  The aim of this paper is to study the critical elliptic equations with
Stein-Weiss type convolution parts $$ \displaystyle-\Delta u
=\frac{1}{|x|^{\alpha}}\left(\int_{\mathbb{R}^{N}}\frac{|u(y)|^{2_{\alpha,
\mu}^{\ast}}}{|x-y|^{\mu}|y|^{\alpha}}dy\right) |u|^{2_{\alpha,
\mu}^{\ast}-2}u,~~~x\in\mathbb{R}^{N}, $$ where the critical exponent is due to
the weighted Hardy-Littlewood-Sobolev inequality and Sobolev embedding. We
develop a nonlocal version of concentration-compactness principle to
investigate the existence of solutions and study the regularity, symmetry of
positive solutions by moving plane arguments. In the second part, the
subcritical case is also considered, the existence, symmetry, regularity of the
positive solutions are obtained.
