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数学进展 - 2020, Vol. 49(1): 53-63
研究论文
临界Hartree方程组基态解的存在性
The Ground State Solution of Critical Hartree System

郑雨, 沈自飞
ZHENG Yu1,*, SHEN Zifei2,**

浙江师范大学数学与计算机科学学院, 金华, 浙江, 321004
College of Mathematics and Computer Science, Zhejiang Normal University, Jinhua, Zhejiang, 321004, P. R. China

收稿日期: 2018-12-22
出版日期: 2020-03-25
2020, Vol. 49(1): 53-63
DOI: 10.11845/sxjz.2018104b


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摘要 本文考虑临界耦合的 Hartree 方程组 $$ \begin{cases} -\Delta u+\lambda u =\int_{\Omega}\frac{|u(z)|^{2_{\mu}^{\ast}} }{|x-z|^{\mu}}\mathrm{d}z|u|^{2_{\mu}^{\ast}-2}u +\beta v, &\quad x\in\Omega,\\ -\Delta v+\nu v =\int_{\Omega}\frac{|v(z)|^{2_{\mu}^{\ast}} }{|x-z|^{\mu}}\mathrm{d}z|v|^{2_{\mu}^{\ast}-2}v +\beta u, &\quad x\in\Omega, \end{cases} $$ 其中 $\Omega$ 是 $\mathbb{R}^N$ 中带有光滑边界的有界区域, $N\geq3,$ $\lambda,\,\nu$ 是常数, 且满足${\lambda,\nu}>-\lambda_1(\Omega),\,\lambda_{1}(\Omega)$ 是$(-\Delta,H^1_0(\Omega))$ 的第一特征值, $\beta>0$是耦合参数, 临界指标$2_{\mu}^{\ast}=\frac{2N-\mu}{N-2}$来源于Hardy-Littlewood-Sobolev不等式, 利用变分的方法证明了临界Hartree方程组基态正解的存在性.
关键词 Hartree方程组Brezis-Nirenberg问题Hardy-Littlewood-Sobolev不等式临界指标    
Abstract:In this paper, we are interested in the following critical coupled Hartree system $$ \begin{cases} -\Delta u+\lambda u =\int_{\Omega}\frac{|u(z)|^{2_{\mu}^{\ast}} }{|x-z|^{\mu}}\mathrm{d}z|u|^{2_{\mu}^{\ast}-2}u +\beta v &\quad \mbox{in} \Omega,\\ -\Delta v+\nu v =\int_{\Omega}\frac{|v(z)|^{2_{\mu}^{\ast}} }{|x-z|^{\mu}}\mathrm{d}z|v|^{2_{\mu}^{\ast}-2}v +\beta u &\quad\mbox{in} \Omega, \end{cases} $$ where $\Omega\subset\mathbb{R}^N\,(N\geq3)$ is a smooth bounded domain, $\lambda, \nu>\lambda_{1}(\Omega)$ are constants, $\lambda_{1}(\Omega)$ is the first eigenvalue of $(-\Delta, H^{1}_{0}(\Omega))$, $\beta>0$ is a coupling parameter, $2_{\mu}^{\ast}=\frac{2N-\mu}{N-2}$ due to the Hardy-Littlewood-Sobolev inequality. By using the variational method, the existence of positive ground state solution of this system is proved.
Key wordsHartree system    Brezis-Nirenberg problem    Hardy-Littlewood-Sobolev inequality    critical exponent
PACS:  O177.91  
通讯作者: E-mail: * yzheng@zjnu.edu.cn; ** szf@zjnu.edu.cn   
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