Existence of positive continuous weak solutions for some semilinear elliptic eigenvalue problems

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 \(K(D)\).

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