Positive solutions to multi-critical Schrödinger equations (2202.07117v1)

Abstract: In this paper, we investigate the existence of multiple positive solutions to the following multi-critical Schr\"{o}dinger equation \begin{equation} \label{p} \begin{cases} -\Delta u+\lambda V(x)u=\mu |u|^{p-2}u+\sum\limits_{i=1}^{k}(|x|^{-(N-\alpha_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 $\lambda,\mu\in \mathbb{R}^+, \, N\geqslant 4$, and $2^*_i=\frac{N+\alpha_i}{N-2}$ with $N-4<\alpha_i<N,\,i=1,2,\cdots,k$ are critical exponents and $2<p<2^_{min}=\min{2^_i:i=1,2,\cdots,k}$. 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 $cat_\Omega(\Omega)$ positive solutions.