On some nonlinear elliptic PDEs with Sobolev–Hardy critical exponents and a Li–Lin open problem

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• 作者：G. Cerami ; X. Zhong ; W. Zou
• 关键词：35J15 ; 35J20 ; 35J91
• 刊名：Calculus of Variations and Partial Differential Equations
• 出版年：2015
• 出版时间：October 2015
• 年：2015
• 卷：54
• 期：2
• 页码：1793-1829
• 全文大小：743 KB
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• 作者单位：G. Cerami (1)
  X. Zhong (2)
  W. Zou (2)
  
  1. Dipartimento di Matematica, Politecnico di Bari, Campus Universitario, Via Orabona 4, 70125, Bari, Italy
  2. Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, China
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Analysis
  Systems Theory and Control
  Calculus of Variations and Optimal Control
  Mathematical and Computational Physics
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1432-0835

文摘

Let \(\Omega \) be a \(C^1\) open bounded domain in \(\mathbb {R}^N, \, N\ge 3,\) with \(0\in \bar{\Omega }.\) We consider the following problem involving Hardy–Sobolev critical exponents: $$\begin{aligned} (P)\qquad \qquad {\left\{ \begin{array}{ll} \textstyle \Delta u+\lambda \frac{u^p}{|x|^{s_1}}+\frac{u^{2^*(s_2)-1}}{|x|^{s_2}}=0 \quad \hbox {in}\,\,\,\Omega ,\\ \textstyle u(x)>0\,\hbox {in}\,\,\,\Omega ; \,\,\, u(x)=0\,\,\quad \hbox {on}\,\,\partial \Omega , \end{array}\right. } \end{aligned}$$