Ground state solutions for periodic discrete nonlinear Schr\xf6dinger equations with saturable nonlinearities

Luyu Zhang,Shiwang Ma

doi:10.1186/s13662-018-1574-2

Abstract

In this paper, we study a class of periodic discrete nonlinear Schrödinger equations with asymptotically linear nonlinearities, and prove the existence of ground state solutions (i.e., nontrivial solutions with the least possible energy) by the linking theorem directly. By weakening some conditions, our conclusions extend some existing results.

Highlights

The discrete nonlinear Schrödinger (DNLS) equations belong to the most important inherently discrete models, playing a crucial role in the modeling of a great variety of phenomena, ranging from solid-state and condensed-matter physics to biology ([1,2,3])
Many works in the literature have considered the existence of discrete solitons of the DNLS equations; see Refs. [7,8,9,10,11,12]
Results are obtained for such equations with superlinear nonlinearity [13,14,15,16,17,18,19] and saturable nonlinearity [20,21,22,23]

Summary

Introduction

The discrete nonlinear Schrödinger (DNLS) equations belong to the most important inherently discrete models, playing a crucial role in the modeling of a great variety of phenomena, ranging from solid-state and condensed-matter physics to biology ([1,2,3]). 3 is devoted to the proof of the existence of ground state solutions for equation (1.2). Zhou treated the discrete nonlinear Schrödinger equation with superquadratic nonlinearity, they required the condition (f4) s → fn(s)/|s| is strictly increasing on (–∞, 0) and (0, ∞) for all n ∈ Z, and obtained ground state solutions by using the generalized Nehari manifold approach developed by Szulkin and Weth [27].

Results

Conclusion