  Studies of superconductivity in multiband correlated electronic systems has
become one of the central topics in condensed matter/materials physics. In this
paper, we present the results of thermodynamic measurements on the
superconducting filled skutterudite system Pr$_{1-x}$Ce$_x$Pt$_4$Ge$_{12}$ ($ 0
\leq x \leq 0.2$) to investigate how substitution of Ce at Pr sites affects
superconductivity. We find that an increase in Ce concentration leads to a
suppression of the superconducting transition temperature from $T_{c}\sim 7.9$
K for $x=0$ to $T_c\sim 0.6$ K for $x=0.14$. Our analysis of the specific heat
data for $x\leq 0.07$ reveals that superconductivity must develop in at least
two bands: the superconducting order parameter has nodes on one Fermi pocket
and remains fully gapped on the other. Both the nodal and nodeless gap values
decrease, with the nodal gap being suppressed more strongly, with Ce
substitution. Ultimately, the higher Ce concentration samples ($x>0.07$)
display a nodeless gap only.
