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I'm pretty sure that this is a stupid question, but I'm having troubles in writing down the energy functional of an elliptic pde. That is, what's the energy functional of the problem $$\begin{cases}-\Delta u = \frac{e^{u^p}}{\int_{D}{e^{u^p}\mathrm{d}x}}&x \in D,\\ u = 0& x \in \partial D,\end{cases}$$ where $D \subset \mathbf{R}^2$ is a bounded domain and $p \ge 1$? I know that if the assigned energy is of the form $f(x, u(x))$, then the energy functional is $E(u) = \frac{1}{2}\int_D{\left|\nabla u\right|^2\mathrm{d}x} - \int_D{\int_0^{u(x)}{f(x, t)\mathrm{d}t}\mathrm{d}x}$, but is there a general expression also when $f(x, u)$ depends "globally" on $u$ and not just "pointwise"?

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A partial answer in the case $p=1$: $$ E(u) = \frac{1}{2} \int_D |\nabla u|^2 \, dx - \ln \left(\int_D e^u \ dx\right). $$

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