In this paper, we consider the following semilinear elliptic systems:
$$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle -\Delta u+V(x)u=F_{u}(x, u, v),\quad \text{ in } \mathbb {R}^{N},\\ -\Delta v+V(x)v=F_{v}(x, u, v),\quad \text{ in } \mathbb {R}^{N},\\ \end{array} \right. \end{aligned}$$
where
$$V:\mathbb {R}^{N}\rightarrow \mathbb {R},~F_{u}(x,u,v)$$
and
$$F_{v}(x,u,v)$$
are periodic in x. We assume that 0 is a right boundary point of the essential spectrum of
$$-\triangle +V$$
. Under appropriate assumptions on
$$...
