Let assume $$Lu=Au+a(x)u=f(x)$$ Where $A$ is elliptic operator.
In Chapter 8 of Smoller book the maximum principle is as follow:
On the other hand, the maximum principle of Han-Lin book says: (note that $a(x)$ is replaced by $c(x)$ in this book)
The second 
The first one says positive maximum and the second one says nonnegative maximum. I can't understand this difference .
For example if I want to show that the solution of the following problem $$ \begin{cases} \Delta u -2u=e^u & \Omega \\ u=0 & \partial \Omega \end{cases} $$ is strictly negative in $\Omega$. The second one works but the first one seems not to work and only says that it has no negative minimum, which means solution is nonpositive.