It is well known that for any smooth domain $\Omega\subset\mathbb{R}^N$ the energy functional (the one for which the Euler-Lagrange equation is our b.v.p.) associated to the following local problem:
$$\begin{cases} -\Delta u(x)=f(x,u(x)),\ x\in\Omega \\ \dfrac{\partial u}{\partial\nu}(x)=0,\ x\in\partial\Omega\end{cases}$$ is given by:
$E:H^1(\Omega)\to\mathbb{R},\ E(u)=\dfrac{1}{2}\displaystyle\int_{\Omega}|\nabla u(x)|^2\ dx-\displaystyle\int_{\Omega} F(x,u(x))\ dx$, where $F(x,t)=\displaystyle\int_0^t f(x,s)\ ds$.
My question is the following: What is the energy associated to the following problem:
$$\begin{cases}-\Delta u(x)=[\mathcal{F}(u)](x), x\in\Omega \\ \dfrac{\partial u}{\partial\nu}=0, x\in\partial\Omega \end{cases}$$
where $\mathcal{F}:H^1(\Omega)\to L^2(\Omega)$ is a nonlocal operator (with any assumptions needed). My aim is to find solutions of the pde as minimizers of $E(u)$ and to use the critical point theory.
P.S.: I know that this is called the inverse problem for the calculus of variations. Is it true that we cannot find such a functional in this non-local case?
