Should I give him more time?

My boyfriend has been dealing with a very sudden onset of depression, however, he has become someone I can't handle being around, he's rude, he hardly acknowledges my existence, he has flat out refused to get help and I've told him how I'm feeling about it all. I try to do things that I know he likes, I try to complement him, I try to be there to support him, I ask him what he needs or wants, and nothing seems to get through. Mind you, I'm a mother. They're not his biological children. But they are involved none the less as he has been living with us for over a year now. My son is autistic and has been calling him dad. But my boyfriend has an obvious preference between my kids and it isn't him. It really really shows on the worst days.

I've tried telling him that I can't stay in a relationship with someone who is unwilling to put in effort to be the best version of themselves. I know it's hard, I have had the "mental illness book" thrown at me and I've been hospitalized a few times. But he refuses to try. At all. And I'm so emotionally drained and mentally exhausted that I'm constantly angry and that's obviously bad for my kids.

Is it worth continuing to ask him to do what I would require from him to continue our relationship?
Should I give him more time?
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