On coupled systems of Schrödinger equations
type: Item
instance of: scholarly article
title: On coupled systems of Schrödinger equations
zbMATH Open document ID: 1232.35063
author: Zhijie Chen
author: Wenming Zou
published in: Advances in Differential Equations
publication date: 2011-10-05T00:00:00Z
review text: The system of linearly coupled semilinear Schrödinger equations \[ \begin{cases} -\Delta u+u &= f(x,u)+\lambda v, \qquad x\in \mathbb{R}^{N},\\ -\Delta v+v &= g(x,v)+\lambda u, \qquad x\in \mathbb{R}^{N},\end{cases} \] is studied. The following two cases are considered: (i) \(f(x,u)=(1+a(x)) |u|^{p-1}u\), \(g(x,v)=(1+a(x)) |v|^{p-1}v\), \(1<p<2^{\star} -1\), where \(2^{\star}\) is the critical Sobolev exponent, and (ii) \(f(x,u)=f(u)\), \(g(x,u)=g(u)\). The authors prove, under some assumptions on \(f\) and \(g\), the existence of a positive bound state in case (i) or of a positive ground state in case (ii) for \(\lambda \in (0,1)\). They also analyse the limit behavior of the positive bound or ground state as \(\lambda \rightarrow 0\) and obtain energy estimates of the state as \(\lambda\) changes. The existence of a positive ground state was previously proved in case (i) in [\textit{A. Ambrosetti, G. Cerami} and \textit{D. Ruiz}, J. Funct. Anal. 254, No. 11, 2816--2845 (2008; Zbl 1148.35080)], and in case (ii) in [\textit{H. Brézis} and \textit{E. H. Lieb}, Commun. Math. Phys. 96, 97--113 (1984; Zbl 0579.35025)]. Nevertheless, the methods of proof used in the paper under review allow a more thorough study of the dependence on \(\lambda\) of the bound or ground states and of their energy.
reviewed by: Mihai Pascu
Mathematics Subject Classification ID: 35J61
Mathematics Subject Classification ID: 35J91
Mathematics Subject Classification ID: 35Q55
Mathematics Subject Classification ID: 35A15
zbMATH DE Number: 5953773
zbMATH Keywords: coupled system of semi-linear Schrödinger equations
zbMATH Keywords: positive bound state
zbMATH Keywords: energy estimates
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 5953773