Attainability of the fractional Hardy constant with nonlocal mixed boundary conditions. Applications

Abstract

The first goal of this paper is to study necessary and sufficient conditions to obtain the attainability of the \textit{fractional Hardy inequality } $${\Lambda }_N\equiv {\Lambda }_N(\Omega ):=\underset{\{\varphi \in {\mathbb{E}}^s(\Omega ,D),\varphi \ne 0\}}{\inf }\frac{\frac{a_{d,s}}{2}{\int }_{{\mathbb{R}}^d}{\int }_{{\mathbb{R}}^d}\frac{|\varphi (x)-\varphi (y){|}^2}{|x-y{|}^{d+2s}}dxdy}{{\int }_{\Omega }\frac{{\varphi }^2}{|x{|}^{2s}}dx},$$ where $\Omega$ is a bounded domain of ${\mathbb{R}}^d$, $0<s<1$, $D\subset {\mathbb{R}}^d\setminus \Omega$ a nonempty open set and $${\mathbb{E}}^s(\Omega ,D)=\left\{u\in H^s({\mathbb{R}}^d):u=0\text{ in }D\right\}.$$ The second aim of the paper is to study the \textit{mixed Dirichlet-Neumann boundary problem} associated to the minimization problem and related properties; precisely, to study semilinear elliptic problem for the \textit{fractional laplacian}, that is, $$P_{\lambda }\equiv \{\begin{array}{rcllcccccc}(-\Delta {)}^{s}u& =& \lambda \frac{u}{|x{|}^{2s}}+u^p& \text{ in }\Omega ,u& >& 0& \text{ in }\Omega ,{\mathcal{B}}_{s}u& :=& u{\chi }_D+{\mathcal{N}}_{s}u{\chi }_N=0& \text{ in }{\mathbb{R}}^d\backslash \Omega ,\end{array}$$ with $N$ and $D$ open sets in ${\mathbb{R}}^d\backslash \Omega$ such that $N\cap D=\varnothing$ and $\overline{N}\cup \overline{D}={\mathbb{R}}^d\backslash \Omega$, $d>2s$, $\lambda >0$ and $0<p\le 2_s^{\ast }-1$, $2_s^{\ast }=\frac{2d}{d-2s}$. 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 (1.5) and (1.6) respectively.