Notes on gap solitons for periodic discrete nonlinear Schrödinger equations

Consider a class of periodic discrete Schrödinger equations L u n − ω u n = σ f nu n, n ∈ Z . Under the condition of f nu n= χ nu n2 u n , by applying the linking theorem, Pankov obtained the existence of gap solitons. Pankov's result was then generalized to various type of nonlinear function f n , but most of studying works were concentrated on the case of f n ( t ) = O ( | t |s 1) as t →0 ( s 1 ⩾ 1 ) . In this paper, by ingeniously using Ekeland variational principle, for the nonlinearity which meets f n ( t ) = O ( | t |s 1) as t →0 ( s 1  ∈ (0,1)), we obtain the existence of gap solitons, which supplements the existing ones and gives a positive answer in part to the open problem proposed by Pankov.