Positive solutions to multi-critical elliptic problems

Abstract

In this paper, we investigate the existence of multiple positive solutions to the following multi-critical elliptic problem

$$\begin{aligned} \left\{ \begin{aligned} -\Delta u&=\lambda |u|^{p-2}u +\sum _{i=1}^k(|x|^{-(N-\alpha _i)}*|u|^{2^*_i})|u|^{2^*_i-2}u\quad \mathrm{in}\quad \Omega ,\\&u\in H^1_0(\Omega )\\ \end{aligned}\right. \end{aligned}$$

(0.1)

in connection with the topology of the bounded domain $\Omega \subset {\mathbb {R}}^N, \,N\ge 4$, 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 $2<p<2^*=\frac{2N}{N-2}$. We show that there is $\lambda ^*>0$ such that if $0<\lambda <\lambda ^*$ problem (0.1) possesses at least $cat_\Omega (\Omega )$ positive solutions. We also study the existence and uniqueness of positive solutions for the limit problem of (0.1).