Suppose I can write a second-order elliptic PDE for an unknown function $u:\mathbb R^n\to\mathbb R$ in the form $$\Delta u(x) + F(\nabla u(x))=0\qquad\forall x\in\mathbb R^n.$$ Under what conditions on the (potentially nonlinear) function $F:\mathbb R^n\to \mathbb R$ does this PDE correspond to the Euler-Lagrange equation of some functional acting on $u$?
For example, if $F(v)\equiv0$ then the resulting PDE $\Delta u=0$ (Laplace's Equation) is the Euler-Lagrange equation corresponding to the Dirichlet energy $u\mapsto \int \|\nabla u(x)\|_2^2\,d\mathrm{Vol}(x)$.
More generically, I'm looking for a functional version of Poincaré's Lemma relevant to second-order PDE. Posts like this one and this one are relevant, but I had trouble translating them into the case I'm interested in.