相关论文: A new look at the fractional Poisson problem via t…

Let $u_s$ denote a solution of the fractional Poisson problem $$ (-\Delta)^s u_s = f\quad\text{ in }\Omega,\qquad u_s=0\quad \text{ on }\mathbb{R}^N\setminus \Omega, $$ where $N\geq 2$ and $\Omega\subset \mathbb{R}^N$ is a bounded domain of…

In this paper, we analyze the $s$-dependence of the solution $u_s$ to the fractional Poisson equation $(-\Delta)^s_{\Omega} = f$ in an open bounded set $\Omega \subset \mathbb{R}^N$. Precisely, we show that the solution map $(0,1)\to…

We study families $u_s$ of functions satisfying the equations $(-\Delta)^s u_s=0$, $s \in (0,1)$ in a smooth bounded open set $\Omega \subset \mathbb{R}^N$. The main purpose of this paper is twofold. First, we provide a detailed analysis of…

We analyze the limit behavior as $s\to 1^-$ of the solution to the fractional Poisson equation $(-\Delta)^s u_s=f_s$, $x\in\Omega$ with homogeneous Dirichlet boundary conditions $u_s\equiv 0$, $x\in\Omega^c$. We show that $\lim_{s\to 1^-}…

This paper is devoted to the Laplacian operator of fractional order $s\in (0,1)$ in several dimensions. We consider the equation $(-\Delta)^su=f(x,u)$ in $\Omega$, $u=0$ in $\Omega^c$ and establish a representation formula for partial…

We introduce a definition of the fractional Laplacian $(-\Delta)^{s(\cdot)}$ with spatially variable order $s:\Omega\to [0,1]$ and study the solvability of the associated Poisson problem on a bounded domain $\Omega$. The initial motivation…

We study the fractional laplacian problem (-\Delta)^s u &=& u^p -\epsilon u^q \quad\text{in }\quad \Omega, u &\in& H^s(\Omega)\cap L^{q+1}(\Omega),u &>&0 \quad\text{in }\quad \Omega, u&=&0 \quad\text{in}\quad \mathbb{R}^N\setminus\Omega,…

In this paper we prove the Pohozaev identity for the semilinear Dirichlet problem $(-\Delta)^s u = f(u)$ in $\Omega$, $u \equiv 0$ in $\mathbb R^n\setminus\Omega$. Here, $s\in(0,1)$, $(-\Delta)^s$ is the fractional Laplacian in $\mathbb…

This work concerns with the existence of solutions for the following class of nonlocal elliptic problems \begin{equation*}\label{00} \left\{ \begin{array}{l} (-\Delta)^{s}u + u = |u|^{p-2}u\;\;\mbox{in $\Omega$},\\ u \geq 0 \quad \mbox{in}…

We study the regularity up to the boundary of solutions to the Neumann problem for the fractional Laplacian. We prove that if $u$ is a weak solution of $(-\Delta)^s u=f$ in $\Omega$, $\mathcal N_s u=0$ in $\Omega^c$, then $u$ is $C^\alpha$…

We provide a Hopf boundary lemma for the regional fractional Laplacian $(-\Delta)^s_{\Omega}$, with $\Omega\subset\mathbb{R}^N$ a bounded open set. More precisely, given $u$ a pointwise or weak super-solution of the equation…

In this paper, we consider the following problem involving fractional Laplacian operator: \begin{equation}\label{eq:0.1} (-\Delta)^{\alpha} u= |u|^{2^*_\alpha-2-\varepsilon}u + \lambda u\,\, {\rm in}\,\, \Omega,\quad u=0 \,\, {\rm on}\, \,…

In this paper we study a Neumann problem for the fractional Laplacian, namely \begin{equation}\left\{ \begin{array}{rcll} \varepsilon^{2s}(- \Delta)^{s}u + u &=& f(u) \ \ &\mbox{in} \ \ \Omega \\ \mathcal{N}_{s}u &=& 0 , \,\, &\text{in}…

We study the regularity up to the boundary of solutions to the Dirichlet problem for the fractional Laplacian. We prove that if $u$ is a solution of $(-\Delta)^s u = g$ in $\Omega$, $u \equiv 0$ in $\R^n\setminus\Omega$, for some…

For $s\in(0,1)$ and an open bounded set $\Omega\subset\mathbb R^n$, we prove existence and uniqueness of absolute minimisers of the supremal functional $$E_\infty(u)=\|(-\Delta)^s u\|_{L^\infty(\mathbb R^n)},$$ where $(-\Delta)^s$ is the…

Let $\Omega \subset \mathbb{R}^N$, $N \geq 2$, be a smooth bounded domain. For $s \in (1/2,1)$, we consider a problem of the form \[ \left\{\begin{aligned} (-\Delta)^s u & = \mu(x)\, \mathbb{D}_s^{2}(u) + \lambda f(x)\,, & \quad \mbox{in}…

This paper deals with the obstacle problem for the fractional infinity Laplacian with nonhomogeneous term $f(u)$, where $f:\mathbb{R}^+ \mapsto \mathbb{R}^+$: $$\begin{cases} L[u]=f(u) &\qquad in \{u>0\}\\ u \geq 0 &\qquad in\, \Omega\\ u=g…

Let $\Omega $ be an open, smooth, bounded subset of $ \Bbb R ^n$. In connection with the fractional Laplacian $(-\Delta )^a$ ($a>0$), and more generally for a $2a$-order classical pseudodifferential operator ($\psi $do) $P$ with even…

We provide closed formulas for (unique) solutions of nonhomogeneous Dirichlet problems on balls involving any positive power $s>0$ of the Laplacian. We are able to prescribe values outside the domain and boundary data of different orders…

We look for solutions of $(-\Delta)^s u+f(u) = 0$ in a bounded smooth domain $\Omega$, $s\in(0,1)$, with a strong singularity at the boundary. In particular, we are interested in solutions which are $L^1(\Omega)$ and higher order with…