Browsing by Subject "Kato class"

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Access status: Open Access ,
Existence of positive continuous weak solutions for some semilinear elliptic eigenvalue problems
(Wydawnictwa AGH, 2022) Zeddini, Noureddine; Sari, Rehab Saeed
Let $D$ be a bounded $C^{1,1}$-domain in $\mathbb{R}^d$, $d \geq 2$. The aim of this article is twofold. The first goal is to give a new characterization of the Kato class of functions $K(D)$ that was defined by N. Zeddini for $d=2$ and by H. Mâagli and M. Zribi for $d \geq 3$ and adapted to study some nonlinear elliptic problems in $D$. The second goal is to prove the existence of positive continuous weak solutions, having the global behavior of the associated homogeneous problem, for sufficiently small values of the nonnegative constants $\lambda$ and $\mu$ to the following system $\Delta u=\lambda f(x,u,v)$, $\Delta v=\mu g(x,u,v)$ in $D$, $u=\phi_{1}$ and $v=\phi_{2}$ on $\partial D$, where $\phi_{1}$ and $\phi_{2}$ are nontrivial nonnegative continuous functions on $\partial D$. The functions $f$ and $g$ are nonnegative and belong to a class of functions containing in particular all functions of the type $f(x,u,v)=p(x)u^{\alpha}h_{1}(v)$ and $g(x,u,v)=q(x)h_{2}(u)v^{\beta} with $\alpha \geq 1$, $\beta \geq 1$, $h_1$, $h_2$ are continuous on $[0,\infty)$ and $p$, $q$ are nonnegative functions in $K(D)$.
Access status: Open Access ,
Nonnegative solutions for a class of semipositone nonlinear elliptic equations in bounded domains of Rn
(Wydawnictwa AGH, 2022) Bachar, Imed; Mâagli, Habib; Eltayeb, Hassan
In this paper, we obtain sufficient conditions for the existence of a unique nonnegative continuous solution of semipositone semilinear elliptic problem in bounded domains of $\mathbb{R}^n$ ($n \geq 2$). The global behavior of this solution is also given.