Good question. It might be useful to divide the problems into several categories.
There are problems of survival. This refers to the way that the boys had to build shelters, learn to find food, learn to hunt, start fires, etc.
Related to these are problems due to isolation. They have only limited choices of what to eat. They have only a finite number of kids.
There are problems related to rescue. Because they have no radio, etc., they have to try to keep the fire going.
This leads to the biggest category: social or cultural problems. The boys have problems early on simply forming a new society. They have problems due to lack of social structures and lack of knowledge. (There are no adults, and no rules they have to follow.) The stresses of isolation and survival combine with the effort of forming a society to create new social problems. Some of the boys don't see things the same way, or don't want to follow the rules. That's why Jack rebels. Some of them go a bit crazy (Simon), and that leads to a larger social problem of murder. There's a civil war going on when the adults finally arrive.