It shall be an old story in PDE. I am looking for a sufficient condition of Dirichlet problem for the existence of the unique viscosity solution of the equation in the form of $$\inf_{a \in [-1,1]} \{b(a) \cdot D u(x) + \frac 1 2 tr (A(a) D^{2} u(x) )\} + f(x) = 0, \ \forall x\in O$$ with boundary condition $u = g$ on $\partial O$.
In particular, I want to explain the following two simple examples.
[ex1] There is no solution for $$- u' + u - 1 = 0, \ \forall x\in (-1,1) \hbox{ with } u (\pm 1) = 0.$$ Although $u(x) = 1 - e^{-1+x}$ is the unique solution of generalized Dirichlet problem according to the generalized definition of section 7 of [User's Guide by Crandall, Ishii, and Lions], it does not meet boundary data in the strong sense. END
[ex2] One can check $u(x) = 1 - e^{-1+|x|}$ is the solution of $$- \inf_{b\in [-1,1]} b u' + u - 1 = 0, \ \forall x\in (-1,1) \hbox{ with } u (\pm 1) = 0.$$ This equation also satisfies comparison principle, and hence it has unique solution. END
[ex3] As of discussion by @Willis Wong below, there is no solution for $$- \inf_{b\in [-1,1]} b u' + u + 1 = 0, \ \forall x\in (-1,1) \hbox{ with } u (\pm 1) = 0.$$
Indeed, all equations satisfy comparison principle. But first and last one have no solution, while second has unique one. According to Perron's method, the existence holds if one can construct sub and supersolution, but the sufficient condition of the existence is still left vague from Perron's method.
[Q.] Is there any reference which can explicitly explain why the [ex1] and [ex3] have no solution but [ex2] does? It's more desirable if the sufficient condition works for the general equation with the second order term in the above.
