  In this paper we establish existence of radial and nonradial solutions to the
system $$
  \begin{array}{ll}
  -\Delta u_1 = F_1(u_1,u_2)
  &\text{in }\mathbb{R}^N,\newline
  -\Delta u_2 = F_2(u_1,u_2)
  &\text{in }\mathbb{R}^N,\newline
  u_1\geq 0,\ u_2\geq 0 &\text{in }\mathbb{R}^N,\newline
  u_1,u_2\in D^{1,2}(\mathbb{R}^N),
  \end{array} $$ where $F_1,F_2$ are nonlinearities with critical behavior.
