We consider the positive solutions of some integral systems related to the static
Hartreetype equations:
$$\{\begin{array}{c}\phantom{\rule{1em}{0ex}}\hfill \\ \phantom{\rule{1em}{0ex}}\hfill \\ u(x)={\int}_{{\mathbb{R}}^{N}}\frac{{u}^{p1}(y)v(y)}{xy{}^{N\tau}}dy\phantom{\rule{1em}{0ex}}\hfill & \text{in}\phantom{\rule{0.33em}{0ex}}{\mathbb{R}}^{N},\hfill \\ & \\ v(x)={\int}_{{\mathbb{R}}^{N}}\frac{{u}^{p}(y)}{\leftx{}^{\alpha}\rightxy{}^{\mu}y{}^{\beta}}dy\phantom{\rule{1em}{0ex}}\hfill & \text{in}\phantom{\rule{0.33em}{0ex}}{\mathbb{R}}^{N},\hfill \\ \phantom{\rule{1em}{0ex}}\hfill \\ \phantom{\rule{1em}{0ex}}\hfill \\ \phantom{\rule{1em}{0ex}}\hfill \end{array}$$ 
where
$N\ge 3$,
$p\ge 1$,
$0<\mu ,\tau <N$,
$\alpha ,\beta \ge 0$ and
$0<\alpha +\beta +\mu \le N$. Firstly, assuming
that the exponent
$p$
belongs to some suitable interval depending on the parameters
$\mu ,\tau ,\alpha ,\beta $, we
are able to prove some nonexistence results for the positive solution. Secondly, we
also establish some qualitative results for the integrable solution of the system like
regularity, symmetry and asymptotic behaviour. As a corollary, we deduce the
corresponding results for the equivalent weighted Hartreetype nonlocal equations.
The results obtained in this paper generalise and complement the existing results in
the literature.
