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

In this work we study the system { − Δ u = a u + b v + F u ( u + , v + ) + f 1 ( x ) in Ω , − Δ v = b u + c v + F v ( u + , v + ) + f 2 ( x ) in Ω , u = 0 , v = 0 on ∂ Ω , where Ω ⊂ R N is bounded with smooth boundary, N ⩾ 3 , F = H + G , where H is a 2 ∗ ≡ 2 N / ( N − 2 ) positively homogeneous function, G is a lower order perturbation, w + = max { w , 0 } and f 1 , f 2 ∈ L r ( Ω ) , r > N . Using the Mountain Pass Theorem we prove existence of two solutions. If N = 3 , 4 and 5, an additional hypothesis over the subcritical term is needed.