论文标题
多临界Schrödinger方程的积极解决方案
Positive solutions to multi-critical Schrödinger equations
论文作者
论文摘要
在本文中，我们研究了以下多临界schrödinger方程的多个积极解决方案\ begin {equation} \ label {p} \ begin {case} -ΔU+λv（x）u =μ| u |^{p-2} u++\ sum \ limits_ {i = 1}^{k} {k}（| x |^|^{ - （n-α_i）}*| u |^| u |^{2^{2^*_ i} \ Mathbb {r}^n，\\ \ qquad \ qquad \ qquad \ qquad \ qquad u \，\ \ in H^1（\ Mathbb {r}^n），\ end {cases} \ end {ement {equication {equation {equication}，其中$λ，μ\ $ 2^*_ i = \ frac {n+α_i} {n-2} $，带有$ n-4 <α_i<n，\，\，i = 1,2，\ cdots，k $是关键指数，$ 2 <p <2^*_ {min} _ {min} = \ {2^\ \ \ \ \ \ {2^*^{2^*_ i：i：i：i = 1,1,2,2，假设$ω= int \，v^{ - 1}（0）\ subset \ mathbb {r}^n $是一个有界的域，我们表明，对于$λ$，大于$λ$，上面的问题至少具有$cat_Ω（ω）$阳性解决方案。
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.
