On existence of minimizers for weighted $L^p$-Hardy inequalities on $C^{1,\gamma }$-domains with compact boundary

Abstract

Let $p\in (1,\infty )$, $\alpha \in \mathbb{R}$, and $\Omega ⊊{\mathbb{R}}^N$ be a $C^{1,\gamma }$-domain with a compact boundary $\partial \Omega$, where $\gamma \in (0,1]$. Denote by ${\delta }_{\Omega }(x)$ the distance of a point $x\in \Omega$ to $\partial \Omega$. Let ${\tilde{W}}_0^{1,p;\alpha }(\Omega )$ be the closure of $C_c^{\infty }(\Omega )$ in ${\tilde{W}}^{1,p;\alpha }(\Omega )$, where

We study the following two variational constants: the weighted Hardy constant

and the weighted Hardy constant at infinity

We show that $H_{\alpha ,p}(\Omega )$ is attained if and only if the spectral gap ${\Gamma }_{\alpha ,p}(\Omega ):={\lambda }_{\alpha ,p}^{\infty }(\Omega )-H_{\alpha ,p}(\Omega )$ is strictly positive. Moreover, we obtain tight decay estimates for the corresponding minimizers. Furthermore, when $\Omega$ is bounded and $\alpha +p=1$, then ${\lambda }_{1-p,p}^{\infty }(\Omega )=0$ (no spectral gap) and the associated operator $-{\Delta }_{1-p,p}$ is null-critical in $\Omega$ with respect to the weight ${\delta }_{\Omega }^{-1}$, whereas, if $\alpha +p<1$, then ${\lambda }_{\alpha ,p}^{\infty }(\Omega )=|\frac{\alpha +p-1}{p}{|}^p>0=H_{\alpha ,p}(\Omega )$ (positive spectral gap) and $-{\Delta }_{\alpha ,p}$ is positive-critical in $\Omega$ with respect to the weight ${\delta }_{\Omega }^{-(\alpha +p)}$.