Multiple solitary wave solutions of nonlinear Schrödinger systems. (English) Zbl 1255.35101

Tian, Rushun; Wang, Zhi-Qiang

Topol. Methods Nonlinear Anal. 37, No. 2, 203-223 (2011).

Summary: Consider the \(N\)-coupled nonlinear elliptic system \[ \begin{cases} -\Delta U_j+U_j=\mu U^3_j+\beta U_j\sum_{k\neq j} U^2_k \;\text{in}\;\Omega, \\ U_j>0\;\text{in}\;\Omega,\;U_j=0 \;\text{on}\;\partial\Omega, \;j=1,\dots,N, \end{cases}\tag{P} \] where \(\Omega\) is a smooth and bounded (or unbounded if \(\Omega\) is radially symmetric) domain in \(\mathbb{R}^n\), \(n \leq 3\). By using a \(Z_N\) index theory, we prove the existence of multiple solutions of (P) and show the dependence of multiplicity results on the coupling constant \(\beta\).

Reviewer: Ma Wen-Xiu (Tampa)

MSC:

35J57	Boundary value problems for second-order elliptic systems
35J61	Semilinear elliptic equations

Keywords:

nonlinear Schrödinger system; Nehari manifold; \(Z_N\)-index theory