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