  We investigate the stability and phase transition of localized modes in
Bose-Einstein Condensates (BECs) in an optical lattice with the discrete
nonlinear Schr\"{o}dinger model by considering both two- and three-body
interactions. We find that there are three types of localized modes, bright
discrete breather (DB), discrete kink (DK), and multi-breather (MUB). Moreover,
both two- and three-body on-site repulsive interactions can stabilize DB, while
on-site attractive three-body interactions destabilize it. There is a critical
value for the three-body interaction with which both DK and MUB become the most
stable ones. We give analytically the energy thresholds for the destabilization
of localized states and find that they are unstable (stable) when the total
energy of the system is higher (lower) than the thresholds. The stability and
dynamics characters of DB and MUB are general for extended lattice systems. Our
result is useful for the blocking, filtering, and transfer of the norm in
nonlinear lattices for BECs with both two- and three-body interactions.
