Periodic Solutions for Second‐Order Partial Difference Equations With Local Superlinear Conditions

ABSTRACT This study uses critical point theory to investigate the existence of periodic solutions for second‐order partial difference equations with local superlinear conditions. It is specifically noted that the nonlinear component taken into consideration in the research displays mixed nonlinear characteristics at the origin, which can be either superlinear or asymptotically linear, but only satisfies local superlinear criteria at infinity. Due to the local superlinear condition, classical method cannot be applied. We develop new idea to address this issue and obtain at least one nonconstant periodic solution. The conclusions drawn not only enrich the relevant research on periodic solutions of second‐order difference equations but also provide a new perspective for exploring the existence of timespace periodic solutions for two‐dimensional discrete Schrödinger equations with local nonlinear growth conditions.