Large Constant-Sign Solutions of Discrete Dirichlet Boundary Value Problems with p-Mean Curvature Operator

In this paper, we consider the existence of infinitely many large constant-sign solutions for a discrete Dirichlet boundary value problem involving p -mean curvature operator. The methods are based on the critical point theory and truncation techniques. Our results are obtained by requiring appropriate oscillating behaviors of the non-linear term at infinity, without any symmetry assumptions.


Introduction
Let Z, N and R denote the sets of integer numbers, natural numbers 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.
We may think problem (D λ, f p ) as being a discrete analog of one-dimensional case of the following problem −div φ p, c ( u) = λ f (x, u), x ∈ Ω ⊂ R n , where div φ p, c ( u) is named p-mean curvature operator, which is a generalization of mean curvature operator; see [1,2]. If p = 1, it reduces to the mean curvature operator. If p = 2, it reduces to the Laplacian operator. The above problem arises from differential geometry and physics such as capillarity; see [3][4][5] and references therein. When p = 1 and f (x, u) = u, the above problem describes the free surface of a pendent drop filled with liquid under gravitational field [4]. In the past decades, several authors have discussed the existence and multiplicity of solutions of Problem (1); see [1,[6][7][8][9][10][11][12]. For example, Chen and Shen in [1] have obtained the existence of infinitely many solutions of Problem (1) with λ = 1 via a symmetric version of Mountain Pass Theorem. When p = 1 and Ω = (0, 1), 1 ), obtaining infinitely many positive solutions when λ belongs to a precise real interval. It is worth noticing that the suitable oscillating behaviors of the nonlinear term f at infinity play a key role. Inspired by [19,32,[35][36][37][38][39][40], the main purpose of this paper is to investigate the existence conditions of infinitely many constant-sign solutions for problem (D λ, f p ), without any symmetry hypothesis. Here, a solution {u(k)} of (D To facilitate the analysis, we have to divide the problem into two categories: 1 ≤ p < 2 and 2 ≤ p < +∞. We believe that this is the first time to discuss the existence of infinitely many solutions for a non-linear second order difference equation with p -mean curvature operator.
A special case of our results is the following.
This paper is organized as follows. In Section 2, we introduce the the suitable Banach space and appropriate functional corresponding to problem (D λ, f p ). To obtain sequences of constant-sign solutions of problem (D λ, f p ), three basic lemmas are introduced. In Section 3, under suitable hypotheses on f , we obtain the existence of infinitely many constant-sign solutions for problem (D λ, f p ). In Section 4, we give two examples to demonstrate our results. Finally, conclusions are given for this paper.

Mathematical Background
To solve problem (D λ, f p ), we naturally select the T-dimensional Banach space ( u(k)) 2 1 2 for all u ∈ X.
Another useful norm on X is In the sequel, we will use the following inequalities.
for every u ∈ X, it can follow from Lemma 2.2 of [42].
where F(k, t) := t 0 f (k, τ) dτ for every t ∈ R and k ∈ Z(1, T). Further, let us denote I λ (u) := Φ(u) − λΨ(u) for u ∈ X. Through standard arguments, we follow that I λ ∈ C 1 (S, R), and the critical points of I λ are exactly the solutions of problem (D λ, f p ). In fact, one has for all u, v ∈ X. Next, we need to establish the following strong maximum principle to obtain the positive solutions of problem (D for any k ∈ Z(1, T). Then, either u > 0 in Z(1, T) or u ≡ 0.
On the other hand, by (5), let k = j, we obtain Combining inequalities (6) and (7), we get that ϕ p, c ( u(j) In the same way, we have the following result to get negative solutions problem (D for any k ∈ Z(1, T). Then, either u < 0 in Z(1, T) or u ≡ 0.
Truncation techniques are usually used to discuss the existence of constant-sign solutions. To the end, we introduce the following truncations of the functions f (k, t) for every k ∈ Z(1, T).
has non-zero solutions, then problem (D λ, f p ) possesses negative solutions. Here, we introduce a lemma (Theorem 4.3 of [38]) which is the main tool used to research problem (D λ, f p ). Lemma 3. Let X be a finite dimensional Banach space and let I λ : X → R be a function satisfying the following structure hypothesis: For all r > 0, put Assume that ϕ ∞ < +∞ and for each λ ∈ (0, 1 ϕ ∞ ) I λ is unbounded from below. Then, there is a sequence {u n } of critical points (local minima) of I λ such that lim n→+∞ Φ(u n ) = +∞.

Main Results
In the following, we will discuss the existence of constant-sign solutions of problem (D λ, f p ). Our purpose is to apply Lemma 3 to the function I ± λ : dτ for every k ∈ Z(1, T) and then exploit Lemma 1 or Lemma 2 to get our results. Let Considering the functional I + λ , we have the following conclusions.
In the following, we take in X the sequence {ω n } defined by putting ω n (k) = c n , for k ∈ Z(1, T). Using again (2), one has Arguing as before and by choosing {ω n } in X as above, we have Since 2 p − λB +∞ + λ 0 < 0, it is clear that lim n→+∞ I + λ (ω n ) = −∞. Considering the above two cases, we follow that I + λ is unbounded from below. According to Lemma 3, there exist a sequence {u n } of critical points (local minima) of I + λ such that lim n→+∞ Φ(u n ) = +∞. Hence, for every n ∈ N, u n is a non-zero solution of problem (D λ, f + p ), by Lemma 1, u n is a positive solution of problem (D λ, f p ). Since Φ is bounded on bounded sets and lim n→+∞ Φ(u n ) = +∞, {u n } must be unbounded. So Theorem 2 holds and the proof is complete. Theorem 3. Let 2 ≤ p < +∞ and f (k, ·) : R → R to be a continuous function with f (k, 0) ≥ 0 for each k ∈ Z(1, T). Assume that Then, for each λ ∈ ( ( Proof. We sketch only the differences with the proof of Theorem 2. For t > 0, make Assume u ∈ X and Noting the inequality (x + y) θ ≤ x θ + y θ , for 0 < θ ≤ 1, x ≥ 0, y ≥ 0 and Hölder inequality, one has Applying (3), we have By the definition of ϕ, we have Using condition(i 2 ), ϕ ∞ ≤ p(T+1) p−1 2 p A +∞ < +∞ holds. Now, we verify that I + λ is unbounded form blow. Fist, assume that B +∞ = +∞. Let {c n } be a sequence of positive numbers, with lim n→+∞ c n = +∞, such that Picking the sequence {ω n } in X by ω n (k) = c n , for k ∈ Z(1, T). Exploiting the inequality (x + y) θ ≤ 2 θ−1 (x θ + y θ ) for θ ≥ 1, x ≥ 0, y ≥ 0 , we get which implies that lim n→+∞ I λ (ω n ) = −∞.
Next, assume that B +∞ < +∞. Since λ > ( √ 2 ) p pB +∞ , we may take 0 > 0 such that ( Then there exists a sequence of positive numbers {c n } such that lim n→+∞ c n = +∞ and Define the sequence {ω n } in S as above, we obtain Thus, we follow that I + λ is unbounded from below. According to Lemmas 1 and 3, we have finished the proof of the theorem.
Similarly, considering the functional I − λ , we can achieve the following results.
Theorem 4. Let 1 ≤ p < 2 and f (k, ·) : R → R to be a continuous function with f (k, 0) ≤ 0 for each k ∈ Z(1, T). Assume that Then, for each λ ∈ ( 2 pB −∞ , has an unbounded sequence of negative solutions. Theorem 5. Let 2 ≤ p < +∞ and f (k, ·) : R → R to be a continuous function with f (k, 0) ≤ 0 for each k ∈ Z(1, T). Assume that Then, for each λ ∈ ( ( has an unbounded sequence of negative solutions. Combining Theorems 2 and 4, we have the following corollary. Corollary 1. Let 1 ≤ p < 2 and f (k, ·) : R → R to be a continuous function with f (k, 0) = 0 for each k ∈ Z(1, T). Assume that Then, for each λ ∈ ( Similarly, combining Theorems 3 and 5, we have the following corollary. Corollary 2. Let 2 ≤ p < +∞ and f (k, ·) : R → R to be a continuous function with f (k, 0) = 0 for each k ∈ Z(1, T). Assume that p ) admits admits two unbounded sequences of constant-sign solutions ( one positive and one negative ). Remark 1. If we let p → 2 − in Theorem 2, we find that the conditions and consequence of Theorem 2 is the same as those of Theorem 3 for p = 2. Moreover the results are consistent with results in [37]. For the special case, p = 1, Theorem 2 reduces to Corollary 2.1 of [35].
In view of 1 ≤ p < 2 , we follow that A + ∞ < 2 p−1 (T+1) Admits an unbounded sequence of positive solutions and an unbounded sequence of negative solutions.

Conclusions
In this paper, we have discussed the Dirichlet boundary value problem of the difference equation with p-mean curvature operator. Some sufficient conditions are derived for the existence of sequences of constant-sign solutions to the problem. Two examples are given to show the effectiveness of our results.
To solve problem (D λ, f p ), we further develop the methods adopted in [23]. The approaches can be used for the boundary value problems of differential equations involving p-mean curvature operator. Therefore, our work has both theoretical and practical significance.