Existence and concentration of ground states for saturable nonlinear Schrödinger equations with intensity functions in R2. (August 2018)

• Record Type: Journal Article
• Authors: Wang, Xiaoming; Lin, Tai-Chia; Wang, Zhi-Qiang
• Is Part Of: Nonlinear analysis. Volume 173(2018)
• Page Start: 19
• Page End: 36
• Publication Date: 2018-08
• Subjects: Schrödinger equations -- Saturable nonlinearity -- Ground states -- Concentrations; 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.2018.03.005
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: In this paper, we study the existence and concentration behavior of semiclassical ground states for a class of saturable nonlinear Schrödinger equations with intensity functions in R 2 : (0.1) − ε 2 Δ v + Γ I ( x ) + v 2 1 + I ( x ) + v 2 v = λ v, for x ∈ R 2 . We show that for sufficiently negatively large coupling constant Γ and sufficiently small ε there exists a family of normalized ground states (i.e., with the L 2 constraint) of the problem. We prove that the family of solutions concentrate around the maxima of the intensity function as ε → 0 . Our method is variational and depends upon a convexity technique together with the concentration-compactness method.