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

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.


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 ðPÞ ÀΔðw p ðΔuðt À 1ÞÞÞ ¼ f ðt; uðtÞÞ; t ∈ ½1; N Z ; uð0Þ ¼ uðN þ 1Þ ¼ 0; where N ≥ 1 is an integer, ½1; N Z is the discrete interval 1; . . . ; N f g , Δ is the forward difference operator defined by Δu(t) 5 u(t þ 1) À u(t), w p (s) 5 jsj pÀ2 s, 1 < p < ∞ and f : ½1; N Z 3 R → R is a continuous function.
By a solution of (P), we mean a function u : ½0; N þ 1 Z → R satisfies both equations of (P). If f(t, 0) 5 0 for any t ∈ ½1; N Z , the constant function u 5 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 Let λ 1 > 0 be the first eigenvalue of the nonlinear eigenvalue problem (P 0 ) corresponding to problem (P) The value of λ 1 is λ 1 ¼ min , 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][3][4][5][6][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][9][10][11][12]). Especially, many researchers have paid attention to studying boundary value problems and spectral theories for the discrete p-Laplacian (see [1,[13][14][15][16][17][18][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.

AJMS
The main results in this paper are the following theorems: (1.2)) is the last eigenvalue of the nonlinear eigenvalue problem (P 0 ). 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.

Preliminary lemmas
In the present paper, we define a vector space E N by Let u ∈ E N , we consider the functional as follows: Fðt; uðtÞÞ: It is easy to see that Φ ∈ C 1 ðE N ; RÞ and its derivative Φ 0 (u) at u ∈ E N is given by By the summation by parts formula and the fact that v(0) 5 v(N þ 1) 5 0, it follows that ΔðjΔuðt À 1Þj p−2 Δuðt À 1ÞÞvðtÞ: Discrete boundary value problem Therefore, Φ 0 can be written as 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 ∈ C 1 ðE; RÞ. 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 0 (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 vB ρ denote its boundary.
Then Φ possesses a critical value c ≥ a given by c ¼ inf (see [21] ) Let E be a real Banach space and Φ ∈ C 1 ðE; RÞ be even, bounded from below, and satisfy the (PS) condition. Suppose that Φ(0) 5 0 and there is a set Ω ⊂ E such that Ω is homeomorphic to S nÀ1 by an odd map and sup u∈Ω ΦðuÞ < 0, where S nÀ1 is the n À 1 dimensional unit sphere. Then, Φ has at least n disjoint pairs of nontrivial critical points.

Eigenvalue problem
We consider the nonlinear eigenvalue problem (P 0 ) corresponding to problem (P): w p ðuðtÞÞvðtÞ; ∀v ∈ E N : Proposition 3.1. (see [22] ) Let E be a real Banach space, G; J ∈ C 1 ðE; RÞ and a set of constraints Suppose that for any u ∈ S, G 0 (u) ≠ 0 and there exists u 0 ∈ S such that J ðu 0 Þ ¼ inf u∈S J ðuÞ. Then there is λ ∈ R such that J 0 (u 0 ) 5 λG 0 (u 0 ).
Proof of Theorem 1.1. Put J ðuÞ ¼ P Nþ1 t¼1 jΔuðt − 1Þj p ; GðuÞ ¼ P N t¼1 juðtÞj p − 1, and It is easy to see that G 0 (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 Clearly λ 0 > 0. From the Proposition 3.1, there exists λ N such that Multiplying (3.1) by u N in the sense of inner product, we obtain Therefore, λ 0 5 λ N is an eigenvalue of the problem (P 0 ). Thus, we have If λ is an eigenvalue of the problem (P 0 ), then there exists u ∈ E N \{0} such that: In particular for v 5 u, we get λ ¼ So, we deduce that λ 1 ≤ λ ≤ λ N . Then, λ N is the last eigenvalue of the problem (P 0 ). The proof of Theorem 1.1 is completed. ,

Using (4.2) and (4.3), we obtain
Since « < λ 1 À η, then Φ(u) → ∞ as kuk → ∞. Thus, Φ is coercive and bounded from below, hence there is a minimum point of Φ at some u 0 ∈ E N i.e. Φðu 0 Þ ¼ inf ΦðuÞ, which is a critical point of Φ and in turn is a solution of problem (P).
Thus, we have ΦðuÞ ≥ α > 0; ∀u ∈ vB ρ 2 : At the same time, we have also proved that there exist constants α > 0 and ρ 2 > 0 such that That is to say, Φ satisfies the condition σ 1 of the mountain pass lemma.
For our setting, clear Φ(0) 5 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.
Let u 1 ∈ E N be a critical point associated to the critical value c of Φ, i.e. Φ(u 1 ) 5 c. Hence, u 1 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 u 2 ∈ E N , i.e. Φðu 2 Þ ¼ sup

ΦðuÞ.
Using the preceding equality and (4.9), we obtain ΦðuÞ > 0: Hence u 2 is a nontrivial solution of problem (P). If u 1 ≠ u 2 , then we have two nontrivial solutions: u 1 and u 2 . Otherwise, suppose u 1 5 u 2 , then inf

ΦðgðsÞÞ:
Thus, we get two different critical points of Φ on E N denoted by v 2 5 g 2 (s 2 ), v 3 5 g 3 (s 3 ) that are nontrivial solutions of problem (P).

AJMS
The proof is completed.
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 NÀ1 be the unit sphere in R N and define T: Ω → S NÀ1 by where Ω (defined in (4.5)). Then, T is an odd homeomorphism between Ω and S NÀ1 , and sup u∈Ω ΦðuÞ < 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. ,