Cheesecakes without carbohydrates to bind them are always going to fall -- it would technically be a cheese souffle.

You could try adding a leavening agent, though. Baking soda sounds about right, just less than a teaspoonful. If you're OK with adding a bit of carbs in, you could also use about 1 tablespoon of corn starch mixed in with your lemon juice and vanilla to make a paste before you add them.

Or google recipes for "Tophenkuchen", it's Austrian cheesecake, and basically what you are describing there. Austrians tend to bake theirs in pie plates, not springform pans, so that also helps with the falling.