In this paper, we investigate the existence of multiple positive solutions to following multi-critical elliptic problem $$\begin{aligned} \left\{ \begin{aligned} -\Delta u&=\lambda |u|^{p-2}u +\sum _{i=1}^k(|x|^{-(N-\alpha _i)}*|u|^{2^*_i})|u|^{2^*_i-2}u\quad \mathrm{in}\quad \Omega ,\\&u\in H^1_0(\Omega )\\ \end{aligned}\right. \end{aligned}$$ (0.1) in connection with topology bounded domain \...