Positive solutions to multi-critical Schrödinger equations

Abstract

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 $\lambda ,\mu \in {\mathbb{R}}^{+},N⩾4$, and $2_i^{\ast }=\frac{N+{\alpha }_i}{N-2}$ with $N-4<{\alpha }_i<N,i=1,2,\cdots ,k$ are critical exponents and $2<p<2_{min}^{\ast }=\min \{2_i^{\ast }:i=1,2,\cdots ,k\}$. Suppose that $\Omega =int{V}^{-1}(0)\subset {\mathbb{R}}^N$ is a bounded domain, we show that for $\lambda$ large, problem above possesses at least $ca{t}_{\Omega }(\Omega )$ positive solutions.

20 pages