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摘要: |
研究了一类带~Hardy-Sobolev~临界指数的奇异~Kirchhoff~型方程
\begin{equation*}
\begin{cases}
-\left(a+b\displaystyle\int_{\Omega}|\nabla u|^2dx\right)\Delta u=\frac{u^{5-2s}}{|x|^{s}}+\lambda u^{-\gamma},& x\in\Omega, \u>0, & x\in\Omega, \u=0, & x\in\partial\Omega,
\end{cases}
\end{equation*}
其中~$\Omega\subset\mathbb{R}^{3}$~是一个有界开区域且具有光滑边界~$\partial\Omega$, $0\in\Omega,$ $a,b\geq0$~且~$a+b>0$, $\lambda>0,0<\gamma<1,0\leq s<1.$ 利用变分方法, 获得了该问题的一个正局部极小解. |
关键词: Kirchhoff型方程 Hardy-Sobolev临界指数 奇异 变分方法 |
DOI: |
分类号:O175.25 |
基金项目:贵州省科技厅联合基金项目(黔科合LH字[2016]7033);贵州省教育厅创新群体重大研究项目(黔教合KY[2016]046); 西华师范大学博士启动资金项目(16E014) |
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Existence of positive solutions for a class of singular Kirchhoff-type equations with critical Hardy-Sobolev exponent |
chenming,liaojiafeng
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Abstract: |
The following singular Kirchhoff-type equations with critical Hardy-Sobolev exponent is considered,
\begin{equation*}
\begin{cases}
-\left(a+b\displaystyle\int_{\Omega}|\nabla u|^2dx\right)\Delta u=\frac{u^{5-2s}}{|x|^{s}}+\lambda u^{-\gamma},& x\in\Omega, \u=0, & x\in\partial\Omega,
\end{cases}
\end{equation*}
where~$\Omega\subset\mathbb{R}^{3}$~is an open bounded domain with smooth boundary $\partial\Omega$, $0\in\Omega,$ $a,b\geq0$ and $a+b>0$, $\lambda>0,0<\gamma<1,0\leq s<1.$ The existence of positive local minimal solutions is obtained by the variational methods. |
Key words: Kirchhoff-type equation critical Hardy-Sobolev exponent singular variational method |