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Let \( \varOmega \subset \mathbb {R}^N \) be a smooth bounded domain with \( 0 \in \varOmega \) , \( N \ge 3 \) , \( 0 \le s < 2 \) , and define the critical Hardy-Sobolev exponent by \( 2^*(s) \triangleq \frac{2(N-s)}{N-2} \) . In this paper, We establish the existence of positive weak solutions of the singular critical problem \( -\varDelta u - \mu \frac{u}{|x|^2} = \frac{|u|^{2^*(s)-2}u}{|x|^s} + \lambda f(x) u \) with Dirichlet boundary conditions on \( \varOmega \) , where \( \lambda \) and \( \mu \) are positive parameters.

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Positive Solutions for Elliptic Problems with Critical Hardy-Sobolev Exponent and Singular Weight

  • 会议论文

  • 发表日期：

    2025年

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  • Abderrahmane Layati,
  • Ali Rimouche

摘要

Let \( \varOmega \subset \mathbb {R}^N \) be a smooth bounded domain with \( 0 \in \varOmega \) , \( N \ge 3 \) , \( 0 \le s < 2 \) , and define the critical Hardy-Sobolev exponent by \( 2^*(s) \triangleq \frac{2(N-s)}{N-2} \) . In this paper, We establish the existence of positive weak solutions of the singular critical problem \( -\varDelta u - \mu \frac{u}{|x|^2} = \frac{|u|^{2^*(s)-2}u}{|x|^s} + \lambda f(x) u \) with Dirichlet boundary conditions on \( \varOmega \) , where \( \lambda \) and \( \mu \) are positive parameters.