  We investigate the question of existence of plurisubharmonic defining
functions for smoothly bounded, pseudoconvex domains in $\mathbb{C}^2$. In
particular, we construct a family of simple counterexamples to the existence of
plurisubharmonic smooth local defining functions. Moreover, we give general
criteria equivalent to the existence of plurisubharmonic smooth defining
functions on or near the boundary of the domain. These equivalent
characterizations are then explored for some classes of domains.
