A shallow shell model is developed for the elastic buckling of a single wall car bon nanotube under a uniform external axial pressure by using the nonlocal elastic theory. Effects of the small size scale are incorporated in the formulation Critical conditions are given under the axial buckling pressure for a single-w all carbon nanotube. Influences of the small size scale on the axial buckling pr essure are found. It is concluded that the axial buckling pressure for a carbon nanotube could be overestimated by the classic shell model without considering t he effect of the small size scale.