We consider the system
$$\begin{aligned} \left\{ \begin{array}{l} -\varepsilon ^{2} \text{ div }(a(x) \nabla u)+u=Q_{u}(u,v)~\quad \text {in } \mathbb {R}^N, \\ -\varepsilon ^{2} \text{ div }(b(x) \nabla v) +v=Q_{v}(u,v)~\quad \text {in } \mathbb {R}^N, \\ u,v \in H^{1}(\mathbb {R}^N),u(x),v(x)>0\,\,\quad \,\,\, \text {for each } x \in \mathbb {R}^N, \end{array} \right. \end{aligned}$$
where
$$\varepsilon >0$$
, *a* and *b* are positive continuous potentials and *Q* is a p-homogeneous function with subcritical growth. In the first place we show existence of a ground state solution for this system. After that, we show existence of multiple solutions involving the category theory and the topology of the sets of minima of the potentials *a* and *b* . Finally, we show a concentration result. More precisely, we show that at the maximum points of each solution, the potentials *a* and *b* converge to their points of minimum points when
$$\varepsilon$$
converges to zero.