Question
What is the corresponding Euler-Lagrange equation that minimizes the energy
$$E[u] = \frac{1}{2}\int_{\Omega} \|\nabla^k u \|^2 \> dx$$
where $\Omega \subseteq \mathbb{R}^n$ and $u: \Omega \mapsto \mathbb{R}$.
Conjecture $$\Delta^{k} u = 0$$
Background
For the standard Dirichlet Energy
$$E[u] = \frac{1}{2}\int_{\Omega} \|\nabla u \|^2 \> dV$$
we can prove that harmonic functions minimize this energy. A standard way to do this is to assume we have a function $w$ such that $\Delta w = 0$ for $x \in \Omega$. This means
$$\int_\Omega \Delta w(w-u) \> dx = 0$$ $$\int_\Omega \nabla u \cdot \nabla w \> dx - \int_\Omega \|\nabla w\|^2 \> dx = 0$$ $$\frac{1}{2} \int_\Omega \|\nabla u\|^2 + \|\nabla w\|^2 \> dx - \int_\Omega \|\nabla w\|^2 \> dx \ge 0$$ $$\frac{1}{2} \int_\Omega \|\nabla u\|^2 \ge \frac{1}{2} \int_\Omega \|\nabla w\|^2 \> dx$$
Since we didn't lose generality, this argument works for any choice of $u$. I suspect a similar technique should show the more general version, but I haven't been able to work out the identities.
The reason I am interested is this more general energy can be used to enforce more continuity at the handles of a mesh deformation. I'm going to keep looking into this and maybe post my own answer if no one beats me to it.
One way to come up with the Euler Lagrange equation associated to an energy functional is to look at the associated critical points. For example, an alternative derivation of Laplace's equation in your $k = 1$ case would proceed as follows. If $u$ minimizes $E$ then all directional derivatives of $E$ at $u$ should vanish. In other words, for every $\varphi\in C_c^\infty(\Omega)$, $u$ should satisfy \begin{eqnarray*} 0 &= & \frac{d}{dt}\bigg|_{t = 0} E(u + t\varphi) \\ & = & \int_\Omega\nabla u\cdot\nabla \varphi\; dx \\ &=& -\int_\Omega \Delta u \varphi\; dx \end{eqnarray*} If this holds for all $\varphi\in C_c^\infty(\Omega)$ then $u$ is harmonic in $\Omega$.
For larger $k$, one can use a similar computation. For example, when $k= 2$ the energy functional is \begin{equation*} E_2[u] = \frac 12 \int_\Omega |\nabla^2u|^2\; dx = \frac 12\int_\Omega\sum_{ij}(\partial_{ij}u)^2\; dx. \end{equation*} If $u$ minimizes $E_2$ then all directional derivatives of $E_2$ at $u$ vanish, so for every $\varphi\in C_c^\infty(\Omega)$ we have \begin{eqnarray*} 0 & = & \frac{d}{dt}\bigg|_{t = 0}E_2[u + t\varphi] \\ & = & \sum_{ij}\int_\Omega \partial_{ij}u\partial_{ij} \varphi\; dx \\ & = & \sum_{ij}\int_\Omega \partial_{ii}\partial_{jj}u\; \varphi\; dx\qquad(\text{do two integration by parts using $\varphi\in C_c^\infty(\Omega)$}) \end{eqnarray*} Since this holds for all $\varphi$, $u$ satisfies \begin{equation*} 0 = \sum_{ij}\partial_{ii}\partial_{jj} u = \Delta^2 u. \end{equation*}