Positive solutions to multi-critical elliptic problems

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 \(\Omega \subset {\mathbb {R}}^N, \,N\ge 4\), where \(\lambda >0\), \(2^*_i=\frac{N+\alpha _i}{N-2}\) \(N-4<\alpha _i0\) such if \(0<\lambda <\lambda ^*\) possesses at least \(cat_\Omega (\Omega )\) solutions. also study uniqueness for limit (0.1).