When nonsatiation or free disposal fail, competitive equilibria are either weak, when the individual budget constraint is imposed with equality, or strong, when the budget constraint is imposed as a weak inequality. Weak competitive equilibria exist under standard assumptions since the aggregate demand function satisfies Walras' law. This is the case even though prices cannot be restricted to a contractible domain. Strong competitive equilibria may fail to exist under standard assumptions. A boundary condition on the aggregate demand function guarantees, nevertheless, the existence of strong competitive equilibria.
Polemarchakis, HM., & Siconolfi, P. (1993). Competitive Equilibria Without Free Disposal Or Nonsatiation. Journal of Mathematical Economics, 22(1), 85-99. https://doi.org/10.1016/0304-4068(93)90032-G (Original work published 1993)