Verb
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.
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!
To endeavour to find or obtain.
(obsolete) To seek the company or assistance of; to frequent; to consult (someone).
(obsolete) To seek to participate in, devote oneself to (something).
Source: en.wiktionary.orgrun-after
I'm amazed when I see mothers wearing high heels. If your kids run off you can't run after them. Claudia Schiffer
Does a man of sense run after every silly tale of hobgoblins or fairies, and canvass particularly the evidence? I never knew anyone, that examined and deliberated about nonsense who did not believe it before the end of his enquiries. David Hume
Those who run after happiness will never be happy. Happiness is something that has to come to the fore from within. Sri Chinmoy
If you run after two hares, you will catch neither. Japanese Proverb
If you run after two hares, you will catch neither. English Proverb
Riches run after the rich, and poverty runs after the poor. French Proverb