  Non-interacting fermions in one dimension can undergo a
localization-delocalization transition in the presence of a quasi-periodic
potential as a function of that potential. In the presence of interactions,
this transition transforms into a Many-Body Localization (MBL) transition.
Recent studies have suggested that this type of transition can also occur in
models with quasi-periodic potentials that possess single particle mobility
edges. Two such models were studied in PRL 115,230401(2015) but only one was
found to exhibit an MBL transition in the presence of interactions while the
other one did not. In this work we investigate the occurrence of MBL in the
presence of weak interactions in five different models with single particle
mobility edges in one dimension with a view to obtaining a criterion for the
same. We find that not all such models undergo a thermal-MBL phase transition
in presence of weak interactions. We propose a criterion to determine whether
MBL is likely to occur in presence of interaction based only on the properties
of the non-interacting models. The relevant quantity $\epsilon$ is a measure of
how localized the localized states are relative to how delocalized the
delocalized states are in the non-interacting model. We also study various
other features of the non-interacting models such as the divergence of the
localization length at the mobility edge and the presence or absence of
`ergodicity' and localization in their many-body eigenstates. However, we find
that these features cannot be used to predict the occurrence of MBL upon the
introduction of weak interactions.
