  Exciton-polariton condensate in semiconductor microcavities constitute a
novel kind of non-equilibrium superfluid. In a recent experiment [P. Stepanov,
{\it et. al.,} Nat. Commun. {\bf 10}, 1038 (2019)], the dispersion relation of
collective excitations in a polariton condensate under the resonant pumping has
been investigated with the emphasis on the role of reservoir of long-lived
excitons in determining the superfluidity. Inspired by such an experimental
advance, we study the superfluidity of a exciton-polaritonn condensate under
non-resonant pumping by calculating the drag force exerted on a classical
impurity moving in a polariton condensate. For a non-resonant pumped polariton
condensate prepared in the gapped phase, due to the reservoir's modes, the drag
force can be large when the velocity of the impurity is small. Besides, as the
velocity increases, the drag force can decrease. For not very large velocity,
the drag force is enhanced if the condensate is tuned to be more dissipative.
When the condensate is close to the transition point between the gapped phase
and the gapless one, the drag force is similar to that of the equilibrium
superfluid. Our present work reveals the effects of the reservoir's modes on
the superfluidity properties of a polariton condensate with the non-resonant
pumping.
