  In this paper, the Riemann problem for the pressureless Euler equations with
a discontinuous source term is considered. The delta shock wave solution is
obtained by combining the generalized Rankine-Hugoniot conditions together with
the method of characteristics for different situations, which reflects the
impact of the source term on the delta shock front. Moreover, during the
construction process of the Riemann solution, some interesting phenomena are
also observed, such as the disappearance of the delta shock wave and the
occurrence of the vacuum state, etc.
