Let $\Omega\subset\mathbb{R}^N$ be a bounded domain and $\Delta^2=-\Delta\circ(-\Delta)$. If $u\in C^4(\overline{\Omega})$ satisfies $$ \left\{ \begin{array}{rl} \Delta^2u\leq 0 &\mbox{ in $\Omega$} \\ u=\Delta u=0 &\mbox{ in $\partial\Omega$} \end{array} \right. $$
then, by applying the maximum principle for harmonic functions two times, we can conclude that $u\leq 0$ and $-\Delta u\leq 0$ in $\Omega$. My question is: If now we change $\Delta^2$ in the above system by the operator $\Delta^2+cI$, where $c\geq 0$ is a fixed number and $Iu=u$, can we conclude the same thing?