Bifurcation from infinity of the Schrödinger equation via invariant manifolds. (December 2021)

• Record Type: Journal Article
• Authors: Li, Chunqiu; Wang, Jintao
• Is Part Of: Nonlinear analysis. Volume 213(2021)
• Publication Date: 2021-12
• Subjects: 37B30 -- 35B32 -- 35K55 -- 58J55; Conley index -- Bifurcation from infinity -- Invariant manifold -- Schrödinger equation -- Parabolic equation; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2021.112490
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: This paper is concerned with the bifurcation from infinity of the nonlinear Schrödinger equation − Δ u + V ( x ) u = λ u + f ( x, u ), x ∈ R N . We treat this problem in the framework of dynamical systems by considering the corresponding parabolic equation on unbounded domains. Firstly, we establish a global invariant manifold for the parabolic equation on R N . Then, we restrict the parabolic equation to this invariant manifold, which generates a system of finite dimension. Finally, we use the Conley index theory and the shape theory of attractors to establish some new results on bifurcations from infinity and multiplicity of solutions of the Schrödinger equation under an appropriate Landesman–Lazer type condition.