Strong maximum principle for a local maximum

Question

Suppose $L$ is a strictly elliptic operator in the non-divergence form with $c \equiv 0$, $u \in C^2(\Omega)\cap C(\Omega)$ and $Lu \le 0$ in $\Omega$. Prove that if $u$ attains a local maximum at an interior point of the connected open set $\Omega$, then $u$ is constant. The result is easy for Harmonic functions, because they are analytic, but hard in the general case. Any hint would be appreciated. Thanks.