TY  - JOUR
ID  - Zeddini2022
LB  - Zeddini2022
AU  - Zeddini, Noureddine
AU  - Sari, Rehab Saeed
TI  - Existence of positive continuous weak solutions for some semilinear elliptic eigenvalue problems
ST  - Existence of positive continuous weak solutions for some semilinear elliptic eigenvalue problems
JO  - Opuscula Math.
JA  - Opuscula Math.
JF  - Opuscula Mathematica
PY  - 2022
VL  - 42
IS  - 3
SP  - 489
EP  - 519
DO  - https://doi.org/10.7494/OpMath.2022.42.3.489
UR  - https://doi.org/10.7494/OpMath.2022.42.3.489
AB  - 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 \(K(D)\).
KW  - Green function
KW  - Kato class
KW  - nonlinear elliptic systems
KW  - positive solution
KW  - maximum principle
KW  - Schauder fixed point theorem
SN  - 1232-9274
ER  -