In this paper, we obtain some results for the existence of infinitely many positive solutions for a coupled discrete boundary value problem. The approach is based on variational methods.

National Natural Science Foundation of China11571084Program for Changjiang Scholars and Innovative Research Team in UniversityIRT-16R161. Introduction

Let Z and R denote the sets of integers and real numbers, respectively. For a,b∈Z, define [a,b]={a,a+1,…,b}, when a≤b.

In this paper, we consider the following coupled discrete boundary value problem:(1)-ΔpkΔuik-1+qkuik=λFuik,u1k,…,umk,ui0=uiN+1=0,for all k∈[1,N], N∈Z+, where 1≤i≤m, and λ is a positive parameter, Δui(k)=ui(k+1)-ui(k) is the forward difference operator, p(k)>0 for all k∈[1,N+1], and q(k)≥0 for all k∈[1,N], F(k,·,…,·) is a C1-function in Rm satisfying F(k,0,…,0)=0 for every k∈[1,N], and Fui denotes the partial derivative of F(k,u1,u2,…,um) with respect to ui for i=1,…,m.

As we know, results for existence of solutions for difference equations have been widely studied because of their applications to various fields of applied sciences, like mechanical engineering, control systems, computer science, economics, artificial or biological neural networks, and many others. Many scholars have studied such problems and main tools are fixed point methods, Brouwer degree theory, and upper and lower solution techniques; see [1–4] and references therein. In recent years, variational methods have been employed to study difference equation and various results have been obtained. See, for instance, [5–22].

More recently, especially, in [23–31], by starting from the seminal papers [32, 33], many results for the existence and multiplicity of solutions for discrete boundary value problems have been obtained also by adopting variational methods.

However, these papers only deal with a single equation. For instance, in [23], by studying the Dirichlet discrete boundary value problem(2)-Δ2uk-1=λfkuk,k∈1,N,u0=uN+1=0,the author obtained the existence of two positive solutions of the problem through appropriate variational methods. In this paper, we consider system (1) with m difference equations by using Ricceri’s variational principle proposed in [33]. In Theorem 4 and Remark 5, we prove the existence of an interval (λ1,λ2) such that, for each λ∈(λ1,λ2), problem (1) admits a sequence of positive solutions which is unbounded in X. To the best of our knowledge, this is the first time to deal with coupled discrete boundary value problems. This method has already been used for the continuous counterparts [34–36]. The monographs [37–39] are related books of the critical point theory and difference equations.

The rest of the paper is organized as follows: Section 1 consists of some definitions and mathematical symbols. In Section 2, we emphasize that a strong maximum principle (Lemma 1) is presented so that if Fui(k,u1,…,ui-1,0,ui+1,…,um)≥0, for all k∈[1,N] and i=1,…,m, our results guarantee the existence of infinitely many positive solutions (Remark 5 in Section 3). Section 3 contains a more precise version (Lemma 3) of Ricceri’s variational principle, the statements and proofs of the main results (Theorem 4 and Remark 5), two corollaries (Corollaries 6 and 7), and an example (Example 10).

Throughout this paper, we let S={u:[0,N+1]→R:u(0)=u(N+1)=0} and X be the Cartesian product of m Banach spaces S,…,S; i.e., (3)X≔S×⋯×S︸mendowed with the norm(4)u=u1,…,um≔∑i=1mui∗21/2,where (5)v∗≔∑k=1N+1pkΔvk-12+∑k=1Nqkvk21/2,forv∈S,which is a norm in S.

Put (6)A≔liminfy→+∞∑k=1Nmaxt1,…,tm∈KyFk,t1,…,tmy2,where (7)Ky≔t1,…,tm∈Rm:∑i=1mti≤y,for every y∈R+, and (8)B≔limsup∑i=1mti2→+∞t1,…,tm∈R+m∑k=1NFk,t1,…,tm∑i=1mti2,where R+m={(t1,…,tm)∈Rm:ti≥0 for all 1≤i≤m}.

2. Preliminaries

First, we establish a strong maximum principle.

Lemma 1.

Fix u∈S such that either u(k)>0 or(9)ifuk≤0then-ΔpkΔuk-1+qkuk≥0.Then either u>0 in [1,N] or u≡0.

Proof.

Let j∈[1,N] such that(10)uj=minuk:k∈1,N.If u(j)>0, then it is clear that u>0 in [1,N].

If u(j)≤0, then by (9), we have(11)-ΔpjΔuj-1≥-Δqjuj≥0,that is (12)pj+1Δuj≤pjΔuj-1.On the other hand, by the definition of u(j), we see that(13)Δuj-1≤0,Δuj≥0.Thus, by (12), we obtain that u(j+1)=u(j)=u(j-1).

By similar arguments applied to u(j-1) and u(j+1) and continuing in this way, we have u(k)=u(N+1)=u(0)=0,∀k∈[1,N].

Remark 2.

Let F:[1,N]×Rm→R be such that Fti(k,t1,…,ti-1,0,ti+1,…,tm)≥0 for all k∈[1,N] and i=1,…,m.

Put(14)Fti∗k,t1,…,tm=Ftik,t1,…,tm,ifti>0,Ftik,t1,…,ti-1,0,ti+1,…,tm,ifti≤0. Clearly, Fti∗(k,·) is continuous in Rm for each k∈[1,N]. Owing to Lemma 1, all solutions of problem(15)-ΔpkΔuik-1+qkuik=λFui∗k,u1k,…,umk,ui0=uiN+1=0,are either zero or positive and hence are also solutions for problem (1). Hence we emphasize that when (15) admits nontrivial solutions, then problem (1) admits positive solutions, independently of the sign of Fui.

3. Main results

Let X be a reflexive real Banach space and let Iλ:X→R be a function satisfying the following structure hypothesis:

(Λ)Iλ(u)≔Φ(u)-λΨ(u) for all u∈X, where Φ,Ψ:X→R are two functions of class C1 on X with Φ coercive; i.e., limu→+∞Φ(u)=+∞, and λ is a real positive parameter.

Provided that infXΦ<r, put(16)φr≔infu∈Φ-1-∞,rsupv∈Φ-1-∞,rΨv-Ψur-Φu, and(17)γ≔liminfr→+∞φr. Clearly, γ≥0. When γ=0, in the sequel, we agree to read 1/γ as +∞.

For the readers’ convenience, we recall a more precise version of Theorem 2.1 of [32] (see also Theorem 2.5 of [33]) which is the main tool used to investigate problem (1).

Lemma 3.

Assume that condition (Λ) holds. If γ<+∞, then, for each λ∈(0,1/γ), the following alternative holds: either

(a1)Iλ possesses a global minimum, or

(a2)there is a sequence {un} of critical points (local minima) of Iλ such that (18)limn→+∞Φun=+∞

Put (19)P≔p1+pN+1,Q≔∑k=1Nqk,p∗≔minpk:k∈1,N+1.Our main result is the following theorem.

Theorem 4.

Assume that

(i) F is nonnegative in [1,N]×R+m

(ii) A<p∗/m(P+Q)(N+1)B, where A and B are given by (6) and (8), respectively

Then, for each λ∈(P+Q/2B,p∗/2m(N+1)A), system (1) admits an unbounded sequence of solutions.

Proof.

For each u=(u1(k),…,um(k))∈X, put (20)Φu≔12u2,Ψu≔∑k=1NFk,u1k,…,umk, and (21)Iλu≔Φu-λΨu.Standard arguments show that Iλ∈C1(X,R) and that critical points of Iλ are exactly the solutions of problem (1). In fact, Φ,Ψ∈C1(X,R); that is, Φ and Ψ are continuously Fréchet differentiable in X. Using the summation by parts formula and the fact that ui(0)=ui(N+1)=0 for any u∈X, we get (22)Φ′uv=limt→0Φu+tv-Φut=∑k=1N+1∑i=1mpkΔuik-1Δvik-1+∑k=1N∑i=1mqkuikvik=∑i=1mpN+1uiNviN+∑i=1mpkΔuik-1vik-11N+1-∑k=1N∑i=1mΔpkΔuik-1vik+∑k=1N∑i=1mqkuikvik=-∑k=1N∑i=1mΔpkΔuik-1vik-qkuikvikand (23)Ψ′uv=limt→0Ψu+tv-Ψut=∑k=1N∑i=1mFuik,u1k,…,umkvik,for any u,v∈X.

Moreover, we obtain (24)Iλ′uv=∑k=1N∑i=1m-ΔpkΔuik-1+qkuik-λFuik,u1k,…,umkvik,for all u,v∈X.

Now we verify that γ<+∞. Let {bn} be a real sequence such that limn→+∞bn=+∞, and(25)limn→+∞∑k=1Nmaxt1,…,tm∈KbnFk,t1,…,tmbn2=A.It follows from [24] that(26)supk∈1,Nuik≤N+11/22p∗1/2ui∗,for 1≤i≤m. And put rn≔2p∗bn2/m(N+1) for all n∈Z+. Hence a computation ensures that ∑i=1m|ui(k)|≤bn whenever u∈Φ-1(-∞,rn).

Taking into account the fact that ∑k=1NF(k,0,…,0)=0, one has (27)φrn=infu∈Φ-1-∞,rnsupv∈Φ-1-∞,rnΨv-Ψurn-Φu=inf∑i=1mui∗2/2<rnsup∑i=1mvi∗2/2<rn∑k=1NFk,v1k,…,vmk-∑k=1NFk,u1k,…,umkrn-∑i=1mui∗2/2≤sup∑i=1mvi∗2/2<rn∑k=1NFk,v1k,…,vmkrn≤mN+12p∗∑k=1Nmaxt1,…,tm∈KbnFk,t1,…,tmbn2. Therefore, since from assumption (ii) one has A<+∞, we obtain(28)γ≤liminfn→+∞φrn≤mN+12p∗A<+∞.

Now fix λ∈(P+Q/2B,p∗/2m(N+1)A). We claim that Iλ is unbounded from below. Let {ξi,n} be m positive real sequences such that limn→+∞(∑i=1mξi,n2)=+∞, and(29)limn→+∞∑k=1NFk,ξ1,n,…,ξm,n∑i=1mξi,n2=B,for all n∈Z+.

For each n∈Z+, let ωi,n(k)≔ξi,n for all k∈[1,N],ωi,n(0)=ωi,n(N+1)=0.

Clearly, ωn=(ω1,n,…,ωm,n)∈X, and (30)ωi,n∗2=∑k=1N+1pkΔωi,nk-12+∑k=1Nqkωi,nk2=p1ωi,n1-ωi,n02+⋯+pN+1ωi,nN+1-ωi,nN2+∑k=1Nqkξi,n2=p1+pN+1+∑k=1Nqkξi,n2=P+Qξi,n2. Therefore, we have (31)Φωn-λΨωn=∑i=1mωi,n∗22-λ∑k=1NFk,ξ1,n,…,ξm,n=P+Q2∑i=1mξi,n2-λ∑k=1NFk,ξ1,n,…,ξm,n, for all n∈Z+.

If B<+∞, let ε∈P+Q/2λB,1. By (29) there exists Nε such that(32)∑k=1NFk,ξ1,n,…,ξm,n>εB∑i=1mξi,n2, for all n>Nε. Moreover, (33)Φωn-λΨωn<P+Q2∑i=1mξi,n2-λεB∑i=1mξi,n2=P+Q2-λεB∑i=1mξi,n2, for all n>Nε. Taking into account the choice of ε, we have(34)limn→+∞Φωn-λΨωn=-∞. If B=+∞, let us consider M>P+Q/2λ. By (29) there exists NM such that(35)∑k=1NFk,ξ1,n,…,ξm,n>M∑i=1mξi,n2, for all n>NM. Moreover, (36)Φωn-λΨωn<P+Q2∑i=1mξi,n2-λM∑i=1mξi,n2=P+Q2-λM∑i=1mξi,n2, for all n>NM. Taking into account the choice of M, in this case we also have(37)limn→+∞Φωn-λΨωn=-∞. Due to Lemma 3, for each λ∈P+Q/2B,p∗/2m(N+1)A, the functional Iλ admits an unbounded sequence of critical points, and the conclusion is proven.

Remark 5.

When Fti(k,t1,…,ti-1,0,ti+1,…,tm)≥0 for all k∈[1,N] and i=1,…,m, owing to Remark 2, the solutions in the conclusion of Theorem 4 are positive.

It is interesting to list some special cases of the above results.

Corollary 6.

Assume that

(i′)F is nonnegative in [1,N]×R+m

(ii′)A<B/m(2+Q)(N+1)

Then, for each λ∈(2+Q/2B,1/2m(N+1)A), the system(38)-Δ2uik-1+qkuik=λFuik,u1k,…,umk,k∈1,N,ui0=uiN+1=0, for 1≤i≤m, admits an unbounded sequence of solutions.

Corollary 7.

Let F:Rm→R be a C1-function and assume that

(i′′)F is nonnegative in R+m

(ii′′)C<p∗/m(P+Q)(N+1)D,

where (39)C≔liminfy→+∞maxt1,…,tm∈KyFt1,…,tmy2and (40)D≔limsup∑i=1mti2→+∞t1,…,tm∈R+mFt1,…,tm∑i=1mti2.Then, for each λ∈(P+Q/2D,p∗/2m(N+1)C), the system(41)-ΔpkΔuik-1+qkuik=λFuiu1k,…,umk,k∈1,N,ui0=uiN+1=0, for 1≤i≤m, admits an unbounded sequence of solutions.

Remark 8.

When Fti(k,t1,…,ti-1,0,ti+1,…,tm)≥0 for all k∈[1,N] and i=1,…,m, owing to Remark 2, the solutions in the conclusion of Corollary 6 are positive.

Remark 9.

When Fti(t1,…,ti-1,0,ti+1,…,tm)≥0 for i=1,…,m, owing to Remark 2, the solutions in the conclusion of Corollary 7 are positive.

Now we give an example to illustrate our results.

Example 10.

Let m=2,p(k)=q(k)=1 and consider the increasing sequence of positive real numbers given by (42)an≔2nn+1/2, for every n∈Z+.

Define the function F:R2→R as follows: If (t1,t2)∈B((an,an),1) for some positive integer n, then(43)Ft1,t2=an21-t1-an2-t2-an22; otherwise, (44)Ft1,t2=0, where B((an,an),1) denotes the open unit ball of center (an,an).

By the definition of F, we see that it is nonnegative in R+2 and F(0,0)=0. Further it is a simple matter to verify that F∈C1(R2). We will denote by Ft1(t1,t2) and Ft2(t1,t2), respectively, the partial derivative of F(t1,t2) with respect to t1 and t2. Now, for every n∈Z+, the restriction F(t1,t2)|B((an,an),1) attains its maximum in (an,an) and one has F(an,an)=(an)2. Obviously,(45)limsupt12+t22→+∞t1,t2∈R+2Ft1,t2t12+t22=12,owing to the fact that(46)limn→+∞Fan,an2an2=12. On the other hand, by setting yn=an+1-1 for every n∈Z+, one has (47)maxt1,t2∈KynFt1,t2=an2,∀n∈Z+.Then(48)limn→+∞maxt1,t2∈KynFt1,t2yn2=0,and hence(49)liminfy→+∞maxt1,t2∈KyFt1,t2y2=0.Finally(50)0=C<12N+2N+1D=14N+2N+1.

The previous observations and computations ensure that all the hypotheses of Corollary 7 are satisfied. Then, for each λ∈(N+2,+∞), the problem(51)-Δ2u1k-1+u1k=λFu1u1k,u2k,k∈1,N,-Δ2u2k-1+u2k=λFu2u1k,u2k,k∈1,N,u10=u1N+1=u20=u2N+1=0,admits an unbounded sequence of solutions.

Taking partial derivative to F(t1,t2) with respect to t1 gives(52)Ft1t1,t2=-4t1-anan21-t1-an2-t2-an2, if (t1,t2)∈B((an,an),1) for some positive integer n; otherwise, Ft1(t1,t2)=0.

It is easy to see that(53)Ft10,t2=0.In a similar way, we obtain(54)Ft2t1,0=0.Consequently, according to Remark 9, problem (51) admits an unbounded sequence of positive solutions.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Authors’ Contributions

Both authors contributed equally to the writing of this paper. Both authors read and approved the final manuscript.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grant No. 11571084) and the Program for Changjiang Scholars and Innovative Research Team in University (Grant No. IRT-16R16).

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