In　this　paper,　we　establish　the　existence　and　uniqueness　theorem　of　entire　solutions　to　the　Lagrangian　mean　curvature　equations　with　prescribed　asymptotic　behavior　at　infinity.　The　phase　functions　are　assumed　to　be　supercritical　and　converge　to　a　constant　in　a　certain　rate　at　infinity.　The　basic　idea　is　to　establish　uniform　estimates　for　the　approximating　problems　defined　on　bounded　domains　and　the　main　ingredient　is　to　construct　appropriate　subsolutions　and　supersolutions　as　barrier　functions.　We　also　prove　a　nonexistence　result　to　show　the　convergence　rate　of　the　phase　functions　is　optimal.