Positive solutions to multi-critical elliptic problems

Abstract

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 $\Omega \subset {\mathbb{R}}^N,N\ge 4$, where $\lambda >0$, $2_i^{\ast }=\frac{N+{\alpha }_i}{N-2}$ 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 $ca{t}_{\Omega }(\Omega )$ positive solutions. We also study the existence and uniqueness of solutions for the limit problem of \eqref{eq:0.1}.