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 {array}{ll} -\Delta u = Q_u(u,v) + 1/2{2^*} H_u(u,v),& {in} \Omega,\vdois\ -\Delta v = Q_v(u,v) + 1/{2^*} H_v(u,v),& {in} \Omega,\vdois\ u=v=0,& {on} \partial\Omega. \leqno{(P)} where $\Omega \subset $\mathbb{R}$^N$isaboundedsmoothdomain,$N \geq 3$,$2^*:=2N/(N-2)$and$Q_u, H_u$and$Q_v$,$H_v$ are the partial derivatives of the homogeneous functions $Q,H\in C^1({\mathbb{R}}_{+}^2,\mathbb{R})$, where $ $\mathbb{R}$^2_+ := [0,\infty) \times [0,\infty)$. In the proofs we apply variational methods and Ljusternik-Schnirelmann theory.