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

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.


Introduction
Denote Z and R the sets of integers and real numbers, respectively. For a, b ∈ Z, de ne Z(a) = {a, a + , · · · }, and Z(a, b) = {a, a + , · · · , b} when a ≤ b.
In this paper, we consider the following Dirichlet problem of the second order nonlinear di erence equation − ϕc( u k− ) + q k ϕc(u k ) = λf (k, u k ), k ∈ Z( , T), u = u T+ = , (1.1) where T is a given positive integer, is the forward di erence operator de ned by u k = u k+ − u k , u k = ( u k ), q k ≥ for all k ∈ Z( , T), ϕc is the mean curvature operator de ned by ϕc(s) = s √ +s [1], λ is a real positive parameter, and f (k, ·) ∈ C(R, R) for each k ∈ Z( , T).
(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 in nity of the potential u 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  [4]. Di erence equations arise in various research elds. Many authors have discussed the existence and multiplicity of solutions for di erence problems by using xed 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 di erence 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 di erence equations [19][20][21][22][23]. For example, in [21], the authors considered the discrete Dirichlet problem (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- (1.4) While the existence results of in nitely 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 ≡ according to the behavior of f at in nity. Compared with di erential equations, there is less work on the boundary value problems of di erence equations involving the mean curvature operator. In this paper, we will consider the existence of in nitely 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 in nity 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 di erence 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 in nitely many positive solutions for (1.1) according to the oscillating behavior of f near at the origin and at in nity, respectively. Finally, in section 4, we present two examples to illustrate our main results.

Preliminaries
In this section, we rst establish the variational framework associated with (1.1). We consider the T- De ne for any u ∈ S. It is clear that Φ and Ψ are two functionals of class C (S, R) whose Gâteaux derivatives at the point u ∈ S are given by Consequently, the critical points of I λ in S are exactly the solutions of the problem (1.1).
Assume that X is a re exive real Banach space and I λ : X → R is a function which satis es the following structure hypothesis: Clearly, δ ≥ and γ ≥ . When δ = (or γ = ), in the sequel, we agree to read δ (or γ ) as +∞.
The following lemma comes from Theorem 7.4 of [24] and will be used to investigate problem (1.1).

Main results
Then, for each λ ∈ ( +Q B , A ), 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 satis ed. Put Noticing that for u ∈ S. Thus, we have It is clear that which implies that By the de nition of φ, we have .
For each n ∈ Z( ), let wn ∈ S given by (wn) k = an for each k ∈ Z( , T), and (wn) = (wn) T+ = . Then, by using (3.1), Φ(wn) = ( + Q)( + a n − ) < rn . Thus, Therefore, by (3.2), we know that γ ≤ lim infn→+∞ φ(rn) ≤ A < +∞. Clearly, u ≡ is a global minimum of Φ. In order to get the conclusion (a ), we need to prove that u ≡ is not a local minimum of I λ . To prove this, we consider two cases: B = +∞ and B < +∞. If B = +∞, let {cn} be a sequence of positive numbers, with limn→+∞ cn = , such that If B < +∞, since λ > +Q B , we can choose ϵ > such that Then we can nd a sequence of positive numbers {cn} satisfying limn→+∞ cn = and Let the sequence {ωn} in S be the same as the case where B = +∞, we get Since I λ ( ) = , by combining the above two cases, we see that u ≡ is not a local minimum of I λ and by Lemma Taking an = for all n ∈ Z( ), then by Theorem 3.1, we have the following corollary.
The proof of Theorem 3.2 is similar to that of Theorem 3.1, we omit it. Let Taking cn = for each n ∈ Z( ), by Theorem 3.2, we have the following corollary.
In order to obtain the positive solutions of (1.1), we need to establish the following strong maximum principle.
then, for each λ ∈ ( +Q B , Ā ), problem (1.1) admits a sequence of positive solutions which converges to zero.
Then  Let ϵ be small enough, such that T max T+ , Then (3.13) holds. By Corollary 3.4, for each λ ∈