  We study the effect of hole-doping and chemical pressure (isovalent doping)
in single crystals of K$_x$Eu$_{1-x}$Fe$_2$As$_2$ and
EuFe$_2$(As$_{1-y}$P$_y$)$_2$, respectively, by measurements of the
thermopower, $S(T)$, and electrical resistivity, $\rho(T)$. The evolution of
$S(T)$ upon doping indicates drastic changes of the electronic configuration at
critical values $x_{\mathrm{cr}}=0.3$ and $y_{\mathrm{cr}}=0.21$, respectively,
as the spin-density-wave transition is completely suppressed and
superconductivity (SC) emerges. For the case of chemical pressure, the
comparison with published ARPES measurements indicates a Lifshitz transition at
$y_{cr}$. The temperature dependences $S(T)/T\propto \log T$ and
$\Delta\rho\propto T$ observed in the normal state above the SC transition
suggest quantum criticality in both systems.
