arXiv subjects

Zeineb Ghardallou

Publications and source records attributed to Zeineb Ghardallou.

3 recordsLinked to original sources

arXiv2018

Large versus bounded solutions to sublinear elliptic problems

Let $L $ be a second order elliptic operator with smooth coefficients defined on a domain $Ω\subset \mathbb{R}^d$ (possibly unbounded), $d\geq 3$. We study nonnegative continuous solutions $u$ to the equation $L u(x) - φ(x, u(x))=0$ on $Ω$, where $φ$ is in the Kato class with respect to the first variable and it grows sublinearly with respect to the second variable. Under fairly general assumptions we prove that if there is a bounded non zero solution then there is no large solution.

math.AP↗

arXiv2018

Sublinear elliptic problems under radiality. Harmonic $NA$ groups and Euclidean spaces

Let $Ł$ be the Laplace operator on $\R ^d$, $d\geq 3$ or the Laplace Beltrami operator on the harmonic $NA$ group (in particular on a rank one noncompact symmetric space). For the equation $ Łu - φ(\cdot,u)=0$ we give necessary and sufficient conditions for the existence of entire bounded or large solutions under the hypothesis of radiality of $φ$ with respect to the first variable. A Harnack-type inequality for positive continuous solutions is also proved.

math.DG↗

arXiv2017

Positive solutions to sublinear elliptic problem

Let $L$ be a second order elliptic operator $L$ with smooth coefficients defined on a domain $Ω$ in $\mathbb{R}^d $, $d\geq3$, such that $L1\leq 0$. We study existence and properties of continuous solutions to the following problem \begin{equation}\label{00} Lu=φ(\cdot,u),% & \hbox{in $Ω$; in the sens of distribution;} \\ \end{equation} in $Ω,$ where $Ω$ is a Greenian domain for $L$ {(possibly unbounded)} in $\mathbb{R}^d$ and $φ$ is a nonnegative function on $Ω\times [0,+\infty [$ increasing with respect to the second variable. By means of thinness, we obtain a characterization of $φ$ for which \eqref{00} has a nonnegative nontrivial bounded solution.

math.AP↗