How to determine if elliptic equation comes from variational problem?

Question

Is there any criteria to determine if a elliptic equation comes from energy minimizing problem? For example, if I have a elliptic equation in divergence form $\nabla \cdot A(x,u,\nabla u)+|\nabla u|=0$ (Supposed that I have found the the energy for the term $\nabla \cdot A(x,u,\nabla u)$). Is this equation has no variational structure? How about the general form $\nabla \cdot A(x,u,\nabla u)+B(x,u,\nabla u)=0$?

Answer 1

If the equation is of the form $$ \nabla \cdot A(x,u,\nabla u) + B(x,u,\nabla u) =0, $$then such an equation can be brought into variational form by integration by parts.

The corresponding natural boundary condition would be of the type $$ A(x,u,\nabla u) \cdot \nu = 0. $$ If the boundary condition has a different form then it might be again difficult to bring the equation into variational form.

An equation of the type $$ -a(x)\Delta u(x)=f(x) $$ with non-smooth $a$ is hard to bring into a variational form, which allows for an easy solution theory.