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

Abstract

In this paper, we consider the periodic discrete nonlinear equation Lu n − ωu n = ±g n (u n ), n ∈ Z, lim |n|→∞ u n = 0, where L is a Jacobi operator, and the nonlinearities g n (s) are asymptotically linear as |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.

Ground state solutions for periodic discrete nonlinear Schrödinger equations with saturable nonlinearities

“…Then a necessary and sufficient condition for the existence of gap solitons of the DNLS equations was given by Chen, Ma and Wang [21] when V n ≡ V 0 (n ∈ Z, 0 < V 0 ∈ R).…”

The existence of positive ground state solutions for the Choquard type equation on groups of polynomial growth

“…In recent years, people paid attention to the analysis on discrete spaces, especially for the nonlinear equations, see for example [26,27,35,13,40,20,28,69,29,34,31,30]. As far as we know, there is no existence results for the nonlinear Choquard equation on graphs as the continuous setting.…”

The ground state solutions of nonlinear Schrödinger equations with Hardy weights on lattice graphs

“…Later, Chen and Ma [6] proved the existence of ground state solitons and the existence of infinitely many pairs of geometrically distinct solitons by the generalized Nehari manifold method developed by Szulkin and Weth [36]. Moreover, Chen and Ma [5,7] established the existence of nontrivial solutions with asymptotically or super linear terms by a variant generalized weak linking theorem. For related works, we refer readers to [21,34,35,37,41].…”