The first goal of this paper is to study necessary and sufficient conditions to obtain the attainability of the \textit{ fractional Hardy inequality } $$\Lambda_{ N}\equiv\Lambda_{ N}(\Omega):=\inf_{ \{ \phi\in \mathbb{ E}^s(\Omega, D), \phi\neq 0\}} \dfrac{ \frac{ a_{ d,s}}{ 2} \displaystyle\int_{ \mathbb{ R}^d} \int_{ \mathbb{ R}^d} \dfrac{ |\phi(x)-\phi(y)|^2}{ |x-y|^{ d+2s}}dx dy} { \displaystyle\int_\Omega \frac{ \phi^2}{ |x|^{ 2s}}\,dx}, $$ 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, $$P_{ \lambda} \, \equiv \left\{ \begin{ array}{ rcll} (-\Delta)^s u &= & \lambda \dfrac{ u}{ |x|^{ 2s}} +u^p & { \text{ in }}\Omega, u & > & 0 &{ \text{ in }} \Omega, \mathcal{ B}_{ s}u&:=&u\chi_{ D}+\mathcal{ N}_{ s}u\chi_{ N}=0 &{ \text{ in }}\mathbb{ R}^{ d}\backslash \Omega, \\ \end{ array}\right. $$ 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 $0<p\le 2_s^*-1$, $2_s^*=\frac{ 2d}{ d-2s}$. 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 (1.5) and (1.6) respectively.
Categories: math.AP