| Summary: | We consider the system of coupled nonlinear Schrodinger equations
$$\displaylines{
-\varepsilon^2\Delta u+a(x) u=H_{u}(x, u, v)+\mu(x) v, \quad
x\in \mathbb{R}^N,\cr
-\varepsilon^2\Delta v+b(x) v=H_{v}(x, u, v)+\mu(x) u, \quad
x\in \mathbb{R}^N,\cr
u,v\in H^1(\mathbb{R}^N),
}$$
where $N\geq 3$, $a, b, \mu \in C(\mathbb{R}^N)$ and
$H_{u}, H_{v}\in C(\mathbb{R}^N\times \mathbb{R}^2, \mathbb{R})$.
Under conditions that $a_0=\inf a=0$ or $b_0=\inf b=0$ and
$|\mu(x)|^2\le \theta a(x)b(x)$ with $\theta\in(0, 1)$
and some mild assumptions on $H$, we show that the system has at least one
nontrivial solution provided that
$0<\varepsilon\le \varepsilon_0$, where the bound $\varepsilon_0$ is
formulated in terms of N, a, b and H.
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