## Abstract

Let $\Omega $ be a bounded domain of ${\mathbb{R}}^{n}$ ($n\ge 1$) containing the origin. In the present paper we establish the weighted Hardy–Sobolev inequalities with sharp remainders. For example, when $\alpha =1-n/p$ and $1<p<+\infty $ hold, we establish the following inequality.

There exist positive numbers ${\Lambda}_{n,p,\alpha},C$, and $R$ such that we have

(0.1) $$\begin{array}{rcl}{\int}_{\mathrm{\Omega}}{|\mathrm{\nabla}u|}^{p}{|x|}^{\alpha p}\phantom{\rule{0.2em}{0ex}}dx& \ge & {\mathrm{\Lambda}}_{n,p,\alpha}{\int}_{\mathrm{\Omega}}\frac{{|u(x)|}^{p}}{{|x|}^{n}}{A}_{1}{(|x|)}^{-p}\phantom{\rule{0.2em}{0ex}}dx\\ \\ & & +C{\int}_{\mathrm{\Omega}}\frac{{|u(x)|}^{p}}{{|x|}^{n}}{A}_{1}{(|x|)}^{-p}{A}_{2}{(|x|)}^{-2}\phantom{\rule{0.2em}{0ex}}dx\end{array}$$

for any $u\in {W}_{\alpha ,0}^{1,p}(\Omega )$. Here ${A}_{1}\left(\right|x\left|\right)=log\frac{R}{\left|x\right|}$, and ${A}_{2}\left(\right|x\left|\right)=log{A}_{1}\left(\right|x\left|\right)$. This is called the critical Hardy–Sobolev inequality with a sharp remainder involving a singular weight ${A}_{1}\left(\right|x\left|{)}^{-p}{A}_{2}\right(\left|x\right|{)}^{-2}$, in the sense that the improved inequality holds for this weight but fails for any weight more singular than this one. Here ${\Lambda}_{n,p,\alpha}$ is a sharp constant independent of each function $u$. Further we establish the Hardy–Sobolev inequalities in the subcritical case $(\alpha >1-n/p)$ and the supercritical case $(\alpha <1-n/p)$.

As an application, we use our improved inequality to determine exactly when the first eigenvalues of the weighted eigenvalue problems for the operators represented by $-div\left(\right|x{|}^{\alpha p}|\nabla u{|}^{p-2}\nabla u)-\mu /\left|x{|}^{n}{A}_{1}\right(\left|x\right|{)}^{-p}|u{|}^{p-2}u$ (the critical case) will tend to zero as $\mu $ increases to ${\Lambda}_{n,p,\alpha}$. This also gives us sufficient conditions for the operators to have the positive first eigenvalue in a certain nontrivial functional framework, and we study the eigenvalue problem in the borderline case.

## Citation

Hiroshi Ando. Toshio Horiuchi. "Missing terms in the weighted Hardy–Sobolev inequalities and its application." Kyoto J. Math. 52 (4) 759 - 796, Winter 2012. https://doi.org/10.1215/21562261-1728857

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