Characterizations of compactness and weighted eigenvalue problem associated with fractional Hardy-type inequalities
Ujjal Das, Rohit Kumar, Abhishek Sarkar
TL;DR
The article investigates a nonlocal fractional Hardy-type framework, establishing when the energy map $W(u)=\int |w||u|^p$ is compact on the fractional Beppo Levi space $\mathcal{D}^{s,p}(\mathbb{R}^N)$ via Maz'ya-type capacities and a concentration-compactness approach. It proves that $W$ is compact if and only if $w$ has an absolutely continuous norm in $\mathcal{H}_{s,p}(\mathbb{R}^N)$, equivalently $w\in \mathcal{H}_{s,p,0}(\mathbb{R}^N)$ with vanishing concentration at zero and infinity. The paper then analyzes a weighted fractional $p$-Laplacian eigenvalue problem with $w=w_{1}-w_{2}$, proving the existence of a positive, simple first eigenvalue and, using Ljusternik-Schnirelmann theory, an infinite sequence of eigenvalues diverging to infinity. The results bridge nonlocal Hardy-type inequalities, capacity theory, and variational eigenvalue techniques, providing sharp compactness criteria and a rich spectral theory for weights in this fractional setting.
Abstract
In this article, we consider the following fractional {Hardy-type} inequality: \begin{align} \label{Fractional Hardy_abst} \int_{\mathbb{R}^N} |w(x)||u(x)|^p \mathrm{d}x \leq C \int_{\mathbb{R}^N \times \mathbb{R}^N} \frac{|u(x)-u(y)|^p}{|x-y|^{N+sp}} \mathrm{d}x\mathrm{d}y:= \|u\|_{s,p}^p\,, \ \forall u \in \mathcal{D}^{s,p}(\mathbb{R}^N), \end{align} where $0<s<1<p<\frac{N}{s}$, and $\mathcal{D}^{s,p}(\mathbb{R}^N)$ is the completion of $C_c^1(\mathbb{R}^N)$ with respect to the {norm} $\|\cdot\|_{s,p}$. We denote the space of admissible {weight function} $w$ in \eqref{Fractional Hardy_abst} by $\mathcal{H}_{s,p}(\mathbb{R}^N)$. Maz'ya-type characterization helps us to define a Banach function norm on $\mathcal{H}_{s,p}(\mathbb{R}^N)$. Using the Banach function space structure and the concentration compactness type arguments, we provide several characterizations for the compactness of the map ${W}(u)= \int_{{\mathbb{R}^N}} |w| |u|^p \mathrm{d}x$ on $\mathcal{D}^{s,p}(\mathbb{R}^N)$. In particular, we prove that ${W}$ is compact on $\mathcal{D}^{s,p}(\mathbb{R}^N)$ if and only if $w \in \mathcal{H}_{s,p,0}(\mathbb{R}^N):=\overline{C_c(\mathbb{R}^N)} \ \mbox{in} \ \mathcal{H}_{s,p}(\mathbb{R}^N)$. Further, we study the following {weighted} eigenvalue problem: \begin{equation*} (-Δ_{p})^{s}u = λw(x) |u|^{p-2}u ~~\text{in}~\mathbb{R}^{N}, \end{equation*} where $(-Δ_{p})^{s}$ is the fractional $p$-Laplace operator and $w = w_{1} - w_{2}~\text{with}~ w_{1},w_{2} \geq 0,$ is such that $ w_{1} \in \mathcal{H}_{s,p,0}(\mathbb{R}^N)$ and $w_{2} \in L^{1}_{loc}(\mathbb{R}^N)$.