Multiplicity of solutions for the discrete boundary value problem involving the p-Laplacian

Abstract

Purpose

The purpose of this paper is the study of existence and multiplicity of solutions for a nonlinear discrete boundary value problems involving the p-laplacian.

Design/methodology/approach

The approach is based on variational methods and critical point theory.

Findings

Theorem 1.1. Theorem 1.2. Theorem 1.3. Theorem 1.4.

Originality/value

The paper is original and the authors think the results are new.

Citation

El Amrouss, A. and Hammouti, O. (2023), "Multiplicity of solutions for the discrete boundary value problem involving the p-Laplacian", Arab Journal of Mathematical Sciences, Vol. 29 No. 1, pp. 73-82. https://doi.org/10.1108/AJMS-02-2021-0050

1. Introduction

The main goal of the present paper is to establish the existence and multiplicity of nontrivial solutions for the following discrete nonlinear boundary value problems

(P)−Δ(ϕp(Δu(t−1)))=f(t,u(t)),t∈[1,N]Z,u(0)=u(N+1)=0,

where N ≥ 1 is an integer, [1,N]Z is the discrete interval 1,…,N, Δ is the forward difference operator defined by Δu(t) = u(t + 1) − u(t), ϕ_p(s) = |s|^p−2s, 1 < p < ∞ and f:[1,N]Z×R→R is a continuous function.

By a solution of (P), we mean a function u:[0,N+1]Z→R satisfies both equations of (P).

If f(t, 0) = 0 for any t∈[1,N]Z, the constant function u = 0 is a trivial solution of problem (P). In this case, the key point is proving the existence of nontrivial solutions for (P). For this purpose, we need to introduce a condition that gives us information about the behaviors of the perturbed function f(t, x) or its primitive F(t, x) near infinity and near zero.

We may think of (P) as a discrete analogue of the following second-order p-Laplacian functional differential equation

−ddtϕpdu(t)dt=f(t,u(t)),t∈[0,1],u(0)=u(1)=0.

Let λ₁ > 0 be the first eigenvalue of the nonlinear eigenvalue problem (P₀) corresponding to problem (P)

(P0)−Δ(ϕp(Δu(t−1)))=λϕp(u(t)),t∈[1,N]Z,u(0)=u(N+1)=0.

The value of λ₁ is λ1=minu∈EN\{0}∑t=1N+1|Δu(t−1)|p∑t=1N|u(t)|p, where E_N defined in (2.1) (see [1]).

The discrete analogue of the Laplacian on Riemannian manifolds, the so-called discrete Laplacian, has been studied intensively for the past few decades (see [2–7]). But, most phenomena on many cases are not expressed by the discrete Laplacian, which is known to be linear because they have a nonlinear flow governed by these intrinsic characteristics. For this reason, a nonlinear operator, called the discrete p-Laplacian, which is a generalization of the discrete Laplacian has recently been studied by many researchers in various fields, for example, dynamical systems, molecular structures, internet webs, image processing and so on (for more details, see [8–12]). Especially, many researchers have paid attention to studying boundary value problems and spectral theories for the discrete p-Laplacian (see [1, 13–19]).

As is well known, critical point theory and variational methods are powerful tools to investigate the existence of solutions of various problems.

In this paper, we shall study the existence and multiplicity of nontrivial solutions of (P), via variational methods and critical point theory.

For convenience, we introduce the following notations.

F∞=lim|x|→∞infmint∈[1,N]ZpF(t,x)|x|p,F∞=lim|x|→∞supmaxt∈[1,N]ZpF(t,x)|x|p,F0=limx→0infmint∈[1,N]ZpF(t,x)|x|p

And we make the following conditions:

• (H₁) there exists η with η < λ₁ such that F ^∞ ≤ η, where

(1.1)λ1=minu∈EN\{0}∑t=1N+1|Δu(t−1)|p∑t=1N|u(t)|p;

• (H₂) there exists δ with δ > λ_N such that F_∞≥ δ, where

(1.2)λN=maxu∈EN\{0}∑t=1N+1|Δu(t−1)|p∑t=1N|u(t)|p;

• (H₃) F₀ > λ_N;

• (H₄) limx→0F(t,x)|x|p=0, ∀t∈[1,N]Z;

• (H₅) f(t, x) is odd in x, i.e. f(t, −x) = −f(t, x) for (t,x)∈[1,N]Z×R.

The main results in this paper are the following theorems:

Theorem 1.1.

λ_N (defined in (1.2)) is the last eigenvalue of the nonlinear eigenvalue problem (P₀).

Theorem 1.2.

Assume that (H₁) and (H₃) hold, then problem (P) has at least one nontrivial solution.

Theorem 1.3.

Assume that (H₂) and (H₄) hold, then problem (P) has at least two nontrivial solutions.

Theorem 1.4.

Assume that (H₁), (H₃) and (H₅) hold, then problem (P) has at least 2N nontrivial solutions.

The rest of this paper is organized as follows. Section 2 contains some preliminary lemmas. The main results will be proved in Sections 3 and 4.

2. Preliminary lemmas

In the present paper, we define a vector space E_N by

(2.1)EN={u:[0,N+1]Z→R∣u(0)=u(N+1)=0},

and for any u ∈ E_N, define ‖u‖=∑t=1N|u(t)|p1/p.

Equipped with ‖ ‖, E_N is an N dimensional Banach space. In fact, E_N is isomorphic to RN.

Let u ∈ E_N, we consider the functional as follows:

(2.2)Φ(u)=1p∑t=1N+1|Δu(t−1)|p−∑t=1NF(t,u(t)).

It is easy to see that Φ∈C1(EN,R) and its derivative Φ′(u) at u ∈ E_N is given by

Φ′(u).v=∑t=1N+1|Δu(t−1)|p−2Δu(t−1)Δv(t−1)−∑t=1Nf(t,u(t))v(t),∀v∈EN.

By the summation by parts formula and the fact that v(0) = v(N + 1) = 0, it follows that

∑t=1N+1|Δu(t−1)|p−2Δu(t−1)Δv(t−1)=|Δu(N)|p−2Δu(N)Δv(N)+∑t=1N|Δu(t−1)|p−2Δu(t−1)Δv(t−1)=−|Δu(N)|p−2Δu(N)v(N)+|Δu(t−1)|p−2Δu(t−1)v(t−1)1N+1−∑t=1NΔ(|Δu(t−1)|p−2Δu(t−1))v(t)=−|Δu(N)|p−2Δu(N)v(N)+|Δu(N)|p−2Δu(N)v(N)−|Δu(0)|p−2Δu(0)v(0)−∑t=1NΔ(|Δu(t−1)|p−2Δu(t−1))v(t)=−∑t=1NΔ(|Δu(t−1)|p−2Δu(t−1))v(t).

Therefore, Φ′ can be written as

Φ′(u).v=∑t=1N−Δ(|Δu(t−1)|p−2Δu(t−1))−f(t,u(t))v(t),∀v∈EN.

Thus, finding solutions of (P) is equivalent to finding critical point of the functional Φ.

Definition 2.1.

Let E be a real Banach space and Φ∈C1(E,R). Recall that Φ is said to satisfy the Palais Smale (PS) condition if every sequence (u_n) ⊂ E, such that Φ(u_n) is bounded and Φ′(u_n) → 0 as n → ∞, has a convergent subsequence. Here, the sequence (u_n) is called a PS sequence.

Let B_ρ denote the open ball in E about 0 of radius ρ and let ∂B_ρ denote its boundary.

Lemma 2.1.

(Mountain pass lemma [20]) Let E be a real Banach space and Φ∈C1(E,R) satisfy the (PS) condition. If Φ(0) = 0 and

• σ1) there exist constants ρ, α > 0 such that Φ|∂Bρ≥α,

• σ2) there exist e ∈ E\B_ρ such that Φ(e) ≤ 0.

Then Φ possesses a critical value c ≥ a given by c=infg∈Γmaxs∈0,1Φ(g(s)),

where

Γ={g∈C(0,1,E)/g(0)=0,g(1)=e}.

Lemma 2.2.

(see [21] ) Let E be a real Banach space and Φ∈C1(E,R) be even, bounded from below, and satisfy the (PS) condition. Suppose that Φ(0) = 0 and there is a set Ω ⊂ E such that Ω is homeomorphic to Sⁿ⁻¹ by an odd map and supu∈ΩΦ(u)<0, where Sⁿ⁻¹ is the n − 1 dimensional unit sphere. Then, Φ has at least n disjoint pairs of nontrivial critical points.

3. Eigenvalue problem

We consider the nonlinear eigenvalue problem (P₀) corresponding to problem (P):

(P0)−Δ(ϕp(Δu(t−1)))=λϕp(u(t)),t∈[1,N]Z,u(0)=u(N+1)=0,

Definition 3.1.

λ∈R is called eigenvalue of (P₀) if there exists u ∈ E_N \{0} such that:

∑t=1N+1ϕp(Δu(t−1))Δv(t−1)=λ∑t=1Nϕp(u(t))v(t),∀v∈EN.

Proposition 3.1.

(see [22]) Let E be a real Banach space, G,J∈C1(E,R) and a set of constraints

S = {u ∈ E ∣ G(u) = 0}. Suppose that for any u ∈ S, G′(u) ≠ 0 and there exists u₀ ∈ S such that J(u0)=infu∈SJ(u). Then there is λ∈R such that J′(u₀) = λG′(u₀).

Proof of Theorem 1.1.

Put J(u)=∑t=1N+1|Δu(t−1)|p,G(u)=∑t=1N|u(t)|p−1, and

S={u∈EN∣G(u)=0}={u∈EN∣‖u‖=1}.

It is easy to see that G′(u) ≠ 0 for any u ∈ S.

If the set S is compact and J is continuous on S, then there exists u_N ∈ S such that

J(uN)=maxu∈SJ(u)=λ′.

Thus,

−J(uN)=minu∈S(−J(u))=−λ′.

Clearly λ′ > 0. From the Proposition 3.1, there exists λ_N such that

J′(uN)=λNG′(uN).

Which means that

(3.1)−Δ(ϕp(ΔuN(t−1)))=λNϕp(uN(t)),t∈[1,N]Z.

Multiplying (3.1) by u_N in the sense of inner product, we obtain

∑t=1N+1|ΔuN(t−1)|p=λN∑t=1N|uN(t)|p,

i.e.

J(uN)=λN‖uN‖p=λN.

Therefore, λ′ = λ_N is an eigenvalue of the problem (P₀).

Thus, we have

λN=maxu∈S∑t=1N+1|Δu(t−1)|p=maxu∈EN\{0}∑t=1N+1|Δu(t−1)‖u‖|p=maxu∈EN\{0}∑t=1N+1|Δu(t−1)|p∑t=1N|u(t)|p.

If λ is an eigenvalue of the problem (P₀), then there exists u ∈ E_N \{0} such that:

∑t=1N+1ϕp(Δu(t−1))Δv(t−1)=λ∑t=1Nϕp(u(t))v(t),∀v∈EN.

In particular for v = u, we get λ=∑t=1N+1|Δu(t−1)|p∑t=1N|u(t)|p.

So, we deduce that λ₁ ≤ λ ≤ λ_N.

Then, λ_N is the last eigenvalue of the problem (P₀).

The proof of Theorem 1.1 is completed. □

4. Proofs of the main results

Proof of Theorem 1.2.

Since F^∞ ≤ η, there exists R₁ > 0 such that

pF(t,x)|x|p≤η+εfor(t,|x|)∈[1,N]Z×R1,+∞,

where 0 < ɛ < λ₁ − η, i.e.

(4.1) F(t,x)≤1p(η+ε)|x|pfor(t,|x|)∈[1,N]Z×R1,+∞.

Then, by (4.1) and the continuity of x → F(t, x), there exists c₁ > 0 such that

(4.2)F(t,x)≤1pη+ε|x|p+c1,∀(t,x)∈[1,N]Z×R.

According to (1.1), we have

(4.3)∑t=1N+1|Δu(t−1)|p≥λ1‖u‖p.

Using (4.2) and (4.3), we obtain

Φ(u)≥1pλ1‖u‖p−1p(η+ε)∑t=1N|u(t)|p−c1N≥1pλ1−(η+ε)‖u‖p−c1N.

Since ɛ < λ₁ − η, then Φ(u) → ∞ as ‖u‖ → ∞. Thus, Φ is coercive and bounded from below, hence there is a minimum point of Φ at some u₀ ∈ E_N i.e. Φ(u0)=infu∈ENΦ(u), which is a critical point of Φ and in turn is a solution of problem (P).

From (H₃), there exists ρ₁ > 0 such that

pF(t,x)|x|p≥F0−ε,for(t,|x|)∈[1,N]Z×0,ρ1,

where 0 < ɛ < F₀ − λ_N, i.e.

(4.4)F(t,x)≥1p(F0−ε)|x|pfor(t,|x|)∈[1,N]Z×0,ρ1.

Put

(4.5)Ω={u∈EN∣‖u‖=ρ1}.

For any u ∈ Ω,

|u(t)|≤‖u‖=ρ1,∀t∈[1,N]Z.

From (1.2), we get

(4.6)∑t=1N+1|Δu(t−1)|p≤λN‖u‖p.

Combining the preceding inequality and (4.4), we have

Φ(u)≤1pλN‖u‖p−1p(F0−ε)‖u‖p=1pλN−(F0−ε)ρ1p<0.

Thus, we obtain

(4.7)supu∈ΩΦ(u)<0.

Hence,

Φ(u0)=infu∈ENΦ(u)≤infu∈ΩΦ(u)≤supu∈ΩΦ(u)<0.

So problem (P) has at least one nontrivial solution.

The proof of Theorem 1.2 is completed. □

Proof of Theorem 1.3.

From the condition (H₄), for ε=λ12p there exists ρ₂ > 0 such that:

(4.8) |F(t,x)|≤λ12p|x|pfor(t,|x|)∈[1,N]Z×0,ρ2.

Let u ∈ E_N with ‖u‖ ≤ ρ₂, then |u(t)|≤ρ2,∀t∈[1,N]Z.

Thus, we have

Φ(u)≥λ1p‖u‖p−λ12p‖u‖p≥λ12p‖u‖p.

Take α=λ12pρ2p>0. Therefore,

(4.9)Φ(u)≥α>0,∀u∈∂Bρ2.

At the same time, we have also proved that there exist constants α > 0 and ρ₂ > 0 such that Φ|∂Bρ2≥α. That is to say, Φ satisfies the condition σ1 of the mountain pass lemma.

For our setting, clear Φ(0) = 0. In order to exploit the mountain pass lemma in critical point theory, we need to verify all other conditions of the mountain pass lemma.

According to the condition (H₂), there exists R₂ > 0 such that

pF(t,x)|x|p≥δ−εfor(t,|x|)∈[1,N]Z×R2,+∞,

where 0 < ɛ < δ − λ_N, i.e.

(4.10)F(t,x)≥1p(δ−ε)|x|pfor(t,|x|)∈[1,N]Z×R1,+∞.

Then, by (4.10) and the continuity of x → F(t, x), there exists c₂ > 0 such that

(4.11)F(t,x)≥1p(δ−ε)|x|p−c2,∀(t,x)∈[1,N]Z×R.

Now, using again (4.6) and (4.11), it follows that,

Φ(u)≤λNp‖u‖p−1p(δ−ε)‖u‖p+c2N

≤1pλN−(δ−ε)‖u‖p+c2N.

Consequently, since ɛ < δ − λ_N, we have

(4.12)Φ(u)→−∞,as‖u‖→∞.

Thus, we can choose u¯ large enough to ensure that Φ(u¯)<0, that is, there exists e=u¯∈E\Bρ2 such that Φ(e) < 0.

From (4.12), Φ is anti-coercive, hence for any PS sequence (u_n) is bounded. In view of the fact that the dimension of E_N is finite, we see that Φ satisfies the (PS) condition.

By the mountain pass lemma, Φ possesses a critical value c≥α=λ12pρ2p>0,

where c=infg∈Γmaxs∈0,1Φ(g(s)) and Γ={g∈C(0,1,E)∣g(0)=0,g(1)=u¯}.

Let u₁ ∈ E_N be a critical point associated to the critical value c of Φ, i.e. Φ(u₁) = c.

Hence, u₁ is a nontrivial solution of problem (P).

Since Φ is anti-coercive and bounded from above, there is a maximum point of Φ at some u₂ ∈ E_N,

i.e. Φ(u2)=supu∈ENΦ(u).

Using the preceding equality and (4.9), we obtain

Φ(u2)=supu∈ENΦ(u)≥supu∈∂Bρ2Φ(u)>0.

Hence u₂ is a nontrivial solution of problem (P).

If u₁ ≠ u₂, then we have two nontrivial solutions: u₁ and u₂.

Otherwise, suppose u₁ = u₂, then infg∈Γmaxs∈0,1Φ(g(s))=supu∈ENΦ(u).

Therefore, we have Φ(u1)≤maxs∈0,1Φ(g(s))≤Φ(u2),∀g∈Γ.

Since u₁ = u₂, we deduce that Φ(u1)=maxs∈0,1Φ(g(s)),∀g∈Γ.

The continuity of Φ(g(s)) with respect to s, Φ(0) = 0 and Φ(u¯)<0 implies that there exists s1∈0,1 such that Φ(u₁) = Φ(g(s₁)). Choose g₂, g₃ ∈ Γ such that

{g2(s)∣s∈0,1}∩{g3(s)∣s∈0,1}=∅,

then there exists s2,s3∈0,1 such that

Φ(g2(s2))=Φ(g3(s3))=Φ(u1)=maxs∈0,1Φ(g(s)).

Thus, we get two different critical points of Φ on E_N denoted by v₂ = g₂(s₂), v₃ = g₃(s₃) that are nontrivial solutions of problem (P).

The proof is completed. □

Proof of Theorem 1.4.

Let Φ be defined by (2.2). Then, it is clear that Φ(0) = 0 and Φ is even by (H₅).

From the proof of Theorem 1.2, Φ is bounded from below, coercive and any PS sequence (u_n) is bounded. In view of the fact that the dimension of E_N is finite, we see that Φ satisfies the (PS) condition.

Let S^N−1 be the unit sphere in RN and define T: Ω → S^N−1 by

T(u)=1ρ1u,

where Ω (defined in (4.5)).

Then, T is an odd homeomorphism between Ω and S^N−1, and supu∈ΩΦ(u)<0 (see (4.7)).

Hence, all the conditions of Lemma 2.2 are satisfied, so Φ has at least 2N nontrivial critical points, which are nontrivial solutions of problem (P).

This completes the proof of Theorem 1.4.□