Existence and non-existence results for DNLS systems with mixed and sign-changing terms

We prove the existence of non-trivial homoclinic solutions for a broad class of discrete nonlinear Schrödinger (DNLS) systems characterized by pinching effects and mixed-growth nonlinearities. Besides, we establish a corresponding non-existence result. Our approach is comprehensive, handling perturbation terms, a mix of sub-linear, general asymptotically linear, and super-linear behaviours, and sign-changing nonlinearities. The analysis is further extended to systems with unbounded potentials, and our assumptions are substantiated through specific examples.