Missing terms in the weighted Hardy–Sobolev inequalities and its application

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) $$\begin{array}{rcl}{\int }_{\mathrm{\Omega }}{|\mathrm{\nabla }u|}^p{|x|}^{\alpha p}dx& \ge & {\mathrm{\Lambda }}_{n,p,\alpha }{\int }_{\mathrm{\Omega }}\frac{{|u(x)|}^p}{{|x|}^n}{A}_1{(|x|)}^{-p}dx\\ \\ & & +C{\int }_{\mathrm{\Omega }}\frac{{|u(x)|}^p}{{|x|}^n}{A}_1{(|x|)}^{-p}{A}_2{(|x|)}^{-2}dx\end{array}$$

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.