It means a court could shut the store down for up to six months, if it finds in favour of the local authority.
The case was initially listed for Monday, September 4 at Leeds Magistrates Court, but has now been adjourned until Thursday, September 21.
The notice, displayed on the shop front, confirmed that the closure order was being sought and outlined the allegations.
Asked for further details on the case, the council’s deputy leader and executive member for resources, Coun Debra Coupar, said: “We take all concerns related to anti-social behaviour extremely seriously as it is something none of our residents should have to put up with.
“Working in partnership with West Yorkshire Police amid the continuing concerns of local residents, businesses and shoppers relating to this premises, Leeds City Council have applied to the magistrates court for a premises closure order.
“We will continue to work to tackle all forms of anti-social behaviour in the city in partnership with West Yorkshire Police.”