Q: does the iPad mini gsm have LTE

Yes, but it might depend upon whether your carrier supports it - from http://www.apple.com/ipad-mini/specs/first-generation/ :

• Model A1454*
  • GSM/EDGE (850, 900, 1800, 1900 MHz)
  • UMTS/HSPA/HSPA+/DC-HSDPA (850, 900, 1900, 2100 MHz)
  • LTE (Bands 2, 4, 5, 17)
• Model A1455*
  • CDMA EV-DO Rev. A and Rev. B (800, 1900, 2100 MHz)
  • GSM/EDGE (850, 900, 1800, 1900 MHz)
  • UMTS/HSPA/HSPA+/DC-HSDPA (850, 900, 1900, 2100 MHz)
  • LTE (Bands 1, 3, 5, 13, 25)

 

*To identify your iPad model number, see http://support.apple.com/kb/HT5452. For details on LTE support, contact your carrier and see www.apple.com/ipad/LTE.

This looks to be a duplicate post: https://discussions.apple.com/thread/5539576?tstart=0

OK