Multiplicity of solutions for homogeneous elliptic systems with critical growth

In this paper we are concerned with the number of nonnegative solutions of the elliptic system{−Δu=Qu(u,v)+12⁎Hu(u,v),in Ω,−Δv=Qv(u,v)+12⁎Hv(u,v),in Ω,u=v=0,on ∂Ω, where Ω⊂RNΩ⊂RN is a bounded smooth domain, N⩾4N⩾4, 2⁎:=2N/(N−2)2⁎:=2N/(N−2) and QuQu, HuHu and QvQv, HvHv are the partial derivatives of the homogeneous functions Q,H∈C1(R+2,R), where R+2:=[0,∞)×[0,∞). In the proofs we apply variational methods and Ljusternik–Schnirelmann theory.