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 $2<p<22^*_{min}$ with $2^*_{min}=\min\{2^*_i, \ i=1,2,\cdot\cdot\cdot, k\}$. We show that there is $\lambda^*>0$ such that if $0<\lambda<\lambda^*$ problem \eqref{eq:0.1} possesses at least $cat_\Omega(\Omega)$ positive solutions. We also study the existence and uniqueness of solutions for the limit problem of \eqref{eq:0.1}.