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I will report on a recent result, in collaboration with A. Malchiodi, concerning a four-dimensional PDE of Liouville type arising in the theory of log-determinants in conformal geometry. The differential operator combines a linear fourth-order part with a quasi-linear second-order one. Since both have the same scaling behavior, compactness issues are very delicate and even the ``linear theory'' is problematic. For the log-determinant of the conformal laplacian and of the spin laplacian we succeed to show existence and logarithmic behavior of fundamental solutions, quantization property for non-compact solutions and existence results via critical point theory.