desert and abandon are very clear synonyms. Giving up does not necessarily have anything to do with abandonment and desertion. abandon : cease to support or look after desert : means the same but desertion has more of a disloyal and treacherous quality give up: to desist from They are all closely related but certain words are used in combination with certain contexts more often than others.

desert is often used for ships or military. "He deserted his post" = he gave up his duty and left his job. abandon is more general, you can "abandon ship" (let the ship sink in the water, leave the ship". you can also abandon your post. you can also metaphorically "abandon" someone by never contacting them again; "he abandoned his friends" "give up" can mean abandon but it can also mean to stop doing something （放棄）, like "he practiced Yoga for many years, but recently he gave it up"