Is it permissible to give Zakat on money for the treatment of a 35-year-old brother who is mentally ill, unemployed, has no source of income, and owns three apartments, due to his lack of income and his need for treatment?

Your brother is wealthy due to his ownership of apartments and is therefore not eligible for zakat. However, if these apartments are sold and their price is insufficient to cover the costs of his treatment, then there is no objection to you paying zakat to him.