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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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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

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