arXiv subjects

Boumediene Abdellaoui

Publications and source records attributed to Boumediene Abdellaoui.

22 records · Page 2Linked to original sources

arXiv2015

The effect of the Hardy potential in some Calderón-Zygmund properties for the fractional Laplacian

The goal of this paper is to study the effect of the Hardy potential on the existence and summability of solutions to a class of nonlocal elliptic problems $$ \left\{\begin{array}{rcll} (-Δ)^s u-λ\dfrac{u}{|x|^{2s}}&=&f(x,u) &\hbox{ in } Ω,\\ u&=&0 &\hbox{ in } \mathbb{R}^N\setminusΩ,\\ u&>&0 &\hbox{ in }Ω, \end{array}\right. $$ where $(-Δ)^s$, $s\in(0,1)$, is the fractional laplacian operator, $Ω\subset \mathbb{R}^N$ is a bounded domain with Lipschitz boundary such that $0\inΩ$ and $N>2s$. We will mainly consider the solvability in two cases: 1) The linear problem, that is, $f(x,t)=f(x)$, where according to the summability of the datum $f$ and the parameter $λ$ we give the summability of the solution $u$. 2) The problem with a nonlinear term $f(x,t)=\frac{h(x)}{t^σ}$ for $t>0$. In this case, existence and regularity will depend on the value of $σ$ and on the summability of $h$. Looking for optimal results we will need a weak Harnack inequality for elliptic operators with \emph{singular coefficients} that seems to be new.

math.AP↗

arXiv2015

Optimal results for the fractional heat equation involving the Hardy potential

In this paper we study the influence of the Hardy potential in the fractional heat equation. In particular, we consider the problem $$(P_θ)\quad \left\{ \begin{array}{rcl} u_t+(-Δ)^{s} u&=&ł\dfrac{\,u}{|x|^{2s}}+θu^p+ c f\mbox{ in } Ω\times (0,T),\\ u(x,t)&>&0\inn Ω\times (0,T),\\ u(x,t)&=&0\inn (\ren\setminusΩ)\times[ 0,T),\\ u(x,0)&=&u_0(x) \mbox{ if }x\inØ, \end{array} \right. $$ where $N> 2s$, $0 1$, $c,ł>0$, $u_0\ge 0$, $f\ge 0$ are in a suitable class of functions and $θ=\{0,1\}$. Notice that $(P_0)$ is a linear problem, while $(P_1)$ is a semilinear problem. The main features in the article are: \begin{enumerate} \item Optimal results about \emph{existence} and \emph{instantaneous and complete blow up} in the linear problem $(P_0)$, where the best constant $Λ_{N,s}$ in the fractional Hardy inequality provides the threshold between existence and nonexistence. Similar results in the local heat equation were obtained by Baras and Goldstein in \cite{BaGo}. However, in the fractional setting the arguments are much more involved and they require the proof of a weak Harnack inequality for a weighted operator that appear in a natural way. Once this Harnack inequality is obtained, the optimal results follow as a simpler consequence than in the classical case. \item The existence of a critical power $p_+(s,λ)$ in the semilinear problem $(P_1)$ such that: \begin{enumerate} \item If $p> p_+(s,λ)$, the problem has no weak positive supersolutions and a phenomenon of \emph{complete and instantaneous blow up} happens. \item If $p< p_+(s,λ)$, there exists a positive solution for a suitable class of nonnegative data. \end{enumerate} \end{enumerate}

math.AP↗

arXiv2012

Some existence and regularity results for porous media and fast diffusion equations with a gradient term

In this paper we consider the problem $$(P)\qquad \{{array}{rclll} u_t-\D u^m&=&|\n u|^q +\,f(x,t),&\quad u\ge 0 \hbox{in} Ω_T\equiv Ω\times (0,T), u(x,t)&=&0 &\quad \hbox{on} \partialΩ\times (0,T) u(x,0)&=&u_0(x),&\quad x\in Ω{array}. $$ where $Ø\subset \ren$, $N\ge 2$, is a bounded regular domain, $1<q\le 2$, and $f\ge 0$, $u_0\ge 0$ are in a suitable class of functions. We obtain some results for elliptic-parabolic problems with measure data related to problem $(P)$ that we use to study the existence of solutions to problem $(P)$ according with the values of the parameters $q$ and $m$.

math.AP↗

arXiv2003

Existence and multiplicity for perturbations of an equation involving Hardy inequality and critical Sobolev exponent in the whole R^N

In order to obtain solutions to problem $$ {{array}{c} -Δu=\dfrac{A+h(x)} {|x|^2}u+k(x)u^{2^*-1}, x\in {\mathbb R}^N, u>0 \hbox{in}{\mathbb R}^N, {and}u\in {\mathcal D}^{1,2}({\mathbb R}^N), {array}. $$ $h$ and $k$ must be chosen taking into account not only the size of some norm but the shape. Moreover, if $h(x)\equiv 0$, to reach multiplicity of solution, some hypotheses about the local behaviour of $k$ close to the points of maximum are needed.

math.AP↗