The Authority Mutants
Now that you're at the top, switch back to the shotgun unless you have a lot of ammo for the pulse gun.
A whole bunch of authority mutants will start pouring out onto the catwalk. They actually aren't that tough, or at least, they aren't much tougher than normal mutants. Two shotgun blasts should kill one. Just remember to watch your ammo and reload when they aren't on top of you. After the waves die out, move up and hit the panel to start initializing the core. Ride the core up to the next level.
When you step off of it, you'll need to shoot two mutants that pop out of the side panels. Once they're dead, go down the ramps and hit the button below to unlock the corridor. You can also grab some pulse ammo for the big gun off of the crate next to the button. A few more mutants will pop out, so blast them. You should be able to switch to the pulse gun again, if you want. There will be some authority soldiers located in the area ahead, so it's good to have a versatile weapon.
Move forward and engage the mutants and soldiers that are ahead of you. There are plenty of pillars and the pulse gun will make short work of the few soldiers, so you should be all right if you've made it this far. Just watch your back and don't overextend yourself against the mutants. You should see a panel at the end. Active it to kill the nearby gate, and then jump onto the lift.
On the next level, watch out for some mutants that will jump in as you walk toward the center of the catwalk. Blast them and go down the sides. Switch to the shotgun and get ready to fight off a few groups of mutants as you look for the button to deactivate the gates. Go ahead and go back to the pulse gun. Clear a path to the center and cross over to run right into a few unfortunate authority soldiers. If you want, you can use a BFG on them to clear most of them out. If not, fall back. A few mutants will spawn behind you, so keep your eyes open as you cut down the basic soldiers in front of you. A heavy will walk out, but the pulse gun should cut him down in seconds.
After they're dead, you can move into the next corridor. A few more mutants will pop out of the panels, but you should know what to do with them by now. Turn to the left to grab some ammo for the pulse gun, and then follow the path to the right to run into a few more mutants and soldiers. Kill the mutants that rush you, and watch for the heavy that will slowly make it to you. Cut him down with the pulse gun, and turn the corner. Shred the basic soldiers that are left behind, and just walk on up to the lift; it will take you to the final core.