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 { − Δ u = Q u ( u , v ) + 1 2 ⁎ H u ( u , v ) , in Ω , − Δ v = Q v ( u , v ) + 1 2 ⁎ H v ( u , v ) , in Ω , u = v = 0 , on ∂ Ω , where Ω ⊂ R N is a bounded smooth domain, N ⩾ 4 , 2 ⁎ : = 2 N / ( N − 2 ) and Q u , H u and Q v , H v are the partial derivatives of the homogeneous functions Q , H ∈ C 1 ( R + 2 , R ) , where R + 2 : = [ 0 , ∞ ) × [ 0 , ∞ ) . In the proofs we apply variational methods and Ljusternik–Schnirelmann theory.