In the state of OH--After my brother-in-law dies, who is legally responsible for supporting my sister? Is it me or her two sons?

If your sister is a minor, and her parents are still living, than her parents are responsible.  If your sister is an adult, then no one is responsible for taking care of her.  Perhaps I do not understand fully your situation.  If you want to share additional facts, please do so.  Feel free to contact me privately.   Simon W. Johnson   swj@swjlawoffice.com Law Office of Simon W. Johnson Martindale, Avvo, Justia, LawGuru LinkedIn, Twitter, Facebook Serving Cleveland and Ohio, 44124