By "main problem," I believe that the question is referring to the central conflict. I have to pick two central conflicts because they are woven together and equally central.
The first conflict is an external, man vs. society conflict. Like The Outsiders, Rumble Fish focuses on a protagonist from a lower class, poor part of town. He is a victim of circumstances in a lot of ways. His parents are not supportive or role models at all, and his family doesn't have strong finances. Consequently, Rusty has turned to gangs and violence. On one hand, he experiences a sort of kinship there, but on the other hand he knows that the gang world is dangerous and won't lead to anything good in the long term.
That leads to the second main conflict. Rusty has an internal conflict of trying to do right by his brother's advice. The Motorcycle Boy got out of the gang life, and Rusty idolizes his brother. Rusty wants to be like him, but Rusty struggles with finding the inner reserves to make the tough choices his brother did.