arXiv subjects
Publications and source records attributed to Kamel Saoudi.
This study is devoted to proving the existence of weak solutions for a nonlinear elliptic problem with Neumann-type boundary data. The problem is driven by a discontinuous power nonlinearity and a nonsmooth prescribed data. Additionally, we aim to derive an estimate that proves the well-posedness of the problem. This estimate serves as an evidence for the uniqueness of the existing solution when the boundary term is ``smooth".
We consider the following convective Neumann systems:\begin{equation*}\left(\mathrm{S}\right)\qquad\left\{\begin{array}{ll}-Δ_{p_1}u_1+\frac{|\nabla u_1|^{p_1}}{u_1+δ_1}=f_1(x,u_1,u_2,\nabla u_1,\nabla u_2) & \text{in}\;Ω,\\ -Δ_{p_2}u_2+\frac{|\nabla u_2|^{p_2}}{u_2+δ_2}=f_2(x,u_1,u_2,\nabla u_1,\nabla u_2)&\text{in}\;Ω, \\ |\nabla u_1|^{p_1-2}\frac{\partial u_1}{\partial η}=0=|\nabla u_2|^{p_2-2}\frac{\partial u_2}{\partial η}&\text{on}\;\partialΩ,\end{array}\right.\end{equation*}where $Ω$ is a bounded domain in $\mathbb{R}^{N}$ ($N\geq 2$) with a smooth boundary $\partialΩ$,$δ_1,\,δ_2 >0$ are small parameters, $η$ is the outward unit vector normal to $\partial Ω,$ $f_1,\,f_2:Ω\times\mathbb{R}^2\times\mathbb{R}^{2N}\rightarrow \mathbb{R}$ are Carathéodory functions that satisfy certain growth conditions, and $Δ_{p_i}$ ($1<p_i<N,$ for $i=1,2$) are the $p$-Laplace operators $Δ_{p_i}u_i=\mathrm{div}(|\nabla u_i|^{p_i-2}\nabla u_i)$,for every $\,u_i\in W^{1,p_i}(Ω).$ In order to prove the existence of solutions to such systems, we use a sub-supersolution method. We also obtain nodal solutions by constructing appropriate sub-solution and super-solution pairs. To the best of our knowledge, such systems have not been studied yet.
Using variational methods, we establish the existence of infinitely many solutions to an elliptic problem driven by a Choquard term and a singular nonlinearity. We further show that if the problem has a positive solution, then it is bounded a.e. in the domain $Ω$ and is Hölder continuous.
Existence of nodal (i.e., sign changing) solutions and constant sign solutions for quasilinear elliptic equations involving convection-absorption terms are presented. A location principle for nodal solutions is obtained by means of constant sign solutions whose existence is also derived. The proof is chiefly based on sub-supersolutions technique together with monotone operator theory.
We study the following singular problem involving the p$(x)$-Laplace operator $Δ_{p(x)}u= div(|\nabla u|^{p(x)-2}\nabla u)$, where $p(x)$ is a nonconstant continuous function, \begin{equation} \nonumber {(\rm P_λ)} \left\{\begin{aligned} - Δ_{p(x)} u & = a(x)|u|^{q(x)-2}u(x)+ \frac{λb(x)}{u^{δ(x)}} \quad\mbox{in}\,Ω,\\ u &>0 \quad\mbox{in}\,Ω, \\ u & =0 \quad\mbox{on}\,\partialΩ.\end{aligned} \right. \end{equation} Here, $Ω$ is a bounded domain in $\mathbb{R}^{N\geq2}$ with $C^2$-boundary, $λ$ is a positive parameter, $a(x), b(x) \in C(\overlineΩ)$ are positive weight functions with compact support in $Ω$, and $δ(x),$ $p(x),$ $q(x) \in C(\overlineΩ)$ satisfy certain hypotheses ($A_{0}$) and ($A_{1}$). We apply the Nehari manifold approach and some new techniques to establish the multiplicity of positive solutions for problem ${(\rm P_λ)}$.
The purpose of this paper is to study nonlinear singular parabolic equations with $p(x)$- Laplacian. Precisely, we consider the following problem and discuss the existence of a non-negative weak solution. \begin{align*} \frac{\partial u}{\partial t}-Δ_{p(x)}u&=λu^{q(x)-1} + u^{-δ(x)}g+ f&&\text{in}~Q_T, u&= 0&&\text{on}~Σ_T, u(0,\cdot)&=u_0(\cdot)&&\text{in}~Ω\nonumber. \end{align*} Here $Q_T=Ω\times(0,T)$, $Σ_T=\partialΩ\times(0,T)$, $Ω$ is a bounded domain in $\mathbb{R}^N$ ($N\geq 2$) with Lipschitz continuous boundary $\partialΩ$, $λ\in(0,\infty)$, $f\in L^1(Q_T)$, $g\in L^\infty(Ω)$, $u_0\in L^r(Ω)$ with $r\geq 2$, $δ:\overlineΩ\rightarrow(0,\infty)$ is continuous, and $p,q\in C(\overlineΩ)$ with $\underset{x\in\overlineΩ}{\max}~p(x)<N$, $q(\cdot)<p^*(\cdot)$. The article is distinguished into two cases according to the choice of $f$ with different range of parameters $p(\cdot)$, $q(\cdot)$.
In this article, we prove the existence of solutions to a nonlinear nonlocal elliptic problem with a singualrity and a discontinuous critical nonlinearity which is given as follows. \begin{align} \begin{split}\label{main_prob} (-Δ)_p^su&=μg(x,u)+\fracλ{u^γ}+H(u-α)u^{p_s^*-1},~\text{in}~Ω u&>0,~\text{in}~Ω, u&=0,~\text{in}~\mathbb{R}^N\setminusΩ, \end{split} \end{align} where $Ω\subset\mathbb{R}^N$ is a bounded domain with Lipschitz boundary, $s\in (0,1)$, $2 0$, $α\geq 0$ is real, $H$ is the Heaviside function, i.e. $H(a)=0$ if $a\leq 0$, $H(a)=1$ if $a>0$ and $p_s^*=\frac{Np}{N-sp}$ is the fractional critical Sobolev exponent. Under suitable assumptions on the function $g$, we prove the existence of solution to the problem. Furthermore, we show that as $α\rightarrow0^+$, the sequence of solutions of $\eqref{main_prob}$ for each such $α$ converges to a solution of the problem for which $α=0$.
This article concerns about the existence of a positive SOLA (Solutions Obtained as Limits of Approximations) for the following singular critical Choquard problem involving fractional power of Laplacian and a critical Hardy potential.
In this paper we study the existence of multiple nontrivial positive weak solutions to the following system of problems. \begin{align*} \begin{split} -Δ_{p}u-Δ_q u &= λf(x)|u|^{r-2}u+ν\frac{1-α}{2-α-β}h(x) |u|^{-α}|v|^{1-β}\,\,\mbox{in}\,\,Ω,\\ -Δ_{p}v-Δ_q v &= μg(x)|v|^{r-2}v+ν\frac{1-β}{2-α-β}h(x) |u|^{1-α}|v|^{-β}\,\,\mbox{in}\,\,Ω,\\ u,v&>0\,\,\mbox{in}\,\,Ω,\\ u= v &= 0\,\, \mbox{on}\,\, \partialΩ\end{split} \end{align*} where (C):~$0<α<1,\;0<β<1,$ $2-α-β<q<\frac{N(p-1)}{N-p}<p<r<p^*$, with $p^*=\frac{Np}{N-p}$. We will guarantee the existence of a solution in the Nehari manifold. Further by using the Lusternik-Schnirelman category we will prove the existence of at least $\text{cat}(Ω)+1$ number of solutions.
In this paper we study the existence of a least energy sign-changing solution to a nonlocal elliptic PDE involving singularity by using the Nehari manifold method, the constraint variational method and Brouwer degree theory.
In this paper, we prove the existence of multiple solutions for a nonlinear nonlocal elliptic PDE involving a singularity which is given as \begin{eqnarray} (-Δ_p)^s u&=& \fracλ{u^γ}+u^q~\text{in}~Ω,\nonumber u&=&0~\text{in}~\mathbb{R}^N\setminusΩ,\nonumber u&>& 0~\text{in}~Ω\nonumber, \end{eqnarray} where $Ω$ is an open bounded domain in $\mathbb{R}^N$ with smooth boundary, $N>ps$, $s\in (0,1)$, $λ>0$, $0<γ<1$, $1<p<\infty$, $p-1<q\leq p_s^{*}=\frac{Np}{N-ps}$. We employ variational techniques to show the existence of multiple positive weak solutions of the above problem. We also prove that for some $β\in (0,1)$, the weak solution to the problem is in $C^{1,β}(\overlineΩ)$.