= Solution
The <weak maximum principle for elliptic operators> here states
$$
Lu\geq0\text{ in }\Omega\quad\Longrightarrow\quad
\max_{\overline\Omega}u\leq\max_{\partial\Omega}u
$$
for $u\in C^2(\Omega)\cap C^0(\overline\Omega)$.
To prove it, suppose the interior maximum exceeds the boundary maximum. Put $q(x)=e^{-\ell x_1}$. Since $c=0$,
$$
Lq=q(\ell^2a^{11}-\ell b^1)
\geq q\ell(\lambda\ell-|b^1|)>0.
$$
For sufficiently small $\varepsilon>0$, $u_\varepsilon=u+\varepsilon q$ still has an interior maximum. At that point its <gradient> vanishes and its <Hessian matrix> is negative semidefinite; ellipticity gives $Lu_\varepsilon\leq0$. But $Lu_\varepsilon=Lu+\varepsilon Lq>0$, a contradiction. This proves the principle.
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