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An improved Hardy-Sobolev inequality and its application. (English) Zbl 0987.35049
Summary: For \(\Omega \subset \mathbb{R}^{n}\), \(n \geq 2\), a bounded domain, and for \(1< p<n\), we improve the Hardy-Sobolev inequality by adding a term with a singular weight of the type \((\frac{1}{\log(1/|x|)})^{2}\). We show that this weight function is optimal in the sense that the inequality fails for any other weight function more singular than this one. Moreover, we show that a series of finite terms can be added to improve the Hardy-Sobolev inequality, which answers a question of Brezis-Vazquez. Finally, we use this result to analyze the behaviour of the first eigenvalue of the operator \( L_{\mu}u:= - (\text{div}(|\nabla u|^{p-2}\nabla u) + \frac{\mu}{|x|^{p}} |u|^{p-2}u)\) as \(\mu\) increases to \((\frac{n-p}{p})^{p}\) for \(1< p < n\).

MSC:
35J30 Higher-order elliptic equations
35P15 Estimates of eigenvalues in context of PDEs
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