Three solutions to Dirichlet problems for second-order self-adjoint difference equations involving p-Laplacian

This paper derives several sufficient conditions for the existence of three solutions to the Dirichlet problem for a second-order self-adjoint difference equation involving p-Laplacian through the critical point theory. Furthermore, by using the strong maximum principle, we prove that the three solutions are positive under appropriate assumptions on the nonlinearity. Finally, we present three examples to confirm our results.


Introduction
where T ∈ N, is the forward difference operator denoted by u(k) = u(k + 1)u(k), 2 u(k) = ( u(k)), φ p is the p-Laplacian operator, namely, φ p (s) = |s| p-2 s, p > 1, f (k, ·) ∈ C(R, R) for each k ∈ Z(1, T). Difference equations have been widely applied as mathematical models in biology, physics, and other research fields [1][2][3][4][5]. For instance, the qualitative analysis of equations with p-Laplacian-like operators has become an important research topic due to the fact that these equations arise in a variety of real world problems such as in the study of non-Newtonian fluid theory and the turbulent flow of a polytrophic gas in a porous medium; see [6][7][8] for more details. Accordingly, the research of the difference equations has attracted much attention in recent years. The upper and lower solution techniques, as well as the fixed point methods [9,10], are useful tools in researching the BVPs of difference equations. In 2003, the second-order difference equation was first studied by Yu and Guo [11] via the critical point theory, and some results on the existence of periodic solutions and subharmonic solutions were obtained. Since then, many researchers have explored the difference equations by mainly using the critical point theory to show a lot of interesting results on BVPs [12][13][14][15][16][17][18][19], periodic solutions [11,[20][21][22], and homoclinic solutions [23][24][25][26][27][28][29][30][31][32][33].
In [21], Yu et al. studied the following second-order difference equation: Based on the critical point theory, some sufficient conditions to prove the existence of periodic solutions of (1.2) were derived. In [31], the variational method was also explored to prove the existence of nontrivial homoclinic orbits for (1.2).
In [34], Ma studied the following homogeneous and linear difference equation: and some results on recessive and dominant solutions were established. In [35], Long, Yu, and Guo considered the disconjugacy and the C-disfocality of (1.3), and several sufficient conditions were derived explicitly in terms of equation coefficients. For p(k) ≡ 1 in (1.1), Jiang and Zhou [36] showed the existence of three solutions of (1.1). The aim of this paper is to prove that the three solutions exist for (1.1) by using different methods. Moreover, by using a new strong maximum principle established in this paper, we prove that the three solutions are positive under suitable conditions. Even in the special case p(k) ≡ 1, the existence results of three nontrivial solutions for (1.1) are new. However, as shown in the last Example 4.3, the conclusion about the three solutions using [36, Theorem 3.1] cannot be acquired.
The rest of this paper is organized as follows. Some preliminaries are presented in Sect. 2. Our main results are given in Sect. 3. Finally, we give three examples to confirm our findings in Sect. 4.

Preliminaries
Let X denote a finite-dimensional real Banach space and let I λ : X → R be a functional satisfying the following structure hypothesis: where λ > 0, , : X → R are two continuous functions of class C 1 on X with is coercive, that is, lim u →∞ (u) = +∞. The following lemma comes from Corollary 3.1 of [37].
, the functionalλ is coercive. Then, for each λ ∈ r , the functionalλ has at least three distinct critical points in X.

Consider the T-dimensional Banach space
Define two functionals on X as follows: Through careful calculations, for each u, v ∈ X, we see that Thus That is, each critical point of the functional I λ corresponds to a solution of (1.1). Therefore, we simplify the solution of (1.1) into the problem of finding the critical points of I λ on X.

Lemma 2.3 ([36])
For any u ∈ X and p > 1, one has Proof Let j ∈ Z(1, T) and Owing to p(j + 1) > 0 and p(j) > 0, it holds that On the other hand, by (2.6), we have N(1, T). Replacing j by j + 1, we can get u(j + 2) = u(j + 1). Continuing this process Thus, u ≡ 0 and the proof is completed.

Main results
For two positive constants c and q, we denote is a continuous function with respect to u, and c 1 , c 2 , and d are positive constants, with Proof Our idea is to use Lemma 2.1 to confirm our conclusion. First, we prove the coercivity of . Since we see that is coercive. Next, we prove that is convex. Consider y = x p for x ≥ 0. By taking the derivative of y with respect to x, we have So, y = x p is convex on [0, +∞). That is, we know that is convex. From the definitions of and , we have Similarly, it holds that p . Taken together, we have and sup u∈ -1 (-∞,ρ 2 ) (u) , , Thus, the proof of Theorem 3.1 is completed. Let for all (k, u) ∈ Z(1, T) × R.
Next, we consider the following discrete system: where α : Z(1, T) → R is a continuous function and y : [0, +∞) → R is a nonnegative continuous function with y(0) = 0.
problem (3.7) has at least two positive solutions.
Next, let us give another theorem as follows:

Theorem 3.2 Suppose that c and d are positive constants such that
Clearly,ū ∈ X. It follows from (3.8) that Moreover, one has . (3.10) Therefore, by (3.9) and (3.10), condition (a 1 ) follows. In order to verify the coercivity of the functionalλ , we first suppose that According to condition (iii), there exists ε such that lim sup |ξ |→+∞ for each ξ ∈ R and k ∈ Z(1, T). According to Lemma 2.3 and due to λ < m(2c) p F c p(T+1) p-1 , one has for each u ∈ X. Therefore, we have for u ≥ 1. This gives lim u →+∞ (u)λ (u) = +∞.
On the other hand, if there is a positive constant h ε such that F(k, ξ ) ≤ h ε , and so arguing as before we have for u ≥ 1. Again, we obtain lim u →+∞ (u)λ (u) = +∞, and thereby condition (a 2 ) holds.
In summary, all the hypotheses in Lemma 2.2 have been demonstrated to be true. Therefore, the functional (u)λ (u) has at least three different critical points for each λ ∈ r , and the proof is completed.

15)
and satisfying ξ p < F c Tc p . Then, for each λ ∈ r , problem (1.1) has at least three positive solutions.
Proof For any k ∈ Z(1, T), (3.14) should be considered with Under Theorem 3.2, one has that (i) is tenable.
Consequently, all the requirements of Theorem 3.2 have been met. Besides, since u ≡ 0 is not a solution to (3.14), it can be inferred that (3.14) has at least three nontrivial solutions. Suppose that u = {u(k)} is a nontrivial solution. Then for any k ∈ Z(1, T), we have u(k) > 0 or Therefore, according to Lemma 2.4, we may refer to u(k) > 0 for k ∈ Z(1, T). This u denotes a positive solution. In addition, once u has been verified as a positive solution to (3.14), it can be also considered as a positive solution of (1.1). Thus, Corollary 3.2 is demonstrated.

Examples
We show three examples to confirm our findings in this section.  Applying Corollary 3.1, for each λ ∈ (0.281B, 5.109B), where B = 1/ 2 k=1 α(k), the following problem: has at least two positive solutions.