Homoclinic solutions of discrete nonlinear Schrödinger equations with unbounded potentials. (January 2022)

• Record Type: Journal Article
• Authors: Zhou, Ben-Xing; Liu, Chungen
• Is Part Of: Applied mathematics letters. Volume 123(2022)
• Publication Date: 2022-01
• Subjects: Homoclinic solutions -- Discrete nonlinear Schrödinger equations -- Saddle point reduction -- Conley index theory; Applied mathematics -- Periodicals; 519.05
• DOI: 10.1016/j.aml.2021.107575
• Languages: English
• ISSNs: 0893-9659

Abstract

Abstract: In this paper, the existence and multiplicity of homoclinic solutions for discrete nonlinear Schrödinger equations − Δ u k + v k u k − ω u k = f k ( u k ) are considered. With asymptotical nonlinearity and unbounded potential V = { v k : k ∈ Z } satisfying lim ∣ k ∣ → ∞ v k = ∞, we achieve the existence of nontrivial homoclinic solution via the saddle point reduction method and Conley index theory. Furthermore, we can prove there are at least two nontrivial homoclinic solutions with extra assumptions about the nonlinearity f k .