Ground State Solutions for the Periodic Discrete Nonlinear Schrödinger Equations with Superlinear Nonlinearities

Abstract

We consider the periodic discrete nonlinear Schrödinger equations with the temporal frequency belonging to a spectral gap. By using the generalized Nehari manifold approach developed by Szulkin and Weth, we prove the existence of ground state solutions of the equations. We obtain infinitely many geometrically distinct solutions of the equations when specially the nonlinearity is odd. The classical Ambrosetti-Rabinowitz superlinear condition is improved.

Multiple solutions for discrete periodic nonlinear Schrödinger equations

“…does not, so Theorem 1.3 also generalizes the result of Ref. 22. We believe that our results are also true for the discrete Schrödinger systems of Hamiltonian type (see Ref.…”

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

“…Remark 2.3 In [16], A. Mai and Z. Zhou treated the discrete nonlinear Schrödinger equation with superquadratic nonlinearity, they required the condition (f 4 ) s → f n (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]. In [15], they considered the following DNLS equation in M dimensional lattices:…”

Homoclinic solutions in periodic difference equations with mixed nonlinearities

“…In recent years, the existence of homoclinic solutions for difference equations has been studied by many authors. For the case where f is with superlinear nonlinearity, we refer to . And for the case where f is with saturable nonlinearity, we refer to .…”

“…The aim of this paper reads as follows. On one hand, was considered in when f is only either superlinear or asymptotically linear at , but superlinear at 0. In this paper, the nonlinear terms will mix superlinear nonlinearities with asymptotically linear ones at both and 0; see Remarks and for details.…”