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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} -Δu & =λ|u|^{p-2}u +\sum_{i=1}^k(|x|^{-(N-α_i)}*|u|^{2^*_i})|u|^{2^*_i-2}u\quad {\rm in}\quad Ω,\\ &u\in H^1_0(Ω)\\ \end{aligned}\right. \end{equation} in connection with the topology of the bounded domain $Ω\subset \mathbb{R}^N, \,N\geq 4$, where $λ>0$, $2^*_i=\frac{N+α_i}{N-2}$ with $N-4<α_i<N,\ \ i=1,2,\cdot\cdot\cdot, k$ 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 $λ^*>0$ such that if $0<λ<λ^*$ problem \eqref{eq:0.1} possesses at least $cat_Ω(Ω)$ positive solutions. We also study the existence and uniqueness of solutions for the limit problem of \eqref{eq:0.1}.