In this paper, we obtain infinitely many geometrically distinct solutions with exponential decay at infinity of the discrete periodic nonlinear Schrödinger equation Lunωun = ϱgn(un), n ∈ ℤ, where ω belongs to a spectral gap of the linear operator L, ϱ = ± 1, and the potential gn(s) is symmetric in s, asymptotically or super linear with more general hypotheses as s for all n ∈ ℤ. Our arguments are based on some abstract critical point theorems about strongly indefinite functional developed recently.

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