Standing waves for nonlinear Hartree type equations: existence and qualitative properties

We consider systems of the form $$\{\begin{array}{l}-\mathrm{\Delta }u+u=\frac{2p}{p+q}(I_{\alpha }\ast \mathrm{\mid }v{\mathrm{\mid }}^q)\mathrm{\mid }u{\mathrm{\mid }}^{p-2}u\text{ in }{\mathbb{R}}^N,\\ -\mathrm{\Delta }v+v=\frac{2q}{p+q}(I_{\alpha }\ast \mathrm{\mid }u{\mathrm{\mid }}^p)\mathrm{\mid }v{\mathrm{\mid }}^{q-2}v\text{ in }{\mathbb{R}}^N,\end{array}$$ for $\alpha \in (0,N)$, $\max \{\frac{2\alpha }{N},1\}<p,q<2^{\ast }$ and $\frac{2(N+\alpha )}{N}<p+q<2_{\alpha }^{\ast }$, where $I_{\alpha }$ denotes the Riesz potential, $$2^{\ast }=\{\begin{array}{l}\frac{2N}{N-2}\text{ for }N\ge 3,\\ +\mathrm{\infty }\text{ for }N=1,2,\end{array}\text{and}{2}_{\alpha }^{\ast }=\{\begin{array}{l}\frac{2(N+\alpha )}{N-2}\text{ for }N\ge 3,\\ +\mathrm{\infty }\text{ for }N=1,2.\end{array}$$ 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 $p+q=2_{\alpha }^{\ast }$ are critical for the existence of solutions.