A COMPLETE CLASSIFICATION OF GROUND-STATES FOR A COUPLED NONLINEAR SCHRODINGER SYSTEM

作者：Liu, Chuangye; Wang, Zhi-Qiang^*

来源：Communications on Pure and Applied Analysis, 2017, 16(1): 115-130.

DOI：10.3934/cpaa.2017005

• Ground states
• synchronized solutions
• coupled Schrodinger systems

In this paper, we establish the existence of nontrivial ground-state solutions for a coupled nonlinear Schrodinger system
-Delta u(j) + u(j) = Sigma(m)(i=1)b(ij)u(i)(2)u(j,) in R-n,
u(j) (x) -> 0 as vertical bar x vertical bar -> infinity, j = 1,2, . . . , m,
where n = 1, 2, 3, m >= 2 and b(ij) are positive constants satisfying b(ij) = b(ij). By nontrivial we mean a solution that has all components non-zero. Due to possible systems collapsing it is important to classify ground state solutions. For m = 3, we get a complete picture that describes whether nontrivial ground state solutions exist or not for all possible cases according to some algebraic conditions of the matrix B = (b(ij)). In particular, there is a nontrivial ground state solution provided that all coupling constants b(ij), i not equal j are sufficiently large as opposed to cases in which any ground-state solution has at least a zero component when b(ij), i not equal j are all sufficiently small. Moreover, we prove that any ground-state solution is synchronized when matrix B = (b(ij)) is positive semi-definite.

• 出版日期2017-1
• 单位天津大学; 华中师范大学