Does iPhone 11 work with a 4G network?

Yes.

One caution: There are many 4G (LTE) bands, so how well your phone will work with a particular 4G service will depend on how many bands your phone and your carrier's 4G network have in common.

In Australia, one of the big carriers is shutting down 3G service before the end of this month – and also has a 4G network that relies heavily on FDD-LTE band 28. The U.S. variant of the iPhone 11 supports many LTE bands, but doesn't support FDD-LTE band 28. So very soon, that variant of the iPhone 11 won't work well (if at all) with that Australian carrier's service.

https://www.reddit.com/r/iphone/comments/d79luu/iphone_11_models_and_their_cellular_bands_overview/

Yes, it does. It's actually called LTE in some regions.

Hi there! Yes, but it also depends on your cellular company’s service in your country, some countries unfortunately don’t support LTDE (Not-Spots), or if your phone doesn’t recognize any SIM card or eSIM try turning you iPhone off > Unplug the slot > Pull the card and wipe it with a dried cloth > Insert in your phone slot the SIM card and then turn on the device.

If your phone has a eSIM activation problem/lock, then the only way is contacting your cellular carrier to see more chances to resolve it.