<p>Through the application of the critical point theory, the problem on the existence of periodic solutions for a class of second-order partial difference equations is investigated when the nonlinear terms are superlinear or asymptotically linear growth at the origin or infinity. Specifically, sufficient conditions for the existence of two nonconstant periodic solutions of this type of equations are shown without the Ambrosetti-Rabinowitz superlinear condition and the symmetry of nonlinearities. Our result not only improves and extends the existing ones in the literature, but also applies to show the existence of space-time periodic solutions of two-dimensional discrete Schrödinger equations with mixed nonlinearities.</p>