  Recently, above-room temperature superconductivity was reported in the Cu
doped lead apatite Pb$_{10-x}$Cu$_x$(PO4)$_6$O, dubbed LK-99. By relaxing the
structure with Cu substitution, we derive a four-band low-energy model with two
3/4 filled bands of predominantly Cu $d_{xz}$ and $d_{yz}$ character and two
filled O $p_x$ and $p_y$ bands. This model is further downfolded to a two-band
Cu-$d_{xz/yz}$ model. Using {\it ab-initio} derived interaction parameters, we
perform dynamical mean field theory calculations to determine the correlated
electronic structure in the normal state. These calculations yield a Mott
insulator at $x=1$ and a strongly correlated non-Fermi liquid metal upon
doping. The very large interaction versus bandwidth ratio $U/W\approx 30$-$50$
and the local moment paramagnetic behavior in the relevant filling regime are
hard to reconcile with diamagnetism and high-temperature superconductivity.
Hence, our calculations suggest that this behavior should come from a component
with a different stoichiometry.
