In this paper, we primarily employ the critical point theory to examine the multiplicity of periodic solutions for a specific class of second-order nonlinear partial difference equations with $\varphi$-Laplacian. We can study the more general $\varphi$-Laplacian, obtain clearer sufficient conditions for the multiplicity of periodic solutions of the equation, and compare our results with existing works. Our result can be applied to exploring the spacetime periodic solutions of the two-dimensional discrete nonlinear Schrödinger equation.