Ambrosetti-Prodi type results for elliptic equations with nonlinear gradient terms on an exterior domain

Abstract

The aim of this paper is to establish an Ambrosetti-Prodi type result for the problem

−Δu = K(|x|)f(|x|, u, |∇u|) + sφ, x ∈ Ω,
αu + β ∂u/∂n |∂Ω = 0,
lim_|x|→∞ u(x) = 0,

where s ∈ R is a parameter, Ω = {x ∈ R^N : |x| > r0}, N ≥ 3, K : [r0,∞) → [0,+∞) is continuous and satisfies ꭍ^∞/_r0  r^N−1K(r)dr < ∞. f : [r0,∞) × R × [0,+∞) → R is continuous. φ ∈ C(Ω) with φ ≩ 0 in Ω.