Is there any relevant legal responsibility for falling on the steps and breaking because of poor lighting of the property?

If the lighting does not reach the normal brightness and is too dim, the property has certain responsibilities. You can negotiate with the property first. If negotiation fails, you can go to court to sue for rights protection.

According to the relevant provisions of the Property Management Regulations, property management enterprises shall provide corresponding services in accordance with the provisions of the property management service contract. If the owner's personal and property safety is damaged due to breach of contract, he shall bear corresponding responsibilities.

The property management company has the legal obligation to "ensure the normal use of street lamps in the corridor of the residential area", but the property management company of the residential area failed to fulfill the above obligations, causing Mr. Deng to fall, and should be liable for damages according to law.