On elliptic equations with Stein-Weiss type convolution parts

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.