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

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):= \{\varphi \in{W}^{1,p}_{\mathrm{loc}}(\Omega) : ( \|{}|\nabla \varphi{}|\|_{L^p(\Omega;\delta_{\Omega}^{-\alpha})}^{p} + \|\varphi\|_{L^p(\Omega;\delta_{\Omega}^{-(\alpha+p)})}^{p})<\infty \}{}. We study the following two variational constants: the weighted Hardy constant \begin{align*} H_{\alpha,p}(\Omega): ={}\inf \bigg\{\int_{\Omega} |\nabla \varphi|^p \delta_{\Omega}^{-\alpha} \mathrm{d}x : \int_{\Omega} |\varphi|^p \delta_{\Omega}^{-(\alpha+p)} \mathrm{d}x{}={}1, \varphi \in \widetilde{W}^{1,p;\alpha}_0(\Omega) \biggr\} , \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 \bar K)} \bigg\{\int_{\Omega\setminus \bar K} |\nabla \varphi|^p \delta_{\Omega}^{-\alpha} \mathrm{d}x : \int_{\Omega\setminus \bar K} |\varphi|^p \delta_{\Omega}^{-(\alpha+p)} \mathrm{d}x=1 \biggr\}. \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. 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)=\bigl|\frac{\alpha+p-1}{p}\bigr|^{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)} .