An Ambrosetti-prodi-type Problem for an Elliptic System of Equations via Monotone Iteration Method and Leray-schauder Degree Theory

In this paper we employ the Monotone Iteration Method and the Leray-Schauder Degree Theory to study an IR-parametrized system of elliptic equations. We obtain a curve dividing the plane into two regions. Depending on which region the parameter is, the system will or will not have solutions. This is an Ambrosetti-Prodi-type problem for a system of equations. 0. Introduction Let us consider a bounded domain Ω ⊆ IRN , N ≥ 2, with boundary ∂Ω of class C2,α , 0 < α ≤ 1. In what follows we shall denote the Cartesian product of Hölder spaces as (Ci,α)2 = Ci,α(Ω)×Ci,α(Ω), 0 < α ≤ 1, i = 0, 1, 2 and (Cα)2 = (C0,α(Ω))2. Remark 0.1. (Ci,α)2 ↪→ (Cj,α)2 compactly if i ≥ j ≥ 0. Let us consider the following system of partial differential elliptic equations (S)  −∆u = f(x, u, v) + h(x), Ω, −∆v = g(x, u, v) + l(x), Ω, u = v = 0, ∂Ω, where h, l ∈ Cα(Ω) and f, g ∈ Cα(Ω × IR × IR) are real functions. By a solution of (S) we mean a vector-function U = (u, v) ∈ (C2,α)2 satisfying both equations in (S). Any function h ∈ Cα(Ω) can be uniquely decomposed as h = tφ1 + h1, where φ1 > 0 is an eigenfunction associated to the first eigenvalue λ1 >