Multiple standing waves for discrete nonlinear Schrödinger systems with general nonlinearities

By using variational methods, under some mild one side local conditions, we obtain infinitely many standing waves and homoclinic solutions for time-dependent and time-independent discrete nonlinear Schrödinger equations and systems with unbounded potentials, respectively. The main novelties of this paper are: (1) the primitive functions of nonlinearities may be sign-changing; (2) the conditions required on the primitive functions of nonlinearities are local in nature, therefore, the primitive functions of nonlinearities may be super-quadratic, asymptotically-quadratic and sub-quadratic at zero or infinity; (3) our results can be applied to many mathematical models.