President Gotabaya Rajapaksa has vowed to restore order, in his first national speech since protests began last month.
Ignoring calls to resign, he offered to cede some powers to parliament and name a prime minister, but set no timetable.
His brother quit as PM on Monday amid fury over soaring prices and shortages.
Sri Lankans are desperate as basic items like food and fuel run out or become unaffordable.