# On the Multiplicity of Solutions for the Discrete Boundary Problem Involving the Singularϕ-Laplacian

Authors: Zihua Qiu
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In this paper, we consider the multiplicity of solutions for a discrete boundary value problem involving the singularϕ-Laplacian. In order to apply the critical point theory, we extend the domain of the singular operator to the whole real numbers. Instead, we consider an auxiliary problem associated with the original one. We show that, if the nonlinear term oscillates suitably at the origin, there exists a sequence of pairwise distinct nontrivial solutions with the norms tend to zero. By our strong maximum principle, we show that all these solutions are positive under some assumptions. Moreover, the solutions of the auxiliary problem are solutions of the original one if the solutions are appropriately small. Lastly, we give an example to illustrate our main results.


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## Body

## 1. Introduction

LetZ and R denote the sets of integers and real numbers, respectively. For a,b∈Z, define Za=a,a+1,⋯ and Za,b=a,a+1,⋯,b when a≤b.In this paper, we consider the following boundary value problem of prescribed mean curvature equations in Minkowski spaces:(1)∇Δuk1−Δuk2+λfk,uk=0,k∈Z1,T,u0=αu1,uT+1=0,where T is a given positive integer, α is a constant in 0,1, ∇ is the backward difference operator defined by ∇uk=uk−uk−1, Δ is the forward difference operator defined by Δuk=uk+1−uk, and fk,·∈CR,R for each k∈Z1,T.In 2019, Chen et al. in [1] considered problem (1) in the case where fk,x=μkxq and α=1, that is,
(2)∇Δuk1−Δuk2+λμkukq=0,k∈Z1,T,Δu0=0=uT+1.By using upper and lower solutions, the Brouwer degree theory, and Szulkin’s critical point theory for convex, lower semicontinuous perturbations ofC1-functions, the authors obtained the intervals of the parameter λ such that problem (2) has zero, one, or two positive solutions. Earlier in 2008, Bereanu and Mawhin in [2] obtained the existence of at least one or two solutions for the boundary value problems of second-order nonlinear differences with singular ϕ-Laplacian by using the Brouwer degree together with fixed point reformulations. For the existence and multiplicity of positive solutions of the associated differential problems to (1), we refer to [3, 4]. And for the boundary value problems of nonsingular differential equations, we refer to [5–9].Difference equations arise in various research fields. For the existence and multiplicity of solutions of boundary value problems of difference equations, the classical methods are fixed point theory, the method of upper and lower solution techniques, Rabinowitz’s global bifurcation theorem, etc. (see [2, 10, 11]). Since 2003, variational methods have been employed to study difference equations [12], by which various results are obtained. See, for example, periodic solutions and subharmonic solutions [13, 14], homoclinic solutions [15–22], heteroclinic solutions [23], and boundary value problems [24–27]. In recent years, boundary value problems of difference equations involving ϕ-Laplacian have aroused extensive attention from scholars; for example, in 2019, Zhou and Ling in [28] considered the following Dirichlet problem of the second-order nonlinear difference equation:
(3)−ΔϕcΔuk−1=λfk,uk,k∈Z1,T,u0=uT+1=0,where ϕc is the mean curvature operator defined by ϕcs=s/1+s2. The authors obtained the existence of infinitely many positive solutions for problem (3). The authors in [29] extended the results of [28] to the following Dirichlet problem:
(4)−ΔϕcΔuk−1+qkϕcuk=λfk,uk,k∈Z1,T,u0=uT+1=0.For the Robin problem of the second-order nonlinear difference equation,(5)−ΔφpΔuk−1+qkφpuk=λfk,uk,k∈Z1,T,Δu0=uT+1=0,where φp is a special ϕ-Laplacian operator [13] defined by φps=psp−2s/21+sp with p≥2; we refer to [30].Up to now, there is less work on the boundary value problems of difference equations involving the singularϕ-Laplcian; the known results are the existence of at least one or two solutions. The aim of this paper is to obtain the existence of infinitely many solutions for problem (1), and problem (1) contains the Dirichlet problem (when α=0) and Robin problem (when α=1) as special cases. The difficulty lies that the domain of the singular ϕ-Laplcian is a finite open interval −a,a, not −∞,+∞. The tool is a critical point result in [31]. However, we can not apply the critical point theory to the problem (1) directly, since the domain of the singular ϕ-Laplcian in (1) is −1,1; we need to extend the domain of the singular ϕ-Laplcian to −∞,+∞. Instead, we consider the auxiliary problem (20) associated with problem (1) in Section 2. We will show that solutions of problem (20) are solutions of problem (1) if the solutions are appropriately small. For general background on difference equations, we refer the reader to monographs [32, 33].This paper is organized as follows. In Section2, an auxiliary problem associated with problem (1) is established, the variational framework associated with this auxiliary problem is established, and the abstract critical point theorem is recalled. In Section 3, our main results are presented. And we also establish a strong maximum principle and obtain the existence of infinitely many positive solutions for (1) according to the oscillating behavior of f at the origin. Finally, in Section 4, an example is given to illustrate our main results.


## 2. Preliminaries

In this section, we will first introduce a lemma (Theorem 2.5 of [31]).LetX be a reflexive real Banach space, and let Iλ:X⟶R be a function satisfying the following structure hypothesis:ΛIλu≔Φu−λΨu for all u∈X, where Φ,Ψ:X⟶R are two functions of class C1 on X with Φ coercive, i.e., lim∥u∥⟶+∞Φu=+∞, and λ is a real positive parameter.IfinfXΦ<r, let
(6)μr≔infu∈Φ−1−∞,rsupv∈Φ−1−∞,rΨv−Ψur−Φu,δ≔liminfr⟶infXΦ+μr.Obviously,δ≥0. When δ=0, in the sequel, we agree to read 1/δ as +∞.Lemma 1.
Assume that the conditionΛ holds, and δ<+∞; then, for each λ∈0,1/δ, the following alternative holds: either
(a1) there is a global minimum of Φ which is a local minimum of Iλ, or
(a2) there is a sequence un of pairwise distinct critical points (local minima) of Iλ, with limn⟶+∞Φun=infXΦ, which weakly converges to the global minimum of Φ.We will use this lemma to investigate problem (1). Now, we establish the variational framework associated with problem (1). We consider the T-dimensional Banach space.
(7)S=u:Z0,T+1⟶R:u0=αu1,uT+1=0endowed with the norm
(8)u≔∑k=1TΔuk21/2.We consider another norm inS, that is,
(9)u∞≔maxuk:k∈Z1,T.For eachu∈S, there exists a τ∈Z1,T, such that
(10)u∞=uτ=∑k=τTΔuk≤∑k=1TΔuk≤T1/2u,thus,
(11)u≥1T1/2u∞.We mention that the equality in (11) holds if we let uk=T+1−kc,k∈Z1,T,u0=αTc, where c is a nonzero constant. In fact, in this case, u=T1/2c and u∞=Tc.We notice that the singular operatorϕs=s/1−s2 in problem (1) only defined for s∈−1,1. In order to use Lemma 1, we need to extend the domain of the singular operator ϕ to −∞,∞. Take
(12)gs=s1−s2,s≤32,2s,s>32.Then,gs is continuous in −∞,+∞, and the primary function of g is given by
(13)Gs=s1+1−s2,s≤32,s2−14,s>32.We define(14)Φu=∑k=0TGΔuk+αGΔu01−α,Ψu=∑k=1TFk,uk,for each u∈S, where Fk,u=∫0ufk,τdτ for every k∈Z1,T. When α=1, we read αGΔu0/1−α as 0 in (14), since
(15)limα⟶1αGΔu01−α=limα⟶1α1−α2u121−α1+1−1−α2u12=0.Put(16)Iλu=Φu−λΨu,for u∈S. Then, Φ and Ψ are two functionals of class C1S,R whose Gâteaux derivatives at the point u∈S are given by
(17)Φ′uv=∑k=0TgΔukΔvk+αgΔu0Δv01−α,Ψ′uv=∑k=1Tfk,ukvk,for all u,v∈S. It is clear that
(18)∑k=0TgΔukΔvk=∑k=0TgΔukvk+1−∑k=0TgΔukvk=∑k=1TgΔuk−1vk−∑k=1TgΔukvk−gΔu0v0=−∑k=1T∇gΔukvk−αgΔu0v1=−∑k=1T∇gΔukvk−αgΔu0Δv01−α,then,
(19)Φ′u−λΨ′uv=−∑k=1T∇gΔuk+λfk,ukvk.Consequently, the critical points ofIλ in S are exactly the solutions of the following boundary value problem:
(20)∇gΔuk+λfk,uk=0,k∈Z1,T,u0=αu1,uT+1=0.Remark 2.
Ifu∈S is a solution of problem (20) with Δuk≤3/2 for k∈Z0,T, then, u is a solution of problem (1).


## 3. Main Results

First, we consider the existence of nontrivial solutions for problem (1). Let
(21)B≔limsupt⟶0+∑k=1TFk,tt2.We have the following:Theorem 3.
Assume that there exist two real sequencesun and vn, with vn>0 and limn⟶+∞vn=0, such that
(22)un21+1−un2+1−αun21+1−1−α2un2<vn22T,n∈Z1,(23)γ≔liminfn⟶∞∑k=1Tmaxt≤vnFk,t−∑k=1TFk,unvn2/2T−un2/1+1−un2−1−αun2/1+1−1−α2un2<2B2−α.
Then, for eachλ∈2−α/2B,1/γ, problem (1) admits a sequence of nontrivial solutions which converges to zero.Remark 4.
A sequenceun in S is said to converge to zero if un⟶0 as n⟶∞.
Proof. TakeΦ and Ψ as defined by (14); we will prove Theorem 3 by using Lemma 1. Since lims⟶+∞Gs=lims⟶+∞s2−1/4=+∞, it is easy to see that limu⟶+∞Φu=+∞, and Λ is satisfied. Put
(24)rn=vn22T.
Sincelimn⟶∞vn=0, there is no harm in assuming that vn≤T; then, rn≤1/2. If u∈S and
(25)Φu=∑k=0TGΔuk+αGΔu01−α<rn,then, GΔuk<1/2 for k∈Z0,T. Noting that Gs≥1/2 for s≥3/2, we see that Δuk<3/2 for k∈Z0,T and Φu takes the form
(26)Φu=∑k=0TΔuk21+1−Δuk2+α1−αu121+1−1−α2u12.
Therefore,(27)12∑k=1TΔuk2≤Φu<rn,which implies that u<2rn. By (11), we have
(28)u∞<2Trn=vn.
According to the definition ofμ, we have
(29)μrn≤infu∈Φ−1−∞,rn∑k=1Tmaxt≤vnFk,t−∑k=1TFk,ukvn2/2T−Φu.
For eachn∈Z1, let wn∈S be defined by wnk=un for every k∈Z1,T and wn0=αwn1,wnT+1=0. Then,
(30)Φwn=un21+1−un2+1−αun21+1−1−α2un2<vn22T=rn,by using (22). Thus,
(31)μrn≤∑k=1Tmaxt≤vnFk,t−∑k=1TFk,wnkvn2/2T−Φwn=∑k=1Tmaxt≤vnFk,t−∑k=1TFk,unvn2/2T−un2/1+1−un2−1−αun2/1+1−1−α2un2.
Therefore, by (23), we know that δ≤liminfn⟶+∞μrn≤γ<+∞.To get our results, we need to show that conclusion (a2) of Lemma 1 holds. Therefore, we want to show that the global minimum u≡0 of Φ is not a local minimum of Iλ. To prove this, we consider two cases: B=+∞ and B<+∞. In the case where B=+∞, let tn be a sequence of positive numbers, with tn∈0,3/2 and limn⟶+∞tn=0, such that
(32)∑k=1TFk,tn≥3tn2λ,n∈Z1.Defining a sequenceθn in S by θnk=tn for k∈Z1,T and θn0=αθn1,θnT+1=0, we have
(33)Iλθn=tn21+1−tn2+1−αtn21+1−1−α2tn2−λ∑k=1TFk,tn=−1+αtn2<0.In the case whereB<+∞, since λ>2−α/2B, we can choose a small number ε∈0,3/2 such that
(34)11+1−ε2+1−α1+1−1−α2ε2<λB−ε.By the definition ofB, we can find a sequence of real numbers τn with τn∈0,ε such that limn⟶+∞τn=0 and
(35)∑k=1TFk,τn≥B−ετn2.Defining a sequenceξn in S by ξnk=τn for k∈Z1,T and ξn0=αξn1,ξnT+1=0, we have
(36)Iλξn=τn21+1−τn2+1−ατn21+1−1−α2τn2−λ∑k=1TFk,τn≤11+1−ε2+1−α1+1−1−α2ε2−λB−ετn2<0.Noticing thatIλ0=0, we see that u≡0 is not a local minimum of Iλ by combining the above two cases. Therefore, by Lemma 1 and Remark 2, we know the conclusion of Theorem 3 holds.Now, let(37)B∗=liminft⟶0+∑k=1Tmaxs≤tFk,st2.Then, there exists a sequencevn of positive numbers with limn⟶+∞vn=0 such that
(38)limn⟶∞∑k=1Tmaxt≤vnFk,tvn2=B∗.Takingun=0 for all n∈Z1, by Theorem 3, we get the following corollary.Corollary 5.
If(39)TB∗<B2−α.
Then, for eachλ∈2−α/2B,1/2TB∗, problem (1) admits a sequence of nontrivial solutions which converges to zero.To obtain the positive solutions of problem (20), we need the following strong maximum principle.Theorem 6.
Assumeu∈S such that either
(40)uk>0or∇gΔuk≤0,for all k∈Z1,T. Then, either uk>0 for all k∈Z1,T or u≡0.Proof. There existsτ∈Z1,T such that
(41)uτ=minuk:k∈N1,T.Ifuτ>0, then uk>0 for all k∈Z1,T and the proof is complete.Ifuτ≤0, then uτ=minuk:k∈N0,T+1. Because Δuτ−1=uτ−uτ−1≤0 and Δuτ=uτ+1−uτ≥0, gs is increasing in s, and g0=0, we have
(42)gΔuτ≥0≥gΔuτ−1.On the other hand, letk=τ; (40) implies
(43)gΔuτ≤gΔuτ−1.By combining (42) with (43), we get gΔuτ=0=gΔuτ−1. That is uτ+1=uτ−1=uτ. If τ+1=T+1, we have uτ=0. Otherwise, τ+1∈N1,T. Replacing τ by τ+1, we get uτ+2=uτ+1. Continuing this process T+1−τ times, we have uτ+j=0 for j∈Z0,T+1−τ. Similarly, we have uj=0 for j∈Z0,τ. Thus, u≡0 and the proof is complete.Now, we are ready to establish the existence of positive solutions for problem (1); we haveCorollary 7.
Iffk,0≥0 for all k∈Z1,T,
(44)A∗≔liminft⟶0+2T∑k=1Tmax0≤s≤t∫0sfk,xdxt2<2B2−α.
Then, for eachλ∈2−α/2B,1/A∗, problem (1) admits a sequence of positive solutions which converges to zero.
Proof. Put(45)f∗k,x=fk,x,ifx>0,fk,0,ifx≤0.Noticing thatfk,0≥0, we see that
(46)max0≤s≤t∫0sf∗k,xdx=max0≤s≤t∫0sfk,xdx,for all t≥0. By Corollary 5, we know that problem (1) with f replaced by f∗ admits a sequence of nontrivial solutions which converges to zero for each λ∈2−α/2B,1/A∗. And by Theorem 6, we know that all these solutions are positive.


## 4. An Example

In this section, we give an example to illustrate our main results.Example 8.
Consider the boundary value problem (1) with
(47)fk,x=fx=x2+2ε+2sinεlnx+εcosεlnx,x≠0,0,x=0,for k∈Z1,T. Then,
(48)Fk,x=Fx=∫0xfsds=x21+ε+sinεlnx,forx>0.
Sincefx≥0 for x≥0, we see that Fx is increasing in x∈0,+∞. Thus,
(49)A∗≔liminft⟶0+2T∑k=1Tmax0≤s≤t∫0sfk,xdxt2=liminft⟶0+2T21+ε+sinεlnt=2T2ε,B≔limsupt⟶0+∑k=1TFk,tt2=limsupt⟶0+T1+ε+sinεlnt=T2+ε.
Letε∈0,2/2T−αT−1; then (44) holds. By Corollary 7, for each λ∈2−α/2T2+ε,1/2T2ε, problem (1) admits a sequence of positive solutions which converges to zero.


## 5. Conclusions

In this paper, we consider a discrete boundary value problem involving the singularϕ-Laplacian. The problem contains the Dirichlet problem (when α=0) and Robin problem (when α=1) as special cases. Since the domain of the singular operator ϕ is −1,1, we can not apply the critical point theory to this problem directly. Therefore, we extend the domain of the singular operator to the whole real numbers and consider an auxiliary problem associated with the original one. The conditions for the multiplicity of positive solutions of the discrete boundary problem are found, and an illustrative example is given. The method in this paper provides a new way to discuss the boundary value problems containing a singular Laplacian.



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