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

Let D be a bounded C1,1-domain in Rd, d ≥ 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 ≥ 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 λ and μ to the following system Δu = λf(x, u, v), Δv = μg(x, u, v) in D, u = ϕ1 and v = ϕ2 on ∂D, where ϕ1 and ϕ2 are nontrivial nonnegative continuous functions on ∂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)uαh1(v) and g(x, u, v) = q(x)h2(u)vβ with α ≥ 1, β ≥ 1, h1, h2 are continuous on [0,∞) and p, q are nonnegative functions in K(D).

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Opracowanie rekordu ze środków MEiN, umowa nr SONP/SP/546092/2022 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2022-2023).