= Solution
For
$$
Lu=a^{ij}D_{ij}u+b^iD_iu+cu,
\qquad c\leq0,
$$
the <weak maximum principle for elliptic operators> states that $Lu\geq0$ implies
$$
\max_{\overline\Omega}u\leq\max\{0,\max_{\partial\Omega}u\}.
$$
In particular, a solution of $Lu=0$ cannot have a positive interior maximum exceeding its boundary maximum.
Because the coefficient matrix is positive definite and the closure of the smooth bounded domain is compact, strict ellipticity supplies a uniform lower bound after restricting to $\overline\Omega$. Rotate and translate coordinates so that $\Omega$ is bounded in the $x_1$ direction, and set $h=e^{\gamma x_1}$. For sufficiently large $\gamma$,
$$
Lh=e^{\gamma x_1}(\gamma^2a^{11}+\gamma b^1+c)>0.
$$
If $u+\varepsilon h$ had a positive interior maximum, its gradient would vanish and its <Hessian matrix> would be negative semidefinite there, giving $L(u+\varepsilon h)\leq0$. This contradicts $L(u+\varepsilon h)=\varepsilon Lh>0$. Comparing on the boundary and sending $\varepsilon\downarrow0$ proves the assertion.
Solved by gpt-5.6-sol high.
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