Definition of "run after"
run after
verb
third-person singular simple present runs after, present participle running after, simple past ran after, past participle run after
To follow quickly, often in an effort to catch or catch up with (someone or something).
Quotations
I saw him run after a gilded butterfly: and when he caught it, he let it go again;
c. 1608–1609 (date written), William Shakespeare, “The Tragedy of Coriolanus”, in Mr. William Shakespeares Comedies, Histories, & Tragedies […] (First Folio), London: […] Isaac Iaggard, and Ed[ward] Blount, published 1623, [Act I, scene iii]
To make a determined effort to win someone's affections or to have a sexual relationship with them.
Quotations
[…] if a young man is wild, and must run after women and bad company, it is better this should be done abroad, as, on his return, he can break off such connections, and begin at home a new man […]
1791, Samuel Johnson, quoted in James Boswell, The Life of Samuel Johnson, London: Charles Dilly, p. 265
To endeavour to find or obtain.
Quotations
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, Abraham John Valpy, Locke’s Essay on the Human Understanding Condensed by a Clergyman, London: Whitaker, Treacher, and Arnot, “Conduct of the Understanding,” p. 409
(obsolete) To seek the company or assistance of; to frequent; to consult (someone).
Quotations
And thus he is so poysoned w[i]t[h] bitternes and obstinacy that he begynneth to hate and to blaspheme God in heauen and seketh vnlaufull meanes to helpe and remedye hym self, lyke as Saul dyd, runnynge after wytches and sothsaiers,
1550, Otto Werdmüller, chapter 28, in Miles Coverdale, transl., A Spyrytuall and Moost Precyouse Pearle, London: Gwalter Lynne