹ > λ1, lim

t→−∞

g(k,t)

|t|^p⁻¹ =0, k∈[1,n], (2.16) thenΦg satisfies the Palais-Smale compactness condition (PS): every sequence(uj)inW such thatΦg(u_j)is bounded andΦg(u_j)→0has a convergent subsequence.

Proof. It suﬃces to show that (uj) is bounded sinceWis finite dimensional, so suppose thatρj:= uj → ∞for some subsequence. We have

o(1)u⁻_j= Φg

u_j,u⁻_j≤ −u⁻_j^p−

n+1

k=1

gk,−u⁻_j(k)u⁻_j(k), (2.17) whereu⁻_j =max{−uj, 0}is the negative part ofuj, so it follows from (2.16) that (u⁻_j) is bounded. So, for a further subsequence,u_j:=u_j/ρ_j converges to someu≥0 inWwith u =1.

We may assume that for eachk, either (uj(k)) is bounded oruj(k)→ ∞. In the former case,u(k)=0 andgk,u_j(k)/ρ^p_j⁻¹→0, and in the latter case,gk,u_j(k)≥0 for largej by (2.16). So it follows from

o(1)= Φ_guj

,v ρ^p_j⁻¹ ⁼

n+1

k=1

ϕ_p∆u_j(k−1)∆v(k−1)−gk,u_j(k) ρ^p_j⁻¹ v(k)

(2.18) that

n+1

k=1

ϕ_p∆u(k −1)∆v(k−1)≥0 ∀v≥0, (2.19)

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and hence,u > 0 in [1,n] byLemma 2.3. Thenu_j(k)→ ∞for eachk, and hence, (2.18) can be written as

n+1

k=1

ϕp

∆uj(k−1)∆v(k−1)−αj(k)uj(k)^p⁻¹v(k)=o(1), (2.20)

where

αj(k)=gk,uj(k)

uj(k)^p⁻¹ ^≥λ, jlarge, (2.21) for someλ > λ1by (2.16).

Choosingvappropriately and passing to the limit shows that eachαj(k) converges to someα(k)≥λand

−∆ϕ_p∆u(k −1)=α(k)u(k) ^p⁻¹, k∈[1,n],

u(0)=0=u(n + 1). (2.22)

This implies that the first eigenvalue of the corresponding weighted eigenvalue problem is given by

u∈minW\{0}

_n+1

k=1∆u(k−1)^p _n

k=1α(k)u(k)^p ⁼1. (2.23)

Then

1≤ _n+1

k=1∆ϕ1(k−1)^p _n

k=1α(k)ϕ1(k)^p ^≤ λ1

λ <1, (2.24)

a contradiction.

3. Proofs The problem

−∆ϕp

∆u(k−1)=a0(k), k∈[1,n],

u(0)=0=u(n+ 1), (3.1)

has a unique solutionu0>0 by Lemmas2.3 and2.4. Fixε∈(0, 1] so small that u:= ε^1/(p⁻¹⁾u0< t0. Then

−∆ϕp

∆u(k−1)−fk,u(k)≤ −(1−ε)a0(k)≤0 (3.2) by (1.2), souis a subsolution of (1.1). Let

f_u(k,t)=







f(k,t), t≥u(k),

fk,u(k), t < u(k). (3.3)

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Proof ofTheorem 1.1. By (1.4), there areλ∈[0,λ1) andT > t0such that

f(k,t)≤λt^p⁻¹, (k,t)∈[1,n]×(T,∞). (3.4) Then

f_u(k,t)







≤a1(k)u(k)⁻^γ+ maxf[1,n]×

t0,T+λt^p⁻¹, t≥0,

≥a0(k), t <0,

(3.5) by (1.2), so the modified problem

−∆ϕ_p∆u(k₋1)=f_uk,u(k), k_∈[1,n],

u(0)=0=u(n+ 1), (3.6)

has a solutionubyLemma 2.4. ByLemma 2.1,u≥u, and hence, also a solution of (1.1).

Proof ofTheorem 1.2. Noting thatt1is a supersolution of (3.6), let

fu(k,t)=





 f_uk,t1

, t > t1,

fu(k,t), t≤t1. (3.7)

By (1.2),

f_u(k,t)







≤a1(k)u(k)⁻^γ+ maxf[1,n]× t0,t1

, t≥0,

≥a0(k), t <0, (3.8)

soΦfuhas a global minimizeru1byLemma 2.4. By Lemmas2.1and2.2,u≤u1< t1, so Φfu=Φfunearu1and hence,u1is a local minimizer ofΦfu. Let

fu1(k,t)=







f(k,t), t≥u1(k),

fk,u1(k), t < u1(k). (3.9) Sinceu1is also a subsolution of (1.1), repeating the above argument withu1in place of u, we see thatΦfu1 also has a local minimizer, which we assume isu1itself, for otherwise we are done. By (1.6), there areλ > λ1andT > t1such that

f(k,t)≥λt^p⁻¹, (k,t)∈[1,n]×(T,∞), (3.10) so

Φfu1

tϕ1

≤ −t^p p

λ λ1−1

+Ct <Φfu1

u1

, t >0 large. (3.11)

SinceΦfu1 satisfies (PS) byLemma 2.5, the mountain-pass lemma now gives a second critical pointu2, which is greater thanu1by Lemmas2.1and2.2.

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References

[1] R. Avery and J. Henderson,Existence of three positive pseudo-symmetric solutions for a one di- mensional discretep-Laplacian, J. Diﬀerence Equ. Appl.10(2004), no. 6, 529–539.

[2] D. Jiang, J. Chu, D. O’Regan, and R. P. Agarwal,Positive solutions for continuous and discrete boundary value problems to the one-dimensionp-Laplacian, Math. Inequal. Appl.7(2004), no. 4, 523–534.

[3] D. Jiang, D. O’Regan, and R. P. Agarwal,A generalized upper and lower solution method for singular discrete boundary value problems for the one-dimensionalp-Laplacian, to appear in J. Appl. Anal.

[4] ,Existence theory for single and multiple solutions to singular boundary value problems for the one-dimensionp-Laplacian, Adv. Math. Sci. Appl.13(2003), no. 1, 179–199.

[5] D. Jiang, L. Zhang, D. O’Regan, and R. P. Agarwal,Existence theory for single and multiple so- lutions to singular positone discrete Dirichlet boundary value problems to the one-dimension p-Laplacian, Archivum Mathematicum (Brno)40(2004), no. 4, 367–381.

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Ravi P. Agarwal: Department of Mathematical Sciences, Florida Institute of Technology, Mel- bourne, FL 32901, USA

E-mail address:[email protected]

Kanishka Perera: Department of Mathematical Sciences, Florida Institute of Technology, Mel- bourne, FL 32901, USA

E-mail address:[email protected]

Donal O’Regan: Department of Mathematics, National University of Ireland, Galway, Ireland E-mail address:[email protected]

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Singular Boundary Value Problems for Ordinary Differential Equations

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