arXiv subjects

Mahmoud Ben Fredj

Publications and source records attributed to Mahmoud Ben Fredj.

3 recordsLinked to original sources

arXiv2015

Blow up boundary solutions of some semilinear fractional equations in the unit ball

For $γ>0$, we are interested in blow up solutions $u\in C^+(B)$ of the fractional problem in the unit ball $B$ \begin{equation}\label{2nov} \left\{\begin{array} {rcll} Δ^{\fracα{2}} u &=& u^γ&\ \text{in }B\\ u &=& 0&\ \text{in }B^c.\end{array}\right. \end{equation} We distinguish particularly two orders of singularity at the boundary: solutions exploding at the same rate than $δ^{1-\fracα{2}}$ ($δ$ denotes the Euclidean distance) and those higher singular than $δ^{1-\fracα{2}}.$ As a consequence, it will be shown that the classical Keller-Osserman condition can not be readopted in the fractional setting.

math.AP↗

arXiv2015

Nonnegative entire bounded solutions to some semilinear equations involving the fractional Laplacian

The main goal is to establish necessary and sufficient conditions under which the fractional semilinear elliptic equation $Δ^{\fracα{2}} u=ρ(x)\,φ(u)$ admits nonnegative nontrivial bounded solutions in the whole space $\mathbb{R}^N$.

math.AP↗

arXiv2012

Comparison of harmonic kernels associated to a class of semilinear elliptic equations

Let $D$ be a smooth domain in $\mathbb{R}^N$, $N\geq 3$ and let $f$ be a positive continuous function on $\partial D$. Under some assumptions on $φ$, it is shown that the problem $Δu=2φ(u)$ in $D$ and $u=f$ on $\partial D$, admits a unique solution which will be denoted by $H_D^φf$. Given two functions $φ$ and $ψ$, our main goal in this paper is to investigate the existence of a constant $c>0$ such that $$\frac{1}{c}H_D^φf\leq H_D^ψf\leq c H_D^φf.$$

math.AP↗