Standing Wave Solutions for the Discrete Coupled Nonlinear Schrödinger Equations with Unbounded Potentials

and Applied Analysis 3 Proof. First we show thatN ̸ = 0. From (15) and (16), we rewrite J (φ, ψ) = 1 2 ( 󵄩󵄩󵄩󵄩φ 󵄩󵄩󵄩󵄩 2 E1 − ω 1 󵄩󵄩󵄩󵄩φ 󵄩󵄩󵄩󵄩 2 l 2) + 1 2 ( 󵄩󵄩󵄩󵄩ψ 󵄩󵄩󵄩󵄩 2 E2 − ω 2 󵄩󵄩󵄩󵄩ψ 󵄩󵄩󵄩󵄩 2 l 2) − 1 4 ∑ n∈Z (a 1 φ 4 n + a 2 ψ 4 n + 2a 3 φ 2 n ψ 2 n ) , (18) I (φ, ψ) = 󵄩󵄩󵄩󵄩φ 󵄩󵄩󵄩󵄩 2 E1 − ω 1 󵄩󵄩󵄩󵄩φ 󵄩󵄩󵄩󵄩 2 l 2 + 󵄩󵄩󵄩󵄩ψ 󵄩󵄩󵄩󵄩 2 E2 − ω 2 󵄩󵄩󵄩󵄩ψ 󵄩󵄩󵄩󵄩 2 l 2 − ∑ n∈Z (a 1 φ 4 n + a 2 ψ 4 n + 2a 3 φ 2 n ψ 2 n ) . (19) Let (φ, ψ) ∈ (E 1 − {0}) × (E 2 − {0}); then by (19) I (tφ, tψ) = t 2 ( 󵄩󵄩󵄩󵄩φ 󵄩󵄩󵄩󵄩 2 E1 − ω 1 󵄩󵄩󵄩󵄩φ 󵄩󵄩󵄩󵄩 2 l 2) + t 2 ( 󵄩󵄩󵄩󵄩ψ 󵄩󵄩󵄩󵄩 2 E2 − ω 2 󵄩󵄩󵄩󵄩ψ 󵄩󵄩󵄩󵄩 2 l 2) − t 4 ∑ n∈Z (a 1 φ 4