I am glad you are questioning yourself about staying with your boyfriend.
While it would be hard to know for certain in your situation, the typical reason people stay in unfulfilling relationships is that they do not have high enough self-esteem and cannot tolerate being alone.
By your own description your boyfriend does not treat you well, makes you cry, lies, and regularly does something to hurt you. This doesn’t sound promising.
Treat yourself better and take care of yourself. It tends to be that when we focus on taking care of ourselves, which includes being okay spending time alone, we are usually able to make better and healthier intimate connections.
My guess is you already know what to do.