Characterizations of compactness and weighted eigenvalue problem associated with fractional Hardy-type inequalities

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} $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_{s,p}$. 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 $W(u)={\int }_{{\mathbb{R}}^N}\mathrm{\mid }w\mathrm{\mid }\mathrm{\mid }u{\mathrm{\mid }}^p\mathrm{d}x$ 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 $w\in {\mathcal{H}}_{s,p,0}({\mathbb{R}}^N):=\overline{C_c({\mathbb{R}}^N)}in{\mathcal{H}}_{s,p}({\mathbb{R}}^N)$. 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 $(-{\mathrm{\Delta }}_p{)}^s$ is the fractional $p$-Laplace operator and $w=w_1-w_2\text{ with }{w}_1,w_2\ge 0,$ is such that $w_1\in {\mathcal{H}}_{s,p,0}({\mathbb{R}}^N)$ and $w_2\in L_{loc}^1({\mathbb{R}}^N)$.