撂下 is a resultative verb compound (resultative complement) meaning 'to put down, set down' literally, and 'to leave behind, abandon' figuratively (e.g., 撂下工作). As a separable verb, it can be split: 撂得下 ('can put down'), 撂不下 ('cannot put down'), or with a duration between the components (e.g., 撂了三天). Use it in contexts where you set something aside physically or give up a responsibility.