Nonlinear Analysis 68 (2008) 194–207 www.elsevier.com/locate/na Critical Ambrosetti–Prodi type problems for systems of elliptic equations D.C. de Morais Filho a,∗ , F.R. Pereira b a Departamento de Matem ´ atica e Estat´ ıstica, Universidade Federal de Campina Grande, CX. Postal 10044, CEP 58109-970 Campina Grande - PB, Brazil b Departamento de Matem ´ atica - ICE, Universidade Federal de Juiz de Fora, CEP 36036-330 Juiz de Fora, Minas Gerais, Brazil Received 11 August 2006; accepted 30 October 2006 Abstract In this paper we study multiple solutions for the non-homogeneous system          −Δu = au + bv + 2α α + β u + α−1 v + β + f , Ω −Δv = bu + d v + 2β α + β u + α v + β−1 + g, Ω u = v = 0, ∂ Ω , (1) where Ω ⊂ R N is a bounded smooth domain; α,β> 1 are real constants, α + β = 2 ∗ , where 2 ∗ = 2 N N −2 , N ≥ 3; s + = max{s, 0} and f , g ∈ L s (Ω ) for some s > N . c  2006 Elsevier Ltd. All rights reserved. MSC: 35J50; 35B33 Keywords: Ambrosetti–Prodi type problems; Systems of elliptic equations; Critical Sobolev exponents 1. Introduction In this paper we study multiple solutions for the system of elliptic equations          −Δu = au + bv + 2α α + β u + α−1 v + β + f , Ω −Δv = bu + d v + 2β α + β u + α v + β−1 + g, Ω u = v = 0, ∂ Ω (1.1) where Ω ⊂ R N is a bounded smooth domain; α,β> 1 are real constants, α + β = 2 ∗ , with 2 ∗ = 2 N N −2 , N ≥ 3; s + = max{s , 0} and f , g ∈ L s (Ω ) for some s > N . ∗ Corresponding author. E-mail addresses: daniel@dme.ufcg.edu.br (D.C. de Morais Filho), fpereira@ice.ufjf.br (F.R. Pereira). 0362-546X/$ - see front matter c  2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2006.10.041