I'm a 36 year old US Citizen. My Mother is in the US on a visitors visa. Can I petition her I-130 and she remain here after visa is expires?

As a US citizen, you can sponsor mom for her adjustment of status case. There are several issues that you need to explore with an immigration attorney. What was mom's intention when she entered the USA. If it was just to visit, when did her intention change to wanting to be a permanant resident and why did it change. Third, how long has mom been here. Does mom have any grounds of inadmissibility that may affect her case. These and other issues should be explored in a private phone consultation with an attorney.