Critical Ambrosetti–Prodi type problems for systems of elliptic equations

In this paper we study multiple solutions for the non-homogeneous system equation(1){−Δu=au+bv+2αα+βu+α−1v+β+f,Ω−Δv=bu+dv+2βα+βu+αv+β−1+g,Ωu=v=0,∂Ω, where Ω⊂RNΩ⊂RN is a bounded smooth domain; α,β>1α,β>1 are real constants, α+β=2∗α+β=2∗, where 2∗=2NN−2,N≥3; s+=max{s,0}s+=max{s,0} and f,g∈Ls(Ω)f,g∈Ls(Ω) for some s>Ns>N.