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 $\Omega\subsetneq \mathbb{R}^N$ 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 $\widetilde{W}^{1,p;\alpha}_0(\Omega)$ be the closure of $C_c^{\infty}(\Omega)$ in $\widetilde{W}^{1,p;\alpha}(\Omega)$, where $$\widetilde{W}^{1,p;\alpha}(\Omega):= \left\{\varphi \in {W}^{1,p}_{\mathrm{loc}} (\Omega) \mid \left( \| \, |\nabla \varphi \, |\|_{L^p(\Omega;\delta_{\Omega}^{-\alpha})}^p + \|\varphi\|_{L^p(\Omega;\delta_{\Omega}^{-(\alpha+p)})}^p\right)<\infty \!\right\}.$$ We study the following two variational constants: the weighted Hardy constant \begin{align*} H_{\alpha,p}(\Omega): =\!\inf \left\{\int_{\Omega} |\nabla \varphi|^p \delta_{\Omega}^{-\alpha} \mathrm{d}x \biggm| \int_{\Omega} |\varphi|^p \delta_{\Omega}^{-(\alpha+p)} \mathrm{d}x\!=\!1, \varphi \in \widetilde{W}^{1,p;\alpha}_0(\Omega) \right\} , \end{align*} and the weighted Hardy constant at infinity \begin{align*} \lambda_{\alpha,p}^{\infty}(\Omega) :=\sup_{K\Subset \Omega}\, \inf_{W^{1,p}_{c}(\Omega\setminus \overline{K})} \left\{\int_{\Omega\setminus \overline{K}} |\nabla \varphi|^p \delta_{\Omega}^{-\alpha} \mathrm{d}x \biggm| \int_{\Omega\setminus \overline{K}} |\varphi|^p \delta_{\Omega}^{-(\alpha+p)} \mathrm{d}x=1 \right\}. \end{align*} 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.