Positive solutions to multi-critical elliptic problems

In this paper, we investigate the existence of multiple solutions to the following multi-critical elliptic problem \begin{equation}\label{eq:0.1} \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 {\rm in}\quad \Omega,\ &u\in H^1_0(\Omega)\ \end{aligned}\right. \end{equation} in connection with the topology of the bounded domain $\Omega\subset $\mathbb{R}$^N, ,N\geq 4$,where$\lambda>0$,$2^_i=\frac{N+\alpha_i}{N-2}$with$N-4<\alpha_i<N,\ \ i=1,2,\cdot\cdot\cdot, k$ are critical Hardy-Littlewood-Sobolev exponents and $2<p<22_{min}^{\ast }$ with $2_{min}^{\ast }=\min \{2_i^{\ast },i=1,2,\cdot \cdot \cdot ,k\}$. We show that there is ${\lambda }^{\ast }>0$ such that if $0<\lambda <{\lambda }^{\ast }$ problem \eqref{eq:0.1} possesses at least $ca{t}_{\mathrm{\Omega }}(\mathrm{\Omega })$ positive solutions. We also study the existence and uniqueness of solutions for the limit problem of \eqref{eq:0.1}.