  We study existence and asymptotic behavior of radial positive solutions of
some fully nonlinear equations involving Pucci's extremal operators in
dimension two. In particular we prove the existence of a positive solution of a
fully nonlinear version of the Liouville equation in the plane. Moreover for
the $\mathcal{M}^-_{\lambda,\Lambda}$ operator, we show the existence of a
critical exponent and give bounds for it.
