ATC TRACON Controller reported aircraft descending below the MVA for the area. Flight crew was issued an alert and climbed to safe altitude; and landed at their destination.

Narrative

Aircraft X checked in on ZZZZZ arrival; destination ZZZ. It is almost a straight-in approach; so I issued direct the IAF (ZZZZZ) and a descent to 9000. The MVA's East of ZZZ and ZZZ1 climb rapidly over the ZZZ2 Mountains. As Aircraft X continued eastbound past the 9000 MVA; I issued a further descent to 6000. I was working both feeder sectors combined; as well as the ZZZ satellite sector. There was quite a bit of low level VFR traffic; and one VFR aircraft on VFR flight following. That VFR arrival to ZZZ4 was about to go through the VFR high traffic area. I issued a couple of traffic calls; answered a coordination from ZZZ Center about a VFR Aircraft Y inbound to the airspace; answered a couple of check-ins; and switched communications to a couple of other aircraft to Final Radar. When I looked back at Aircraft X; they were in a 5700 MVA; descending through 5500. I issued 6000; they questioned it; and again issued 5700. I then issued a low altitude alert as they continued descending. They were as low as 4000 before leveling off then climbing back up. The MVA's drop dramatically past that; but not for a couple of miles. The MSAW never alarmed; as the aircraft was past most of the terrain; and the hills were dropping off rapidly. After the event I watched/listened to a replay. Aircraft X read back 6000; and between then and the event I did not issue 4000 to another aircraft. When the pilot called they said they had 4000 in the autopilot; and weren't sure when they changed it. ZZZ TRACON has been chronically understaffed for years. The Controller I relieved recommended splitting the sector. Before I had a chance to assess that it got busy again. The sectors probably should have been split; but there were no people available. There was a briefing about the upcoming software outage; so was somewhat critical to finish. If I hadn't been at the end of a 6-day workweek and had a sector split off I would likely have caught it sooner. The shift had X CPC's; and should have had 12. The shift guide should probably be 14. We're lucky to have 14 between the day and eve shifts combined.