  We study a general class of fully nonlinear boundary-degenerate elliptic
equations that admit a trivial solution. Although no boundary conditions are
posed together with the equations, we show that the operator degeneracy
actually generates an implicit boundary condition. Under appropriate
assumptions on the degeneracy rate and regularity of the operator, we then
prove that there exist no bounded solutions other than the trivial one. Our
method is based on the arguments for uniqueness of viscosity solutions to state
constraint problems for Hamilton-Jacobi equations. We obtain similar results
for fully nonlinear degenerate parabolic equations. Several concrete examples
of the equations that satisfy the assumptions are also given.
