Positive solutions of the discrete Dirichlet problem involving the mean curvature operator

Jiaoxiu Ling; Zhan Zhou

doi:10.1515/math-2019-0081

Article Open Access

Positive solutions of the discrete Dirichlet problem involving the mean curvature operator

Jiaoxiu Ling and Zhan Zhou

Published/Copyright: September 14, 2019

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$Open Mathematics$

From the journal Open Mathematics Volume 17 Issue 1

Abstract

In this paper, by using critical point theory, we obtain some sufficient conditions on the existence of infinitely many positive solutions of the discrete Dirichlet problem involving the mean curvature operator. We show that the suitable oscillating behavior of the nonlinear term near at the origin and at infinity will lead to the existence of a sequence of pairwise distinct nontrivial positive solutions. We also give two examples to illustrate our main results.

Keywords: discrete Dirichlet problem; mean curvature operator; infinitely many positive solutions; critical point theory

MSC 2010: 39A11

1 Introduction

Denote Z and R the sets of integers and real numbers, respectively. For a, b ∈ Z, define Z(a) = {a, a + 1, ⋯}, and Z(a, b) = {a, a + 1, ⋯, b} when a ≤ b.

In this paper, we consider the following Dirichlet problem of the second order nonlinear difference equation

−△ϕc(△uk−1)+qkϕc(uk)=λf(k,uk),k∈Z(1,T),u0=uT+1=0, (1.1)

where T is a given positive integer, △ is the forward difference operator defined by △ u_k = u_k+1 − u_k, △² u_k = △(△ u_k), q_k ≥ 0 for all k ∈ Z(1, T), ϕ_c is the mean curvature operator defined by ϕ_c (s) = s1+s2 [1], λ is a real positive parameter, and f(k, ⋅) ∈ C(R, R) for each k ∈ Z(1, T).

When q_k ≡ 0, problem (1.1) may be regarded as the discrete analog of the following one-dimensional prescribed curvature problem

−ϕc(u′)′=λf(t,u),t∈R,u(0)=u(1)=0. (1.2)

In 2007, based on the variational methods and a regularization of the action functional associated with the curvature problem, Bonheure etc. in [2] obtained the existence and multiplicity of positive solutions of (1.2) according to the behaviour near at the origin and at infinity of the potential ∫0u f(t, s)ds. In [3], Bonanno, Livrea and Mawhin obtained an explicit interval Λ of positive parameters, such that, for every λ ∈ Λ, problem (1.2) admits at least one non trivial nonnegative solution u_λ. For the corresponding case of higher dimensions to problem (1.2), we refer to [4].

Difference equations arise in various research fields. Many authors have discussed the existence and multiplicity of solutions for difference problems by using fixed point theory, the method of upper and lower solution techniques, Rabinowitz’s global bifurcation theorem etc., see [5, 6, 7]. Since 2003, critical point theory has been employed to study difference equations [8], by which various results have been obtained. See, for example, [9, 10, 11, 12, 13, 14, 15, 16, 17, 18]. In recent years, variational methods have been used to study the boundary value problems of difference equations [19, 20, 21, 22, 23]. For example, in [21], the authors considered the discrete Dirichlet problem

−△2uk−1=λf(k,uk),k∈Z(1,T),u0=uT+1=0. (1.3)

By using critical point theory, the authors obtained the existence of at least two positive solutions for (1.3). In [22], the authors extended the results of [21] to the following discrete boundary value problem with p-Laplacian

−△ϕp(△uk−1)+qkϕp(uk)=λf(k,uk),k∈Z(1,T),u0=uT+1=0. (1.4)

While the existence results of infinitely many solutions of (1.4) were also established in [20]. Very recently, the authors in [23] considered the existence of positive solutions of (1.1) for the special case q_k ≡ 0 according to the behavior of f at infinity.

Compared with differential equations, there is less work on the boundary value problems of difference equations involving the mean curvature operator. In this paper, we will consider the existence of infinitely many positive solutions of (1.1) by means of a critical point result in [24], see also [25]. The results show that the suitable oscillating behavior of the nonlinear term f near at the origin and at infinity will lead to the existence of a sequence of pairwise distinct nontrivial positive solutions for problem (1.1). We refer the reader to monographs [26, 27] for the general background on difference equations.

This paper is organized as follows. In section 2, the variational framework associated with (1.1) is established, and the abstract critical point theorem is recalled. In section 3, our main results are presented. In particular, we establish a strong maximum principle and obtain the existence of infinitely many positive solutions for (1.1) according to the oscillating behavior of f near at the origin and at infinity, respectively. Finally, in section 4, we present two examples to illustrate our main results.

2 Preliminaries

In this section, we first establish the variational framework associated with (1.1). We consider the T-dimensional Banach space S = {u : Z(0, T + 1) → R : u₀ = u_T+1 = 0} endowed with the norm

∥u∥:=∑k=0T△uk212.

Define

Φ(u)=∑k=0T1+(△uk)2−1+∑k=1Tqk1+uk2−1,Ψ(u)=∑k=1TF(k,uk), (2.1)

for every u ∈ S, where F(k, u) = ∫0u f(k, τ) dτ is the primary function of f(k, u) with F(k, 0) = 0 for each k ∈ Z(1, T). Let

Iλ(u)=Φ(u)−λΨ(u),

for any u ∈ S. It is clear that Φ and Ψ are two functionals of class C¹ (S, R) whose Gâteaux derivatives at the point u ∈ S are given by

Φ′(u)(v)=∑k=0Tϕc(△uk)△vk+∑k=1Tqkϕc(uk)vk,

and

Ψ′(u)(v)=∑k=1Tf(k,uk)vk,

for all u, v ∈ S. Since

∑k=0Tϕc(△uk)△vk=∑k=0Tϕc(△uk)vk+1−∑k=0Tϕc(△uk)vk=∑k=1Tϕc(△uk−1)vk−∑k=1Tϕc(△uk)vk=−∑k=1T△(ϕc(△uk−1))vk, (2.2)

then,

[Φ′(u)−λΨ′(u)](v)=∑k=1T−△(ϕc(△uk−1))+qkϕc(uk)−λf(k,uk)vk.

Consequently, the critical points of I_λ in S are exactly the solutions of the problem (1.1).

Assume that X is a reflexive real Banach space and I_λ : X → R is a function which satisfies the following structure hypothesis:

(Λ) Assuming that Φ, Ψ : X → R are two functions of class C¹ on X with Φ coercive, i.e. lim_{∥u∥→+∞} Φ(u) = +∞. Let I_λ(u) := Φ(u) − λΨ(u) for each u ∈ X, where λ is a real positive parameter.

If inf_X Φ < r, let

φ(r):=infu∈Φ−1(−∞,r)(supv∈Φ−1(−∞,r)Ψ(v))−Ψ(u)r−Φ(u),

and

δ:=lim infr→(infXΦ)+φ(r),γ:=lim infr→+∞φ(r).

Clearly, δ ≥ 0 and γ ≥ 0. When δ = 0 (or γ = 0), in the sequel, we agree to read 1δ (or 1γ ) as +∞.

The following lemma comes from Theorem 7.4 of [24] and will be used to investigate problem (1.1).

Lemma 2.1

Assume that the condition (Λ) holds. We have

If δ < + ∞, then, for each λ ∈ (0, 1δ ), the following alternative holds: either
there is a global minimum of Φ which is a local minimum of I_λ, or
there is a sequence {u_n} of pairwise distinct critical points (local minima) of I_λ, with lim_n→+∞ Φ(u_n) = inf_X Φ, which weakly converges to a global minimum of Φ.
If γ < +∞, then, for each λ ∈ (0, 1γ ), the following alternative holds: either
I_λ possesses a global minimum, or
there is a sequence {u_n} of critical points (local minima) of I_λ such that limn→+∞ Φ(u_n) = +∞.

3 Main results

Put

B:=lim supt→0+∑k=1TF(k,t)t2,D:=lim supt→+∞∑k=1TF(k,t)t,q∗:=min{qk:k∈Z(1,T)},Q:=∑1Tqk.

First, if considering the oscillating behavior of f near at the origin, we have

Theorem 3.1

Assume that there exist two real sequences {a_n} and {b_n}, with b_n > 0 and lim_n→+∞ b_n = 0, such that

(2+Q)1+an2−1<(1+q∗)1+min4T+1,1bn2−1, (3.1)

for n ∈ Z(1), and

A:=lim infn→∞∑k=1Tmax|t|≤bnF(k,t)−∑k=1TF(k,an)(1+q∗)1+min4T+1,1bn2−1−(2+Q)1+an2−1<2B2+Q. (3.2)

Then, for each λ∈(2+Q2B,1A), problem (1.1) admits a sequence of nontrival solutions which converges to zero.

Proof

To prove Theorem 3.1, we will need to use Lemma 2.1. Firstly, (Λ) is clearly satisfied. Put

rn=(1+q∗)1+min4T+1,1bn2−1.

Fix u ∈ S such that Φ(u) = Φ₁(u) + Φ₂(u) < r_n, where

Φ1(u)=∑k=0T1+(△uk)2−1,Φ2(u)=∑k=1Tqk1+uk2−1.

Put

vk=1+(△uk)2−1

for each k ∈ Z(0, T). Then ∑k=0Tvk=Φ1(u) and

∑k=0T(△uk)2=∑k=0T(vk2+2vk)≤∑k=0Tvk2+2∑k=0Tvk=Φ12(u)+2Φ1(u),

which implies that

Φ1(u)≥∥u∥2+1−1.

Noticing that

∥u∥∞≤T+12∥u∥,

by Lemma 2.2 in [28], where

∥u∥∞:=max{|uk|:k∈Z(1,T)},

for u ∈ S. Thus, we have

Φ1(u)≥4T+1∥u∥∞2+1−1.

It is clear that

Φ2(u)≥q∗∥u∥∞2+1−1.

Therefore

(1+q∗)1+min4T+1,1∥u∥∞2−1≤Φ1(u)+Φ2(u)<rn,

which implies that

∥u∥∞2<maxT+14,1rn1+q∗2+2rn1+q∗=bn2.

By the definition of φ, we have

φ(rn)≤infu∈Φ−1(−∞,rn)∑k=1Tmax|t|≤bnF(k,t)−∑k=1TF(k,uk)(1+q∗)1+min4T+1,1bn2−1−Φ(u).

For each n ∈ Z(1), let w_n ∈ S given by (w_n)_k = a_n for each k ∈ Z(1, T), and (w_n)₀ = (w_n)_T+1 = 0. Then, by using (3.1),

Φ(wn)=(2+Q)(1+an2−1)<rn.

Thus,

φ(rn)≤∑k=1Tmax|t|≤bnF(k,t)−∑k=1TF(k,(wn)k)(1+q∗)1+min4T+1,1bn2−1−Φ(wn)=∑k=1Tmax|t|≤bnF(k,t)−∑k=1TF(k,an)(1+q∗)1+min4T+1,1bn2−1−(2+Q)1+an2−1. (3.3)

Therefore, by (3.2), we know that γ ≤ lim inf_n→+∞ φ(r_n) ≤ A < +∞.

Clearly, u ≡ 0 is a global minimum of Φ. In order to get the conclusion (a₂), we need to prove that u ≡ 0 is not a local minimum of I_λ. To prove this, we consider two cases: B = +∞ and B < +∞. If B = +∞, let {c_n} be a sequence of positive numbers, with lim_n→+∞ c_n = 0, such that

∑k=1TF(k,cn)≥(2+Q)cn2λ,forn∈Z(1).

Defining a sequence {ω_n} in S by (ω_n)_k = c_n for each k ∈ Z(1, T) and (ω_n)₀ = (ω_n)_T+1 = 0, we have

Iλ(ωn)=(2+Q)1+cn2−1−λ∑k=1TF(k,cn)≤2+Q2cn2−(2+Q)cn2=−2+Q2cn2<0.

If B < +∞, since λ > 2+Q2B , we can choose ε₀ > 0 such that

2+Q−2λ(B−ϵ0)<0.

Then we can find a sequence of positive numbers {c_n} satisfying lim_n→+∞ c_n = 0 and

(B−ϵ0)cn2≤∑k=1TF(k,cn)≤(B+ϵ0)cn2.

Let the sequence {ω_n} in S be the same as the case where B = +∞, we get

Iλ(ωn)=(2+Q)1+cn2−1−λ∑k=1TF(k,cn)≤2+Q−2λ(B−ϵ0)2cn2<0.

Since I_λ(0) = 0, by combining the above two cases, we see that u ≡ 0 is not a local minimum of I_λ and by Lemma 2.1, we conclude Theorem 3.1 holds.

Now, let

A∗=lim inft→0+2maxT+14,1∑k=1Tmax|s|≤tF(k,s)(1+q∗)t2.

Then there exists a sequence {b_n} of positive numbers with lim_n→+∞ b_n = 0 such that

lim infn→∞∑k=1Tmax|t|≤dnF(k,t)(1+q∗)1+min4T+1,1bn2−1=A∗.

Taking a_n = 0 for all n ∈ Z(1), then by Theorem 3.1, we have the following corollary.

Corollary 3.1

Assume that

A∗<2B2+Q, (3.4)

then, for each λ∈(2+Q2B,1A∗), problem (1.1) admits a sequence of nontrivial solutions which converges to zero.

Now, considering the oscillating behavior of f at infinity, we have

Theorem 3.2

Assume that there exist two real sequences {c_n}, {d_n}, and lim_n→+∞ d_n = +∞, such that

(2+Q)1+cn2−1<(1+q∗)1+min4T+1,1dn2−1, (3.5)

for n ∈ Z(1), and

C:=lim infn→∞∑k=1Tmax|t|≤bnF(k,t)−∑k=1TF(k,cn)(1+q∗)1+min4T+1,1dn2−1−(2+Q)1+cn2−1<D2+Q. (3.6)

Then, for each λ∈(2+QD,1C), problem (1.1) admits an unbounded sequence of solutions.

The proof of Theorem 3.2 is similar to that of Theorem 3.1, we omit it.

Let

C∗=lim inft→+∞maxT+14,1∑k=1Tmax|s|≤tF(k,s)(1+q∗)t.

Then there exists a sequence {d_n} of positive numbers with lim_n→+∞ d_n = +∞ such that

lim infn→∞∑k=1Tmax|t|≤dnF(k,t)(1+q∗)1+min4T+1,1dn2−1=C∗.

Taking c_n = 0 for each n ∈ Z(1), by Theorem 3.2, we have the following corollary.

Corollary 3.2

Assume that

C∗<D2+Q, (3.7)

then, for each λ∈(2+QD,1C∗), problem (1.1) admits an unbounded sequence of solutions.

In order to obtain the positive solutions of (1.1), we need to establish the following strong maximum principle.

Theorem 3.3

Let u ∈ S be such that either

uk>0or−△ϕc(△uk−1)+qkϕc(uk)≥0, (3.8)

for all k ∈ Z(1, T). Then, either u_k > 0 for all k ∈ Z(1, T) or u ≡ 0.

Proof

Assume that

um=min{uk:k∈N(1,T)},

for some m ∈ Z(1, T).

If u_m > 0, we see that u_k > 0 for all k ∈ Z(1, T) and we complete the proof.

If u_m ≤ 0, then u_m = min{u_k : k ∈ N(0, T + 1)}. Since Δ u_m−1 = u_m − u_m−1 ≤ 0 and Δ u_m = u_m+1 − u_m ≥ 0, and ϕ_c(s) is increasing in s with ϕ_c(0) = 0. We have

ϕc(△um)≥0≥ϕc(△um−1). (3.9)

On the other hand, by (3.8), we see that − △ (ϕ_c (△ u_m−1)) ≥ −q_mϕ_c(u_m) ≥ 0, which implies

ϕc(△um)≤ϕc(△um−1). (3.10)

By combining (3.9) with (3.10), we get ϕ_c(△ u_m) = 0 = ϕ_c(△ u_m−1). That is u_m+1 = u_m−1 = u_m. If m + 1 = T + 1, we get u_m = 0. Otherwise, m + 1 ∈ N(1, T). In this case, we replace m by m + 1 and have u_m+2 = u_m+1. We get u_m = u_m+1 = u_m+2 = ⋯ = u_T+1 = 0 by continuing this process T + 1 − m times. Similarly, we have u_m = u_m−1 = u_m−2 = ⋯ = u₀ = 0. Thus u ≡ 0 and we complete the proof.

Now, considering the existence of positive solutions of problem (1.1), we have

Corollary 3.3

Assume that f(k, 0) ≥ 0 for all k ∈ Z(1, T), and

A¯:=lim inft→02maxT+14,1∑k=1Tmax0≤s≤t∫0sf(k,τ)dτ(1+q∗)t2<2B2+Q, (3.11)

then, for each λ∈(2+Q2B,1A¯), problem (1.1) admits a sequence of positive solutions which converges to zero.

Proof

Let

f∗(k,t)=f(k,t),ift>0,f(k,0),ift≤0. (3.12)

Since f(k, 0) ≥ 0, we see that

max0≤|s|≤t∫0sf∗(k,τ)dτ=max0≤s≤t∫0sf(k,τ)dτ,

for all t ≥ 0. According to Corollary 3.1, we see that problem (1.1) with f replaced by f^* admits a sequence of nontrivial solutions which converges to zero for each λ∈(2+Q2B,1A¯). And by Theorem 3.3, all these solutions are positive.

Arguing as in Corollary 3.3, we have

Corollary 3.4

Assume that f(k, 0) ≥ 0 for all k ∈ Z(1, T), and

C¯:=lim inft→+∞maxT+14,1∑k=1Tmax0≤s≤t∫0sf(k,τ)dτ(1+q∗)t<D2+Q, (3.13)

then, for each λ∈(2+QD,1C¯), problem (1.1) admits an unbounded sequence of positive solutions.

4 Examples

In this section, we give two examples to illustrate our main results.

Example 4.1

Consider the boundary value problem (1.1) with

f(k,u)=f(u)=u(2+2ε+2cos⁡(εln⁡|u|)−εsin⁡(εln⁡|u|)),u≠0,0,u=0, (4.1)

for each k ∈ Z(1, T). Then,

F(k,u)=F(u)=∫0uf(τ)dτ=u2(1+ε+cos⁡(εln⁡u)),foru>0.

Since f(u) ≥ 0 for u ≥ 0, we see that F(u) is increasing in u ∈ [0, +∞). Let

αn=exp−2nπε,βn=exp−2nπ+πε.

Then lim_n→+∞ α_n = 0 = lim_n→+∞ β_n, and

F(αn)αn2=2+ε,max0≤s≤βnF(s)βn2=F(βn)βn2=ε,

which implies that

A¯≤2TmaxT+14,11+q∗ε,B≥2+ε.

Let ε be small enough, such that

2TmaxT+14,11+q∗ε<2(2+ε)2+Q.

Then (3.11) holds. By Corollary 3.3, for each λ∈2+Q2(2+ε),(1+q∗)min{4T+1,1}2Tε, problem (1.1) admits a sequence of positive solutions which converges to zero.

Example 4.2

Consider the boundary value problem (1.1) with

f(k,u)=f(u)=1+ϵ+cos⁡(ϵln⁡(|u|+1))−ϵsin⁡(ϵln⁡(|u|+1)), (4.2)

for each k ∈ Z(1, T). Then,

F(k,u)=F(u)=∫0uf(τ)dτ=(1+u)(1+ϵ+cos⁡(ϵln⁡(u+1)))−2−ϵ,

for u ≥ 0. Since f(u) ≥ 0 for u ≥ 0, we see that F(u) is increasing in u ∈ [0, +∞). Let

γn=exp2nπϵ−1,ηn=exp2nπ+πϵ−1.

Then lim_n→+∞ γ_n = +∞ = lim_n→+∞ η_n, and

limn→+∞F(γn)γn=2+ϵ,limn→+∞max0≤s≤ηnF(s)ηn=limn→+∞F(ηn)ηn=ϵ,

which implies that

C¯≤TmaxT+14,11+q∗ϵ,D≥2+ϵ.

Let ϵ be small enough, such that

TmaxT+14,11+q∗ϵ<2+ϵ2+Q.

Then (3.13) holds. By Corollary 3.4, for each λ∈2+Q2+ϵ,(1+q∗)min4T+1,1Tϵ, problem (1.1) admits an unbounded sequence of positive solutions.

Acknowledgements

The authors would like to thank the referees for their valuable comments and suggestions. 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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Received: 2019-04-17

Accepted: 2019-07-31

Published Online: 2019-09-14

This work is licensed under the Creative Commons Attribution 4.0 Public License.

Articles in the same Issue

https://doi.org/10.1515/math-2019-0081

Keywords for this article

discrete Dirichlet problem; mean curvature operator; infinitely many positive solutions; critical point theory

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