Is it permissible for me to pray at home due to knee pain and stiffness resulting from an accident, given that there is an ascent to the mosque, and knowing that I am able to go to the university by car?

Attending congregational prayers in the mosque is for acts of obedience, and it is not obligatory according to the majority of those who deem congregational prayer obligatory. Thus, if you pray in congregation in your home, you have fulfilled your duty. If you miss the congregational prayer due to an excuse such as illness, you are not sinful, and it is hoped that you will be rewarded for having intended to attend it.