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 \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

Fractional heat equation involving Hardy-Leray Potential

In this paper we analyse the existence and non-existence of non-negative solutions to a non-local parabolic equation with a Hardy-Leray type potential. More precisely, we consider the problem $$\begin{cases} (w_t-\Delta w)^s=\frac{\lambda}{|x|^{2s}} w+w^p +f, &\text{ in }\mathbb{R}^N\times (0,+\infty),\\ w(x,t)=0, &\text{ in }\mathbb{R}^N\times (-\infty,0], \end{cases}$$ where $N> 2s$, $0<s<1$ and $0<\lambda<\Lambda_{N,s}$, the optimal constant in the fractional Hardy-Leray inequality. In particular we show the existence of a critical existence exponent $p_{+}(\lambda, s)$ and of a Fujita-type exponent $F(\lambda,s)$ such that the following holds: - Let $p>p_+(\lambda,s)$. Then there are not any non-negative supersolutions. - Let $p<p_+(\lambda,s)$. Then there exist local solutions while concerning global solutions we need to distinguish two cases: - Let $1< p\le F(\lambda,s)$. Here we show that a weighted norm of any positive solution blows up in finite time. - Let $F(\lambda,s)<p<p_+(\lambda,s)$. Here we prove the existence of global solutions under suitable hypotheses

A nonlocal concave-convex problem with nonlocal mixed boundary data

The aim of this paper is to study a nonlocal equation with mixed Neumann and Dirichlet external conditions. We prove existence, nonexistence and multiplicity of positive energy solutions and analyze the interaction between the concave-convex nonlinearity and the Dirichlet-Neumann data