Homoclinic solutions of periodic discrete nonlinear Schrödinger equations with local growth conditions

We study the existence and multiplicity of nontrivial homoclinic solutions for a class of periodic discrete nonlinear Schrödinger (DNLS) equations when the nonlinearities can be local superlinear or local asymptotically linear at infinity by means of the critical point theory. Principal challenges of the resulting problem entail the restriction of local growth conditions and the boundedness of Cerami sequences. We incorporate weak*-compactness for bounded sequences, unique continuation of discrete Schrödinger equations, and Lusternik-Schnirelman theory under symmetries to solve the problem. This allows us for the first time to derive a ground state and infinitely many homoclinic solutions with such assumptions, improving the relevant works in the literature. Our result implies the existence of infinitely many exponentially localized states of periodic discrete Schrödinger equations with nonlinear impurity. Our work also answers a problem of A. Pankov from the 2010's on the existence of nontrivial homoclinic solutions for periodic DNLS equations.