From Longman Dictionary of Contemporary Englishgo after something/somebody phrasal verb1 TRY TO DO OR GET somethingto follow or chase someone or something because you want to catch them Joe went after her to make sure she was unhurt.2 to try to get something I can’t decide whether to go after the job or not. → go→ See Verb table
Examples from the Corpus
go after • Definitely the place to go after a good bike ride.• Ask him to show you what you need for refreshment and never go after a solution of your own design.• He'd have to go after him now, get the wire off him and fix the door.• It is all about the winning and they both simply go after it in their way that gets them there.• He saw a shaky-looking Halberstadt turn eastwards and went after it.• One story goes back to Warsaw, where his mother went after she left her village.• So I went after the real issues.• But the law that goes after those who hire illegally is controversial among businesses.From Longman Business Dictionarygo after phrasal verb [transitive]1 go after something to try to obtain something, such as a new job, a business contract etcSega plans to go after the core of the video game playing market — children eight to 12 years old.2 go after somebody to take actions to punish someone who has done something wrong or illegalRules are needed to go after brokers who sell unsuitable investments. → go→ See Verb table 