Existence of a Ground-State and Infinitely Many Homoclinic Solutions for a Periodic Discrete System with Sign-Changing Mixed Nonlinearities

We study the existence and multiplicity of homoclinic solutions for a class of periodic discrete systems with sign-changing mixed nonlinearities that can be super-quadratic or asymptotically quadratic both at infinity and the origin. The arising problem engages two major difficulties: one is that the corresponding variational equation is strongly indefinite and the other is that common techniques cannot be directly used to verify the linking structure and confirm the boundedness of Cerami sequences. New arguments including weak*-compactness are applied to settle these two major difficulties. This allows us to prove a ground-state as well as infinitely many geometrically distinct solutions and derive a necessary and sufficient condition for a special case. To the best of our knowledge, this is the first attempt on the existence and multiplicity of homoclinic solutions for such a discrete problem with sign-changing mixed nonlinearities. Our result also considerably improves well-known ones in the literature. Furthermore, our weaker conditions may be applicable to other variational problems.