The Ambrosetti–Prodi problem for gradient elliptic systems with critical homogeneous nonlinearity

In this work we study the system{−Δu=au+bv+Fu(u+,v+)+f1(x)in Ω,−Δv=bu+cv+Fv(u+,v+)+f2(x)in Ω,u=0,v=0on ∂Ω, where Ω⊂RNΩ⊂RN is bounded with smooth boundary, N⩾3N⩾3, F=H+GF=H+G, where H   is a 2∗≡2N/(N−2)2∗≡2N/(N−2) positively homogeneous function, G   is a lower order perturbation, w+=max{w,0}w+=max{w,0} and f1,f2∈Lr(Ω)f1,f2∈Lr(Ω), r>Nr>N. Using the Mountain Pass Theorem we prove existence of two solutions. If N=3,4N=3,4 and 5, an additional hypothesis over the subcritical term is needed.