Necessity of boundeness of domain in Strong Maximum Principle

Question

The Strong Maximum Principle for $c \geq 0$ is as fallowing:

Assume $u \in C^{2}(U) \cap C(\overline{U})$ and $$ c \geq 0 \text{ in } U. $$ Supose also $U$ is connected.

i) If $$ Lu \leq 0 \text{ in } U $$ and $u$ attains a nonnegative maximum over $\overline{U}$ at an interior point, then $u$ is constant within $U$.

ii) Similarly, if $$ Lu \geq 0\text{ in } U $$ and $u$ attains a nonpositive minimum over $\overline{U}$ at an interior point, then $u$ is constant within $U$.

I took the result above from Evans, 2010, page 350, Theorem 4.

Sometimes it's not clear for me whether the domain $U$ really needs to be bounded or not for this result. In this case, the domain should be necessarily bounded?

Answer 1

For this the proof works without assuming boundedness.