Let $F(x,u,Du,D^2u)=0$ for $x\in\Omega$ be our equation of interest. One way of defining a viscosity subsolution is this:

We say $u$ is a viscosity subsolution iff for any test function $\phi\in C^2(\Omega)$ such that $u-\phi$ has a LOCAL maximum at $x_0$ then $F(x_0,D\phi(x_0),D^2\phi(x_0))\leq 0$.

However in this paper, definition 2.1 on page 6

http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.52.8750&rep=rep1&type=pdf

the author replace the word local with GLOBAL. I believe this is fairly common in optimal control/stopping literature.

I think the answer is no, as long as we can solve this seemingy simple analysis question

but I have struggled to do this.