  We study the conformal plate buckling equation (Laplace--Beltrami)^2 u =1,
where the L-B operator is for the metric g = e^{2u}g_0, with $g_0$ the standard
Euclidean metric on R^2. This conformal elliptic PDE of fourth order is
equivalent to the nonlinear system of elliptic PDEs of second order, Delta u
+K_g e^(2u)=0, Delta K_g + e^(2u)=0, with x in R^2, describing a conformally
flat surface with a Gauss curvature function K_g that is generated
self-consistently through the metric's conformal factor.
