Small Solutions of the Perturbed Nonlinear Partial Discrete Dirichlet Boundary Value Problems with ( p , q ) -Laplacian Operator

: In this paper, we consider a perturbed partial discrete Dirichlet problem with the ( p , q ) Laplacian operator. Using critical point theory, we study the existence of inﬁnitely many small solutions of boundary value problems. Without imposing the symmetry at the origin on the nonlinear term f , we obtain the sufﬁcient conditions for the existence of inﬁnitely many small solutions. As far as we know, this is the study of perturbed partial discrete boundary value problems. Finally, the results are exempliﬁed by an example.


Introduction
Let Z and R denote the sets of integers and real numbers, respectively. Denote Z(a, b) = {a, a + 1, · · · , b} when a ≤ b.
Bonanno et al. [20] in 2016 considered the following discrete Dirichlet problem: N), and acquired at least two positive solutions of (2).
In 2020, Wang and Zhou [25] considered discrete Dirichlet boundary value problem as follows: The difference equations studied above involve only one variable. However, the difference equations containing two or more variables are less studied, and such difference equations are called partial difference equations. Recently, partial difference equations were widely used in many fields. Boundary value problems of partial difference equations seem to be challenging problem that has attracted many mathematical researchers [26,27].
In 2015, Heidarkhani and Imbesi [26] adopted two critical points theorems to establish multiple solutions of the partial discrete problem as shown below: with Dirichlet boundary conditions (1). Recently, in 2020, Du and Zhou [27] studied a partial discrete Dirichlet problem as follows: with Dirichlet boundary conditions (1). Inspired by the above research, we found that the perturbed partial difference equations had rarely been studied, so this paper aims at studying small solutions of the perturbed partial discrete Dirichlet problems with the (p, q)-Laplacian operator. Here, the perturbed partial difference equations mean that the term with the parameter µ in the right hand of the equation for the problem (D λ,µ ) is very small. A solution y(s, t) of (D λ,µ ) is called a small solution if the norm y(s, t) is small. In fact, without the symmetric assumption on the origin for the nonlinear term f , we can still verify that problem (D λ,µ ) possesses a sequence of solutions which converges to zero by using the Lemma 2. Moreover, by Lemma 1, we can show that all of these solutions are positive. Furthermore, by truncation techniques, we obtain two sequences of constant-sign solutions, which converge to zero (with one being positive and the other being negative). As far as we know, our study takes the lead in addressing small solutions of the perturbed partial discrete Dirichlet problems with the (p, q)-Laplacian operator.
The rest of this paper is organized as follows. In Section 2, we establish the variational framework linked to (D λ,µ ) and recall the abstract critical point theorem. In Section 3, we give the main results. In Section 4, we provide an example to demonstrate our results. We make a conclusion in the last section.

Proposition 1.
The following inequality holds: Proof. According to the result of ( [27], Proposition 1), we have the following: When y ∞ > 1, according to (9), we have the following: that is, When y ∞ ≤ 1, according to (9), we have the following: that is, In summary, we have the following: is equivalent to . Thus, we reduce the existence of the solutions of (D λ,µ ) to the existence of the critical points of Φ − λΨ on Y.
Here, we present the main tools used in this paper.

Lemma 2 (Theorem 4.3 of [28])
. Let X be a finite dimensional Banach space and let I λ : X → R be a function satisfying the following structure hypothesis: (H) I λ (u) := Φ(u) − λΨ(u) for all u ∈ X, where Φ, Ψ : X → R be two continuously Gâteux differentiable functions with Φ coercive, i.e., lim u →+∞ Φ(u) = +∞, and such that inf For all r > 0, put the following: Assume that ϕ 0 < +∞ and for every λ ∈ (0, 1 ϕ 0 ), 0 is not a local minima of functional I λ . Then, there is a sequence {u n } of pairwise distinct critical points (local minima) of I λ such that lim n→+∞ u n = 0.

Main Results
In this section, the existence of constant-sign solutions of problem (D λ,µ ) is discussed. Our aim is to use Lemma 2 for the function I ± λ : X → R, for each (s, t) ∈ Z(1, a) × Z (1, b). Then, we apply Lemma 1 or Corollary 1 to obtain our results. Let l(s, t).
In addition, put the following: It should be pointed out that if the denominator is 0, we regard 1 0 as +∞.

Remark 1.
When the nonlinear terms f and g are symmetric on the origin, i.e., f (·, −y) = − f (·, y), g(·, −y) = −g(·, y), it is easy to obtain infinitely many small solutions to problem (D λ,µ ) by using the critical point theory with symmetries. However, in this paper, we obtain infinitely many small solutions to problem (D λ,µ ) without the symmetry on f .
Similarly, we obtain the following results.
Then, for every λ ∈ 2bq+2aq+pl , problem (D λ,µ ) has a sequence of negative solutions which converges to zero.

Conclusions
In this paper, we studied the existence of small solutions of perturbed partial discrete Dirichlet problems with the (p, q)-Laplacian operator. Unlike the results in [25], we obtained some sufficient conditions of the existence of infinitely many small solutions, as shown in Theorems 1-3. Firstly, according to Theorem 4.3 of [28] and Lemma 1 of this paper, we obtained a sequence of positive solutions, which converges to zero in Theorem 1. Furthermore, by truncation techniques, we acquired two sequences of constant-sign solutions, which converge to zero (with one being positive and the other being negative). Secondly, the Corollaries 2-4 was acquired when λ = 1. Finally, as a special case of Theorem 1, we obtained a sequence of positive solutions, which converges to zero in Theorem 4. The existence of large constant-sign solutions of partial difference equations with the (p, q)-Laplacian operator will be discussed by the method used in this paper as our future research direction.