As already said by Kosh, you are trying to solve an *inverse problem in the calculus of variation*: in its classical formulation, given a system of PDE, the problem consists in *finding a functional whose Euler-Lagrange equations are exactly the given PDE(s)*.

To my knowledge, a necessary and sufficient condition for a system of PDE (jointly with its boundary/Cauchy conditions) to be the Euler-Lagrange equation(s) of a Lagrangian functional (meaning functional which is the integral of a function whose arguments, apart from a "independent" variable $\mathbf{x}$, are the unknown function $\phi$ and its lower order derivatives) is still unknown.
However, it is possible to approach the problem from a more general yet concrete point of view: consider a general operator
$$
\phi\mapsto\mathsf{N}(\phi)(\mathbf x)-f(\mathbf{x}),\label{op}\tag{OP}
$$
where

- $\mathsf N$ is a linear or nonlinear operator (possibly your differential operator, an integral operator, a system of PDEs etc.),
- $\phi\in\operatorname{Dom}(\mathsf N)$, the domain of $\mathsf N$ in an appropriate function space, and
- $f\in\operatorname{Range}(\mathsf N)$ the range of $\mathsf N$ again in an appropriate function space.

The kernel of this operator is characterized by the following equation

$$
\mathsf N(\phi)(\mathbf x)=f(\mathbf{x}).\label{1}\tag{1}
$$
If there exists a general functional $F$ such that its functional derivative vanish on the set of solution(s) of \eqref{1}, i.e.
$$
\left.\frac{\mathrm{d}}{\mathrm{d}\varepsilon}F(\phi+\varepsilon\psi)\right|_{\varepsilon =0}\!\!\!=0\;\;\,\forall\psi \in \operatorname{Dom}(\mathsf N)\iff \phi(\mathbf x)\text{ is a solution of \eqref{1},}\label{gvf}\tag{GVF}
$$
the problem \eqref{1} is said to admit an *extended variational formulation* and the operator \eqref{op} is said to be a *potential operator* (or *conservative operator* according to [2] §3 p. 152).

Enzo Tonti (see [1], chapter III §11.1 p. 94 and chapter IV §§17.1-17.4, pp. 168-162 and the recent paper [2] or the Author's web site, where a brief survey of the result is given) proved that, under fairly general conditions, a such a functional $F$ always exists. Precisely, Tonti's theorem says that *every linear / nonlinear problem \eqref{1}, provided some natural hypotheses are assumed, admits an extended variational formulation*. The proof is constructive in that it shows that it is possible to explicitly construct a compact self-adjoint invertible linear operator $\mathsf K$ such that the functional
$$
F(\phi)=\int\limits_\Omega \mathsf N(\phi)\mathsf K\big(\mathsf N(\phi)-f\big)\mathrm{d}\mathbf{x}\label{2}\tag{2}
$$
satisfies \eqref{gvf}: moreover, if $\mathsf{K}$ is also a *positive definite operator* (i.e. $\langle v, \mathsf{K}v\rangle>0$ for all functions $v\in \operatorname{Dom}(\mathsf K)$ such that $v\not\equiv 0$), the the solutions of \eqref{1} are the minimum points of the functional \eqref{2} ([2] §4 theorem 2 pp. 155-156). Thus applying the techniques described in [1] and [2] you can surely find variational formulation for your problem, even if it cannot be expressed as the Euler-Lagrange equation(s) of an appropriate energy functional.

**Notes**

When $\phi, f$ belong to function spaces on sufficiently regular domains $\Omega$, then the Green's operator associated to the Green's function for a any symmetric linear PDE on $\Omega$ can be chosen as the needed $\mathsf{K}$ operator. When $\Omega$ is not regular, there are nevertheless other possible choices: some of them are described in §5, pp. 156-161 of [2].

The analytical approach to the inverse problem of the calculus of variations is perhaps comprehensively described in the wonderful monograph [1], which is however up to date up to 1989: on the other hand [2] surveys nicely the theory of potential operator up to the more recent development due to the Author.

- In remark 17.1 of [1] chapter IV, §17.2 pp. 171-172, Filippov gives a brief but very interesting survey on the conditions of potentiality for a nonlinear differential operator (ordinary or partial): perhaps, if something more tailored to the form of a given equation respect to \eqref{2} is needed, it would be worth to have a look at the references cited there.

**References**

[1] Filippov, Vladimir Mikhailovich, *Variational principles for nonpotential operators*. With an appendix by the author and V. M. Savchin, Transl. from the Russian by J. R. Schulenberger. Transl. ed. by Ben Silver, Translations of Mathematical Monographs, 77. Providence, RI: American Mathematical Society (AMS). pp. xiii+239 (1989), ISBN: 0-8218-4529-2. MR1013998, ZBL0682.35006.

[2] Tonti, Enzo, "Extended variational formulation", Vestnik Rossiĭskogo Universiteta Druzhby Narodov, Seriya Matematika 2, No. 2, 148-162 (1995). ZBL0965.35036.