Attainability of the fractional hardy constant with nonlocal mixed boundary conditions: Applications

The first goal of this paper is to study necessary and sufficient conditions to obtain the attainability of the fractional Hardy inequality

$\Lambda_{N}\equiv \Lambda_{N}(\Omega): = \inf\limits_{\{\varphi\in \mathbb{E}^{s}(\Omega, D), \varphi \ne0\}}\dfrac{\frac{a_{d, s}}{2}\displaystyle\int_{\mathbb R^d}\int_{\mathbb R^d}\dfrac{|\varphi(x)-\phi(y)|^{2}}{|x-y|^{d+2s}}dx dy}{\displaystyle\int_{\Omega}\frac{\varphi^2}{|x|^{2s}}\, dx}, $

where $\Omega$ is a bounded domain of $\mathbb R^d$, $0<s<1$, $D\subset \mathbb R^d\setminus \Omega$ a nonempty open set, $N = (\mathbb R^d\setminus \Omega)\setminus\overline{D}$ and

$\mathbb{E}^{s}(\Omega, D) = \{ u \in H^s(\mathbb R^d):\, u = 0 \text{ in } D\}.$

The second aim of the paper is to study the mixed Dirichlet-Neumann boundary problem associated to the minimization problem and related properties; precisely, to study semilinear elliptic problem for the fractional Laplacian, that is,

${P_\lambda } \equiv \left\{ {\begin{array}{*{20}{l}}{{{\left( { - \Delta } \right)}^s}u\;\;\; = \;\;\;\lambda \frac{u}{{|x{|^{2s}}}} + {u^p}}&{{\rm{in}}\;\Omega ,}\\{\;\;\;\;\;\;\;\;\;u\;\;\; > \;\;\;0}&{{\rm{in}}\;\Omega ,}\\{\;\;\;\;\;\;{{\cal B}_s}u\;\;\;: = \;\;u{\chi _D} + {{\cal N}_s}u{\chi _N} = 0}&{{\rm{in}}\;{{\mathbb {R}}^d}\backslash \Omega ,}\end{array}} \right.$

with $N$ and $D$ open sets in $\mathbb R^{d}\backslash\Omega$ such that $N \cap D = \emptyset$ and $\overline{N}\cup \overline{D} = \mathbb R^{d}\backslash\Omega$, $d>2s$, $\lambda> 0$ and $<p\le 2_s^*-1_s^* = \frac{2d}{d-2s}$. We emphasize that the nonlinear term can be critical.

The operators $(-\Delta)^s $, fractional Laplacian, and $\mathcal{N}_{s}$, nonlocal Neumann condition, are defined below in (7) and (8) respectively.