  The periodic Anderson impurity model on the Lieb lattice is studied by the
slave-boson mean-field approximation in the strong interaction limit. The
electron structure of conduction electrons on the Lieb lattice features both
the band flatness and soft gap at low energy. With these features conduction
electrons can form both the soft-gap and the molecular Kondo singlets with the
magnetic impurities, and this leads to a competition between the soft-gap and
the molecular Kondo effects in the lattice. We find a selective Kondo strong
coupling, where at selected sites the magnetic impurities are strongly coupled
to conduction electrons, and at the remaining sites they are decoupled from the
lattice. The selective Kondo strong coupling occurs between the full local
moment and the full strong coupling regimes, and it yields an effective lattice
depletion. At low temperature the selection of the Kondo strong coupling is
dominant at those lattice sites, where the local density of states of
conduction electrons exhibits the flat-band feature, independently of the
impurity parameters. Rich phase diagrams for different model parameters are
obtained.
