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

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} $\Vert \cdot {\Vert }_{s,p}$. 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 $(-\hat{I}{”}_p{)}^s$ 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 $w_2\in L_{loc}^1({\mathbb{R}}^N)$.