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For a class of Bellman equations in bounded domains we prove that sub- andsupersolutions whose growth at the boundary is suitably controlled must beconstant. The ellipticity of the operator is assumed to degenerate at theboundary and a condition involving also the drift is further imposed. We applythis result to stochastic control problems, in particular to an exit problemand to the small discount limit related with ergodic control with stateconstraints. In this context, our condition on the behavior of the operatornear the boundary ensures some invariance property of the domain for theassociated controlled diffusion process.

Nonexistence of nonconstant solutions of some degenerate Bellman equations and applications to stochastic control