  We consider critical points of a class of functionals on compact
four-dimensional manifolds arising from Regularized Determinants for
conformally covariant operators, whose explicit form was derived in [10],
extending Polyakov's formula. These correspond to solutions of elliptic
equations of Liouville type that are quasilinear, of mixed orders and of
critical type. After studying existence, asymptotic behaviour and uniqueness of
fundamental solutions, we prove a quantization property under blow-up, and then
derive existence results via critical point theory.
