  In one dimension, noninteracting particles can undergo a
localization-delocalization transition in a quasiperiodic potential. Recent
studies have suggested that this transition transforms into a many-body
localization (MBL) transition upon the introduction of interactions. It has
also been shown that mobility edges can appear in the single particle spectrum
for certain types of quasiperiodic potentials. Here, we investigate the effect
of interactions in two models with such mobility edges. Employing the technique
of exact diagonalization for finite-sized systems, we calculate the level
spacing distribution, time evolution of entanglement entropy, optical
conductivity, and return probability to detect MBL. We find that MBL does
indeed occur in one of the two models we study, but the entanglement appears to
grow faster than logarithmically with time unlike in other MBL systems.
