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 $N-4<α_i 0$ 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}.
math.AP↗