A Corporate Jet Captain reported engine failure during initial climb and diversion to a nearby airport.

Narrative

In ZZZ; we had a normal engine start and a normal taxi out from the FBO ramp. It was FO's leg to fly. We lined up on and confirmed Runway XX and I transferred the controls to him. He set the engine power and proceeded down the runway with no abnormalities.Around 125 knots and after V1; the left engine rolled back to near idle. FO inputted right rudder and we rotated. The engine momentarily seemed to come back to life and immediately followed with another rollback rocking the airplane back and forth. We climbed up and with the chaos we did not get the landing gear up and continued straight out. Around 500ft (my best guess) FO asked if I would take the controls and I did. We still had the gear down and flaps 10° at this point and made a very quick decision to return to Runway XX as the left engine did not have any control input when manipulating the thrust lever. I took the radios and attempted to continue the climb to 2000 with little luck; the power setting being held got us to 1500ft and 160kts. I called for flaps 30° as we were going to attempt to get on the ground as soon as possible coupled with the lower airspeed. As we were getting ready to make a base turn; FO got the landing numbers in the box and stated that we were still overweight. With no emergency equipment at the field and having briefed ZZZ1 as a Take-Off alternate; we quickly made the decision to limp the airplane to Runway X at ZZZ1. We informed the Tower of this and asked for a heading and the weather at ZZZ1. Enroute to ZZZ1; FO ran all the numbers and put ILS Runway X in the box and the landing performance when we discovered we would need approximately 4000ft of runway. About that time ZZZ Approach asked if we'd like Runway XY for a quicker arrival to which we agreed to. We landed and were able to taxi the airplane to FBO under our own power. In hindsight; we made some mistakes along the way and should have brought the gear up; cleaned up; and ran some more checklist before making any decisions. However; in the heat of the moment along with the fact that the engine came and went; we decided a quick return to the airport was our best interest and left everything the way it was to speed up the landing process. The decision to go to ZZZ1 ended up solely being the longer runway option (which we did not end up using) as well as the rescue equipment. With all the chaos going on with the runway swaps and decision making; we did not run the L ENG OVER EXCEEDANCE QRH to its completion which would have required us to shut the engine down due to the Lack of power responsiveness. However; FO did read through it as we joined the RNAV XY and we made the decision to leave it running as we were about to land and there was no threat of a fire.