The Eridium Injectors are the "life support" for the Siren Angel, Handsome Jack's daughter. They must be shot to trigger the shielding. After that you need to wait for Angel to override Jack's controls.
With Roland's help the Vault Hunter needs to run around, shooting them down. In doing so they are stopping the Eridium feed to Angel who has been charging the Key. The Core is also defended by a large number of Loaders. There are also Core Defense Turrets that appear along the outside of the containment field of Angel.