On existence of minimizers for weighted $L^{p}$-Hardy inequalities on $C^{1,γ}$-domains with compact boundary - Das, Ujjal; Pinchover, Yehuda; Devyver, Baptiste (Journal of spectral theory, 2025)
Let p \in (1,∞) , α\in \mathbb{R} , and Ω\subsetneq \mathbb{R}^{N} be a C^{1,γ} -domain with a compact boundary \partial Ω , where γ\in (0,1] . Denote by δ_{Ω}(x) the distance of a point x\in Ω to \partial Ω . Let \widetilde{W}^{1,p;α}_{0}(Ω) be the closure of C_{c}^{∞}(Ω) in \widetilde{W}^{1,p;α}(Ω) , where \widetilde{W}^{1,p;α}(Ω):= \{φ \in{W}^{1,p}_{loc}(Ω) : ( \|{}|\nabla φ|\|_{L^p(Ω;δ_{Ω}^{-α})}^{p} + \|φ\|_{L^p(Ω;δ_{Ω}^{-(α+p)})}^{p})<∞ \}{}. We study the following two variational constants: the weighted Hardy constant \begin{align*} H_{α,p}(Ω): ={}\inf \bigg\{\int_{Ω} |\nabla φ|^p δ_{Ω}^{-α} dx : \int_{Ω} |φ|^p δ_{Ω}^{-(α+p)} dx{}={}1, φ \in \widetilde{W}^{1,p;α}_0(Ω) \biggr\} , \end{align*} and the weighted Hardy constant at infinity \begin{align*} λ_{α,p}^{∞}(Ω) :=\sup_{K\Subset Ω}{} \inf_{W^{1,p}_{c}(Ω\setminus \bar K)} \bigg\{\int_{Ω\setminus \bar K} |\nabla φ|^p δ_{Ω}^{-α} dx : \int_{Ω\setminus \bar K} |φ|^p δ_{Ω}^{-(α+p)} dx=1 \biggr\}. \end{align*} We show that H_{α,p}(Ω) is attained if and only if the spectral gap Γ_{α,p}(Ω):= λ_{α,p}^{∞}(Ω)-H_{α,p}(Ω) is strictly positive. Moreover, we obtain tight decay estimates for the corresponding minimizers. Furthermore, when Ω is bounded and α+p=1 , then λ_{1-p,p}^{∞}(Ω)=0 (no spectral gap) and the associated operator -Δ_{1-p,p} is null-critical in Ω with respect to the weight δ_{Ω}^{-1} , whereas, if α +p < 1 , then λ_{α,p}^{∞}(Ω)=\bigl|\frac{α+p-1}{p}\bigr|^{p}>0=H_{α,p}(Ω) (positive spectral gap) and -Δ_{α,p} is positive-critical in Ω with respect to the weight δ_{Ω}^{-(α+p)} .