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

Let $p\in (1,\mathrm{\infty })$, $\alpha \in \mathbb{R}$, and $\mathrm{\Omega }⊊{\mathbb{R}}^N$ be a $C^{1,\gamma }$-domain with a compact boundary $\mathrm{\partial }\mathrm{\Omega }$, where $\gamma \in (0,1]$. Denote by ${\delta }_{\mathrm{\Omega }}(x)$ the distance of a point $x\in \mathrm{\Omega }$ to $\mathrm{\partial }\mathrm{\Omega }$. Let ${\tilde{W}}_0^{1,p;\alpha }(\mathrm{\Omega })$ be the closure of $C_c^{\mathrm{\infty }}(\mathrm{\Omega })$ in ${\tilde{W}}^{1,p;\alpha }(\mathrm{\Omega })$, where $${\tilde{W}}^{1,p;\alpha }(\mathrm{\Omega }):=\{\phi \in W_{\mathrm{l}\mathrm{o}\mathrm{c}}^{1,p}(\mathrm{\Omega })\mid (\mathrm{\parallel }\mathrm{\mid }\mathrm{\nabla }\phi \mathrm{\mid }{\mathrm{\parallel }}_{L^p(\mathrm{\Omega };{\delta }_{\mathrm{\Omega }}^{-\alpha })}^p+\mathrm{\parallel }\phi {\mathrm{\parallel }}_{L^p(\mathrm{\Omega };{\delta }_{\mathrm{\Omega }}^{-(\alpha +p)})}^p)<\mathrm{\infty }\text{\hspace{0.17em}⁣}\}\mathrm{.}$$ We study the following two variational constants: the weighted Hardy constant \begin{align*} H_{α,p}(Ω): =!\inf \left{\int_Ω |\nabla \varphi|^p δ_Ω^{-α} $\mathrm{d}$x \biggm| \int_Ω |\varphi|^p δ_Ω^{-(α+p)} $\mathrm{d}$x!=!1, \varphi \in \widetilde{W}^{1,p;α}0(Ω) \right} , \end{align*} and the weighted Hardy constant at infinity \begin{align*} λ{α,p}^{\infty}(Ω) :=\sup_{K\Subset Ω}, \inf_{W^{1,p}{c}(Ω\setminus \overline{K})} \left{\int{Ω\setminus \overline{K}} |\nabla \varphi|^p δ_Ω^{-α} $\mathrm{d}$x \biggm| \int_{Ω\setminus \overline{K}} |\varphi|^p δ_Ω^{-(α+p)} $\mathrm{d}$x=1 \right}. \end{align*} We show that $H_{\alpha ,p}(\mathrm{\Omega })$ is attained if and only if the spectral gap ${\mathrm{\Gamma }}_{\alpha ,p}(\mathrm{\Omega }):={\lambda }_{\alpha ,p}^{\mathrm{\infty }}(\mathrm{\Omega })-H_{\alpha ,p}(\mathrm{\Omega })$ is strictly positive. Moreover, we obtain tight decay estimates for the corresponding minimizers.