A question on the first derivative $\dfrac{\partial u}{\partial n} =0$ on $\partial\Omega$?

Question

I am studying the existence solution of an elliptic system has Hamiltionian type $$\begin{cases} - \Delta u = v|v|^{p-1}\quad\mbox{ in } \Omega\\ -\Delta v = u|u|^{q-1} \quad\mbox{ in } \Omega\\ u= v=0 \quad\mbox{ on } \partial\Omega.\end{cases}$$ By the structure of this system on the boundary $\partial\Omega$ we get the second derivative $\dfrac{\partial^2 u}{\partial^2 n} =0,$ where $n$ is the normal vector on boundary. My question as: from $u=0$ on $\partial\Omega$ and $\dfrac{\partial^2 u}{\partial^2 n} =0,$ we can obtain the first derivative $\dfrac{\partial u}{\partial n} =0$ on $\partial\Omega$ ? Thank.