  We analyze the ground state of a Bose-Einstein condensate in the presence of
higher-order interaction (HOI), modeled by a modified Gross-Pitaevskii equation
(MGPE). In fact, due to the appearance of HOI, the ground state structures
become very rich and complicated. We establish the existence and non-existence
results under different parameter regimes, and obtain their limiting behaviors
and/or structures with different combinations of HOI and contact interactions.
Both the whole space case and the bounded domain case are considered, where
different structures of ground states are identified.
