Positive solutions to multi-critical Schrödinger equations

In this paper, we investigate the existence of multiple positive solutions to the following multi-critical Schrödinger equation \begin{equation} \label{p} \begin{cases} -Δu+λV(x)u=μ|u|^{p-2}u+\sum\limits_{i=1}^{k}(|x|^{-(N-α_i)}* |u|^{2^*_i})|u|^{2^*_i-2}u\quad \text{in}\ \mathbb{R}^N,\\ \qquad\qquad\qquad u\,\in H^1(\mathbb{R}^N), \end{cases} \end{equation} where $λ,μ\in \mathbb{R}^+, \, N\geqslant 4$, and $2^*_i=\frac{N+α_i}{N-2}$ with $N-4<α_i<N,\,i=1,2,\cdots,k$ are critical exponents and $2<p<2^*_{min}=\min\{2^*_i:i=1,2,\cdots,k\}$. Suppose that $Ω=int\,V^{-1}(0)\subset\mathbb{R}^N$ is a bounded domain, we show that for $λ$ large, problem above possesses at least $cat_Ω(Ω)$ positive solutions.

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