The late Walter Kaufmann, a leading authority on existentialism, observed in his work Existentialism that Dostoyevsky was given to writing about characters who were very different than he was, or at least who did not share his beliefs. Dostoyevsky was a Christian, and many of the existentialist ideas that he gave to his characters, the athiest Vanya for example, are not his own.

Are you asking about this book or all of his books?  Being his last novel, it makes sense that this one would best reflect the culmination and full development of his work.  Since one of the main questions is about existence, and existence of God, I'd say it is existential.