Moses Herzog, the protagonist of Saul Bellow's 1964 novel Herzog, is an intellectual in that he is a scholar; he faces crisis (and creates some as well) and he takes an intellectual approach to dealing with his problems.
The book is written as a series of letters, written mentally by Herzog to people in his life and people to whom he feels a connection. Herzog takes this approach rather than directly confront people, and although it takes a shock to his mind to break him out of his self-involvement, the act of thinking through problems and engaging in dialogues with his mental constructs of people allows him to better understand himself. Herzog is a teacher and was once a "noted scholar," so his dialogues are based both in his own experience and in that of his knowledge.
The crises (plural of crisis) that he faces are both in his own life -- his humiliation at being cuckolded by ex-wife Madeleine (even as he committed adultery), his inability to focus on teaching, and his relationship with his daughter. He also creates a crisis in his head, convincing himself that Madeleine and her paramour mean to harm his daughter, which shows his delusions from constant introspection. Eventually, Herzog is able to understand both the people around him and his own self-involvement, and begins to heal from his mental instabilities.