This paper is devoted to dealing with the existence and uniqueness of positive solutions for the following coupled nonlinear Schrödinger systems with multi-parameters
$ \begin{equation*} \begin{cases} &- \varDelta u = \lambda u - \mu_1 u^3 + \beta_1 uv^2,\quad \rm {in}\ \Omega,\\ &- \varDelta v = \lambda v - \mu_2 v^3 + \beta_2 u^2v,\quad \rm {in}\ \Omega,\\ &u, v > 0\quad \rm {in}\ \Omega,\quad u, v = 0\quad \rm {on}\ \partial \Omega, \end{cases} \end{equation*} $
on the range of $ \lambda $ and the different coupling constants $ \beta_1, \beta_2 $, where $ \Omega \subset \mathbb{R}^N $ $ (N \geqslant 1) $ is a bounded smooth domain, $ \lambda > 0 $ and $ \mu_1 \leqslant \mu_2 $. Under some conditions, we establish some interesting positive solutions structure theorems in the $ \beta_1 \beta_2 $-plane, especially we obtain the new structure theorems for the cases that $ \mu_1 $ and $ \mu_2 $ have different signs or they are negative. In addition, we get interesting uniqueness results via synchronized solutions techniques.
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