Non-periodic discrete nonlinear Schr#246;dinger equations with unbounded potentials and general temporal frequencies: In nitely many solitons

Abstract

We study the non-periodic discrete nonlinear Schrodinger equation -Δ u n + v n u n - ωu n = g n ( u n ), n ∈ Z, V = { v n } n ∈Z and g n are non-periodic, v n → +∞ as | n | → +∞ and the temporal frequency ω ∈ R is allowed to satisfy any one of the following three cases: (1) ω belongs to a finite spectral gap of the operator -Δ + V ; (2) ω < inf σ (-Δ + V ); (3) ω ∈ σ (-Δ + V ), where σ (-Δ + V ) denotes the spectrum of -Δ + V . We replace some global conditions by some local conditions (at infinitely or at zero) and obtain infinitely many nontrivial solitons of this equation with super linear nonlinearities by a variant fountain theorem. In particular, we also obtain the existence of nontrivial exponentially decaying solitons.