arXiv subjects
Publications and source records attributed to Abdelaaziz Sbai.
In this article, we investigate the existence and uniqueness of a positive solution for a class of singular nonlinear elliptic problem with boundary condition. Our result holds in fractional Orlicz-Sobolev spaces.
In this paper we are going to prove existence for positive solutions of the following Schrödinger-Maxwell system of singular elliptic equations: begin{equation} \left\{\begin{array}{l} u \in W_{0}^{1,2}(Ω):-\operatorname{div}\left(a(x) \nabla u\right)+ψ|u|^{r-2} u=\frac{f(x)}{u^θ}, ψ\in W_{0}^{1,2}(Ω):-\operatorname{div}(M(x) \nabla ψ)=|u|^{r} \end{array}\right. \end{equation} where $Ω$ is a bounded open set of $\mathbb{R}^{N}, N>2,$ $r>,1,$ $u>0,$ $ψ>0,$ $0 < θ<1$ and $f$ belongs to a suitable Lebesgue space. In particular, we take advantage of the coupling between the two equations of the system by demonstrating how the structure of the system gives rise to a regularizing effect on the summability of the solutions.
In this paper we study the existence and regularity of solutions to the following singular problem \begin{equation} \left\{ \begin{array}{lll} &-\displaystyle\mbox{div} \big(a(x,u)|\nabla u|^{p-2}|\nabla u|\big) + |u|^{s-1}u =\frac{f}{u^γ} &\mbox{ in } Ω\\ &u>0 &\mbox{ in }Ω\\ &u=0 &\mbox{ on } δΩ\end{array} \right. \end{equation} proving that the lower order term $u|u|^{s-1}$ has some regularizing effects on the solutions in the case of an elliptic operator with degenerate coercivity.
In this paper, we prove existence and regularity results for solutions of some nonlinear Dirichlet problems for an elliptic equation defined by a degenerate coercive operator and a singular right hand side. \begin{equation}\label{01} \left\{ \begin{array}{lll} -\displaystyle\mbox{div}( a(x,u,\nabla u))&=\displaystyle\frac{f}{u^γ} & \mbox{ in } Ω\\ u&>0 &\mbox{ in }Ω\\ u&=0 &\mbox{ on } δΩ\end{array} \right. \end{equation} where $Ω$ is bounded open subset of $I\!\!R^{N}(N\geq2),$ $γ>0$ and $ f$ is a nonnegative function that belongs to some Lebesgue space.