preprint

Inserted: 3 feb 2024
Last Updated: 12 jun 2024

Year: 2023

Abstract:

In this article, we consider the following weighted fractional Hardy inequality:

$ \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), $

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 seminorm $||\cdot||_{s,p}$. We denote the space of admissible $w$ in the above inequality 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 eigenvalue problem:

$ (-\Delta_{p})^{s}u = \lambda w(x) |u|^{p-2}u ~~\text{in}~\mathbb{R}^{N},$

where $(-\Delta_{p})^{s}$ 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 $w_{2} \in L^{1}_{loc}(\mathbb{R}^N)$.