WWYD in this situation?

You need to get legal advice, but very broadly if you stay in the house, you’ll have to buy him out (at whatever percentage of the equity that is agreed to be fair - either through mediation or by the courts - which may or may not be 50:50 depending on your facts and circumstances) and take on the mortgage in your name only.

If you can’t afford to buy him out and take out a sole mortgage then you’ll have to sell the house and divide the equity.

It’s becoming very rare for the courts to allow one party to the divorce to stay in the former matrimonial home without buying out the other, because it puts the other at such a disadvantage.