Standing waves for discrete Schrödinger equations in infinite lattices with saturable nonlinearities

In this paper, we consider the periodic discrete nonlinear equation{Lun−ωun=±gn(un),n∈Z,lim|n|→∞⁡un=0, where L   is a Jacobi operator, and the nonlinearities gn(s)gn(s) are asymptotically linear as |s|→∞|s|→∞. In the two different cases (ω is a spectral endpoint of L, or it belongs to a finite spectral gap of L), we obtain the existence of nontrivial solitons of this equation by using variational methods. In particular, a necessary and sufficient condition is obtained for the existence of gap solitons of the nonlinear equation. Here, solitons appear when we look for standing waves of some discrete nonlinear Schrödinger equations.