If I have an energy of the form: $E(u) = \int_{\Omega}\|D^mu(x)\|^2\,dx$ then I get the Euler-Lagrange equations: $\Delta^m u(x) = 0$. Where $D^m$ is the $m$-dimensional tensor of partial derivatives of $u$ (e.g. $m=1\implies \nabla u$, $m=2\implies H_{ij} = \partial_{ij}u$) and $\|\cdot\|^2$ is the sum of the squares of the elements. The $\Delta^m$ is formed by the even derivatives in the even terms from the Taylor expansion: $\left(\sum_i h_i\partial_i\right)^{2m}$, that is essentially the terms: $(\sum_i h_i^2\partial_{ii})^m$.
What would be an energy for PDEs involving odd derivatives if such an energy exists? Let's take a very simple example in 1D:
$$\left(\frac{d^{2k+1}}{dx^{2k+1}}u\right)(x) = 0$$
Basically is the above an Euler-Lagrange equation arising from some energy?