Homoclinic solutions of discrete nonlinear Schrödinger equations with asymptotically or super linear terms

In this paper, we study the following periodic discrete nonlinear Schrödinger equation-Δun+(∊n-ω)un=γχngn(un),n∈Z,where the nonlinearity gngn is asymptotically or super linear. The sufficient conditions on the existence and on the nonexistence of nontrivial solitons was established by using critical point theory.