In this paper, we study discrete elliptic Dirichlet problems. Applying a variational technique together with Morse theory, we establish several results on the existence and multiplicity of nontrivial solutions. Finally, two examples and numerical simulations are provided to illustrate our theoretical results.

In this paper, we consider the existence and multiplicity of nontrivial solutions to second order partial difference equation with Dirichlet boundary conditions by Morse theory. Given suitable conditions, we establish multiple results that the problem admits at least two nontrivial solutions. Moreover, we provide five examples to illustrate applications of our theorems.

By combining variational techniques with the saddle point theorem, we investigate the existence and nonexistence of periodic solutions to second-order partial difference equations involving p-Laplacians. Our obtained results generalize and complement some known ones. Finally, we display some examples and numerical simulations to show the validity of our main results.

In this paper, we investigate the existence and nonexistence of positive solutions to a class of nonlocal partial difference equations via a variant version of the mountain pass theorem. The conditions in our obtained results release the classical (AR) condition in some sense.

In this article, applying variational technique as well as critical point theory, we establish a series of criteria to ensure the existence and multiplicity of nontrivial periodic solutions to a second-order nonlinear partial difference equation. Our results generalize some known results. Moreover, numerical stimulations are presented to illustrate applications of our major findings.

By variational technique coupled with the mountain pass lemma and fountain theorem, we investigate a second-order partial difference equation involving the mean curvature operator in the present paper. We establish a number of criteria to guarantee the existence of multiple nontrivial homoclinic solutions. Our results generalize and improve some known ones. Additionally, two examples are provided to demonstrate applications of our obtained results.