The answer to this question really depends on what kind of egoist position one is talking about. It seems that the question is referring to psychological egoism, which is basically a descriptive position, as opposed to ethical egoism, which is normative or prescriptive. The easiest problem to point out about this position is that it does not, on the surface, seem to explain some human behaviors which are clearly altruistic and self-sacrificial. Another problem is that it tends to be reductionist in terms of human motives, establishing false equivalencies between actions. So if a person sacrificed their own life to save that of another, an egoist would say they were motivated by their own self-interest. Yet a person who killed another person to save their own skin would be, fundamentally, acting from the exact same motive. There is also much evidence from researchers that suggests that people are motivated by altruism in many of their actions. Some of these studies argue that humans have actually evolved altruistic motives.