Is it permissible for a mother to sell an apartment she owns to give its price to her son who takes care of her, and is this amount considered a gift exclusively for him without his other siblings, or do they have the right to claim it when the inheritance is distributed?

Caring for an elderly and weak mother is a great door to good deeds. The Holy Qur’an and the Sunnah of the Prophet have urged honoring one's parents, especially the mother. Do not be saddened by your siblings' lack of assistance, for Allah has singled you out for this good. As for your mother's desire to give you the price of her apartment without your other siblings, there is no harm in that if she is in full possession of her mental faculties, because your need and large family justify singling you out for this gift.