preprint
Inserted: 3 feb 2024
Last Updated: 3 feb 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 \eqref{Fractional Hardyabst} 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: \begin{equation}
(-\Delta{p}){s}u = \lambda w(x)
u
{p-2}u ~~\text{in}~\mathbb{R}{N},
\end{equation} 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 $w_{2} \in
L^{1}_{loc}(\mathbb{R}^N)$.