 Exponential perturbation proof of the weak maximum principle with drift

For $Lu=a^{ij}{D}_{ij}u+b^i{D}_{i}u$ with $a^{ij}{\xi }_i{\xi }_j\ge \lambda |\xi {|}^2$ and $|b^1|<\lambda \ell$, the function $q(x)=e^{-\ell x_1}$ satisfiesIf $Lu\ge 0$ had a strict interior maximum above the boundary maximum, then $u+\epsilon q$ would retain an interior maximum for small $\epsilon >0$, where ellipticity gives $L(u+\epsilon q)\le 0$. This contradicts its strictly positive image under $L$.