Infinitely many solutions for the discrete Schrödinger equations with a nonlocal term.

In the present paper, we consider the following discrete Schrödinger equations − (a + b ∑ k ∈ Z | Δ u k − 1 | 2) Δ 2 u k − 1 + V k u k = f k (u k) k ∈ Z , where a, b are two positive constants and V = { V k } is a positive potential. Δ u k − 1 = u k − u k − 1 and Δ 2 = Δ (Δ) is the one-dimensional discrete Laplacian operator. Infinitely many high-energy solutions are obtained by the Symmetric Mountain Pass Theorem when the nonlinearities { f k } satisfy 4-superlinear growth conditions. Moreover, if the nonlinearities are sublinear at infinity, we obtain infinitely many small solutions by the new version of the Symmetric Mountain Pass Theorem of Kajikiya. [ABSTRACT FROM AUTHOR]