Three positive solutions for a nonlinear partial discrete Dirichlet problem with ( p , q ) -Laplacian operator

In this paper, we prove the existence of three solutions to a partial diﬀerence equation with ( p , q )-Laplacian operator by using critical point theory. Furthermore, based on the strong maximum principle, we prove that the three solutions are positive under appropriate nonlinearity assumptions. Finally, we also give an example to illustrate our main results.


Introduction
Let Z and R denote the sets of integers and real numbers, respectively. Define Z(a, b) = {a, a + 1, . . . , b} for a ≤ b.
It should be noted that the above-mentioned difference equations have only one variable. However, difference equations involving two or more variables have rarely been studied and are called partial difference equations. In recent years, the partial difference equations have been extensively employed in various domains. However, it should be mentioned that the boundary value problem of the partial difference equation is a challenging problem attracting many mathematical researchers [35,36].
Heidarkhani and Imbesi [35] in 2015 considered the partial discrete Dirichlet problem with boundary conditions (1.1) and proved the existence of at least three solutions of (1.8).
Lately, Du and Zhou [36] in 2020 studied the partial discrete Dirichlet problem (s λ ): with boundary conditions (1.1) and proved the existence of multiple solutions of (s λ ). Compared with the results of the partial difference equations with p-Laplacian, those with (p, q)-Laplacian have rarely been studied. Thus in this paper, we demonstrate the existence of three solutions to a partial difference equation with (p, q)-Laplacian operator by using different methods. Furthermore, based on the strong maximum principle, we prove that the three solutions are positive under appropriate nonlinearity assumptions.
The main tool of this paper is as follows.
, the functionλ is coercive. Then, for each λ ∈ r , the functionalλ has at least three distinct critical points in X.
The rest of this paper is organized as follows. In Sect. 2, we establish the variational framework associated with ( λ ). In Sect. 3, we present our main results. Finally, in Sect. 4, we present an example illustrating our main results.

Preliminaries
In this section, we establish the variational framework associated with ( λ ). We consider the cd-dimensional Banach space We also define the other norm for w ∈ V . Obviously, , ∈ C 1 (V , R), that is, 1 , 2 , and are continuously Fréchet differentiable in V , and Therefore for all w, s ∈ V , Obviously, w is a critical point of the functionalλ in V if and only if it is a solution of problem ( λ ). Therefore we reduce the existence of solutions of ( λ ) to the existence of the critical points ofλ on V .

Main results
Now we state the following theorem.
By the assumed conditions we have  , l), w(k, l)) cd (4 p Therefore there is a positive constant s such that According to [36, Proposition 1], we have Combining (3.1) with (3.5), we have Thus we obtain lim w →+∞ (w)λ (w) = +∞, that is, I λ is coercive. Therefore condition (a 2 ) of Lemma 1.1 is verified.
Thus we have proved that all assumptions of Lemma 1.1 are satisfied, so that the functional (w)λ (w) possesses at least three distinct critical points. Since w = 0 is not a solution of problem ( λ ), it possesses at least three nontrivial solutions. Therefore the proof of Theorem 3.1 is completed.
From Theorem 3.1 we have the following: Suppose that the following conditions are satisfied: Then for every λ ∈ , problem ( λ ) possesses at least three positive solutions.
Note that From Lemma 2.1 we can conclude that w > 0 for (k, l) ∈ Z(1, c) × Z(1, d). Problem ( λ + ) possesses at least three positive solutions. Since problem ( λ + ) shares the same solutions with problem ( λ ), the latter possesses at least three positive solutions. Therefore the proof of Corollary 3.2 is completed.
As a particular case of problem ( λ ), we consider the following problem ( vg ):