Ground state solutions for periodic Discrete nonlinear Schrödinger equations

Xionghui Xu; Jijiang Sun; Xionghui Xu; Jijiang Sun

doi:10.3934/math.2021755

AIMS Mathematics

2021, Volume 6, Issue 12: 13057-13071. doi: 10.3934/math.2021755

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Research article

Ground state solutions for periodic Discrete nonlinear Schrödinger equations

Xionghui Xu ,
Jijiang Sun ^,

Department of Mathematics, Nanchang University, Nanchang 330031, China

Received: 23 June 2021 Accepted: 07 September 2021 Published: 14 September 2021
MSC : Primary: 35Q55; Secondary: 35Q51, 39A12, 39A70

In this paper, we consider the following periodic discrete nonlinear Schrödinger equation

$ \begin{equation*} Lu_{n}-\omega u_{n} = g_{n}(u_{n}), \qquad n = (n_{1}, n_{2}, ..., n_{m})\in \mathbb{Z}^{m}, \end{equation*} $

where $ \omega\notin \sigma(L) $(the spectrum of $ L $) and $ g_{n}(s) $ is super or asymptotically linear as $ |s|\to\infty $. Under weaker conditions on $ g_{n} $, the existence of ground state solitons is proved via the generalized linking theorem developed by Li and Szulkin and concentration-compactness principle. Our result sharply extends and improves some existing ones in the literature.
- discrete nonlinear Schrödinger equation,
- ground state,
- superlinear,
- asymptotically linear,
- periodic potential
Citation: Xionghui Xu, Jijiang Sun. Ground state solutions for periodic Discrete nonlinear Schrödinger equations[J]. AIMS Mathematics, 2021, 6(12): 13057-13071. doi: 10.3934/math.2021755

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Abstract

In this paper, we consider the following periodic discrete nonlinear Schrödinger equation

$ \begin{equation*} Lu_{n}-\omega u_{n} = g_{n}(u_{n}), \qquad n = (n_{1}, n_{2}, ..., n_{m})\in \mathbb{Z}^{m}, \end{equation*} $

where $ \omega\notin \sigma(L) $(the spectrum of $ L $) and $ g_{n}(s) $ is super or asymptotically linear as $ |s|\to\infty $. Under weaker conditions on $ g_{n} $, the existence of ground state solitons is proved via the generalized linking theorem developed by Li and Szulkin and concentration-compactness principle. Our result sharply extends and improves some existing ones in the literature.

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