arXiv subjects

Ahmed Attar

Publications and source records attributed to Ahmed Attar.

3 recordsLinked to original sources

arXiv2017

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 } $$Λ_{N}\equivΛ_{N}(Ω):=\inf_{\{ϕ\in \mathbb{E}^s(Ω, D), ϕ\neq 0\}} \dfrac{\frac{a_{d,s}}{2} \displaystyle\int_{\mathbb{R}^d} \int_{\mathbb{R}^d} \dfrac{|ϕ(x)-ϕ(y)|^2}{|x-y|^{d+2s}}dx dy} {\displaystyle\int_Ω\frac{ϕ^2}{|x|^{2s}}\,dx}, $$ where $Ω$ is a bounded domain of $\mathbb{R}^d$, $0 & 0 &{\text{ in }} Ω, \mathcal{B}_{s}u&:=&uχ_{D}+\mathcal{N}_{s}uχ_{N}=0 &{\text{ in }}\mathbb{R}^{d}\backslash Ω, \\ \end{array}\right. $$ with $N$ and $D$ open sets in $\mathbb{R}^d\backslashΩ$ such that $N \cap D=\emptyset$ and $\overline{N}\cup \overline{D}= \mathbb{R}^d \backslashΩ$, $d>2s$, $λ> 0$ and $0<p\le 2_s^*-1$, $2_s^*=\frac{2d}{d-2s}$. We emphasize that the nonlinear term can be critical. The operators $(-Δ)^s $, fractional laplacian, and $\mathcal{N}_{s}$, nonlocal Neumann condition, are defined below in (1.5) and (1.6) respectively.

math.AP↗

arXiv2017

Existence of positive solutions to a nonlinear elliptic system with nonlinearity involving gradient term

In this work we analyze the existence of solutions to the nonlinear elliptic system: \begin{equation*} \left\{ \begin{array}{rcll} -Δu & = & v^q+\a g & \text{in }Ω, \\ -Δv& = &|\nabla u|^{p}+łf &\text{in }Ω, \\ u=v&=& 0 & \text{on }\partial Ω,\\ u,v& \geq & 0 & \text{in }Ω, \end{array}% \right. \end{equation*} where $Ω$ is a bounded domain of $\ren$ and $p\ge 1$, $q>0$ with $pq>1$. $f,g$ are nonnegative measurable functions with additional hypotheses and $\a, ł\ge 0$. As a consequence we show that the fourth order problem \begin{equation*} \left\{ \begin{array}{rcll} Δ^2 u & = &|\nabla u|^{p}+\tildeł \tilde{f} &\text{in }Ω, \\ u=\D u&=& 0 & \text{on }\partial Ω,\\ \end{array}% \right. \end{equation*} has a solution for all $p>1$, under suitable conditions on $\tilde{f}$ and $\tildeł$.

math.AP↗

arXiv2016

On fractional p-laplacian parabolic problem with general data

In this article the problem to be studied is the following $$ (P) \left\{ \begin{array}{rcll} u_t+(-\D^s_{p}) u & = & f(x,t) & \text{ in } Ø_{T}\equiv Ω\times (0,T), \\ u & = & 0 & \text{ in }(\ren\setminusØ) \times (0,T), \\ u & \ge & 0 & \text{ in }\ren \times (0,T),\\ u(x,0) & = & u_0(x) & \mbox{ in }Ø, \end{array}% \right. $$ where $Ω$ is a bounded domain, and $(-\D^s_{p})$ is the fractional p-Laplacian operator defined by $$ (-\D^s_{p})\, u(x,t):=P.V\int_{\ren} \,\dfrac{|u(x,t)-u(y,t)|^{p-2}(u(x,t)-u(y,t))}{|x-y|^{N+ps}} \,dy$$ with $1<p<N$, $s\in (0,1)$ and $f, u_0$ are measurable functions. The main goal of this work is to prove that if $(f,u_0)\in L^1(Ø_T)\times L^1(Ø)$, problem $(P)$ has a weak solution with suitable regularity. In addition, if $f_0, u_0$ are nonnegative, we show that the problem above has a nonnegative entropy solution. In the case of nonnegative data, we give also some quantitative and qualitative properties of the solution according the values of $p$.

math.AP↗