  For a class of Bellman equations in bounded domains we prove that sub- and
supersolutions whose growth at the boundary is suitably controlled must be
constant. The ellipticity of the operator is assumed to degenerate at the
boundary and a condition involving also the drift is further imposed. We apply
this result to stochastic control problems, in particular to an exit problem
and to the small discount limit related with ergodic control with state
constraints. In this context, our condition on the behavior of the operator
near the boundary ensures some invariance property of the domain for the
associated controlled diffusion process.
