run after

verb6

1. •
   To follow quickly, often in an effort to catch or catch up with (someone or something).

   • That dog will get hurt if he continues to run after cars.
2. •
   To make a determined effort to win someone's affections or to have a sexual relationship with them.

   • She runs after any man in uniform!
3. •
   To endeavour to find or obtain.

   • The letting the mind run after similes, though it may be useful for the explaining our thoughts to others, yet is by no means so for settling true notions of any thing in ourselves — 1831
4. •
   (no longer used) To seek to participate in, devote oneself to (something).

   • They will not be ruled by God, nor bridled by his word. They wil follow their owne swinge. They will run after their owne lusts and pleasures. — 1607
5. •
   (no longer used) To be well-liked and much admired.

   • She is run after by many people.
6. •
   (no longer used) To seek the company or assistance of; to frequent; to consult (someone).

   • So, they were as mad, upon their running after Quacks, and Mountebanks, and every practising old Woman, for Medicines and Remedies […] — 1722