  The Yu-Shiba-Rusinov (YSR) states in kagome superconductors have been studied
recently in theory and experiments. However, more details of local
superconductivity around the YSR states in kagome superconductors and the
existence of the related first-order phase transition remain unclear. Based on
the self-consistent Bogoliubov-de Gennes equations for the $s-$wave
superconducting kagome Hubbard model, we show that first-order phase
transitions occur when varying the spin-magnetic interaction $J$ of the
magnetic impurity, and discontinuities of hysteresis loops in the plots of the
local pair potential versus $J$ happen at the upper and lower critical
interactions $J_{\rm c1}$ and $J_{\rm c2}$, rather than the conventional
critical interaction $J_c$. The minimal energy of the stable YSR state $E_{\rm
YSR, min}$ is non-zero for the chemical potential at the van Hove
singularities, Dirac point, and the flat band. Furthermore, $J_{\rm c1}$,
$J_{\rm c2}$, and $E_{\rm YSR, min}$ are dependent on the chemical potential
and the superconducting coupling interaction.
