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.
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