We consider systems of the form \(\begin{aligned} \left\{ \begin{array}{l} -\Delta u + u = \dfrac{2p}{p+q}(I_\alpha *|v|^{q})|u|^{p-2}u \ \ \text { in } \mathbb {R}^N, \\ -\Delta v + v = \dfrac{2q}{p+q}(I_\alpha *|u|^{p})|v|^{q-2}v \ \ \text { in } \mathbb {R}^N, \end{array} \right. \end{aligned}\) for \(\alpha \in (0, N)\) , \(\max \left\{ \frac{2\alpha }{N}, 1\right\}< p, q < 2^*\) and \(\frac{2(N+\alpha )}{N}< p+ q < 2^{*}_{\alpha }\) , where \(I_\alpha \) denotes the Riesz potential, \(\begin{aligned} 2^* = \left\{ \begin{array}{l}\frac{2N}{N-2} \ \ \text {for} \ \ N\ge 3,\\ +\infty \ \ \text {for} \ \ N =1,2, \end{array}\right. \quad \text {and} \quad 2^*_{\alpha } = \left\{ \begin{array}{l}\frac{2(N+\alpha )}{N-2} \ \ \text {for} \ \ N\ge 3,\\ +\infty \ \ \text {for} \ \ N =1,2. \end{array} \right. \end{aligned}\) This type of systems arises in the study of standing wave solutions for a certain approximation of the Hartree theory for a two-component attractive interaction. We prove existence and some qualitative properties for ground state solutions, such as definite sign for each component, radial symmetry and sharp asymptotic decay at infinity, and a regularity/integrability result for the (weak) solutions. Moreover, we show that the straight lines \(p+q=\frac{2(N+\alpha )}{N}\) and \( p+ q = 2^{*}_{\alpha }\) are critical for the existence of solutions.
