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 $$\begin{aligned} \displaystyle -\Delta u =\frac{1}{|x|^{\alpha }}\left( \int _{\mathbb {R}^{N}}\frac{|u(y)|^{2_{\alpha , \mu }^{*}}}{|x-y|^{\mu }|y|^{\alpha }}dy\right) |u|^{2_{\alpha }^{*}-2}u,\quad x\in \mathbb {R}^{N}, \end{aligned}$$ where exponent due weighted Hardy–Littlewood–Sobolev inequality and Sobolev embedding. We develop a nonlocal version concentration-compactness principle investigate existence solutions regularity, symmetry positive by moving plane arguments. In second part, subcritical case also considered, existence, symmetry, regularity are obtained.