My iPhone 11 Pro Max’s battery ,I assume, is older and not maintaining a charge all day long like it did when new…same use.

Hello ~ Yes… there is a replacement battery for your iPhone and take in in to Apple or an authorized service provider. Take a look here:

iPhone Repair and Service - Apple Support

~Katana-San~

Depends. If it's under 80% battery health and it's still under warranty or AppleCare, then yes. Otherwise it's going to require paying for a battery replacement service.