In this paper, we investigated the existence of solutions to the discrete boundary value problem involving the singular $\varphi$-Laplacian. First, we extended the domain of the singular operator to the entire real line, which leads to an auxiliary problem corresponding to the original one. Then, by using critical point theory combined with the strong maximum principle, we obtained a series of conditions for the existence of one positive solution or multiple positive solutions for the original problem. Finally, three numerical examples were provided to illustrate the effectiveness of our results.

We consider a $2m$th-order nonlinear $p$-Laplacian difference equation containing both advance and retardation. Using the critical point theory, we establish some new and weaker criteria on the existence of homoclinic solutions with mixed nonlinearities.