Suppose the derivative of a functional is given by $\int_{\Omega}(\vec{v}.\nabla u)|\nabla u|^{p-2} \phi=\int_{\Omega}\nabla.(u\vec{v})|\nabla u|^{p-2} \phi$ for $\phi\in W_0^{1,p}(\Omega)$
\begin{equation*} \int_{\Omega}(\vec{v}.\nabla u)|\nabla u|^{p-2} \phi=\int_{\Omega}\nabla.(u\vec{v})|\nabla u|^{p-2} \phi,~\phi\in W_0^{1,p}(\Omega) \end{equation*}
where the vector field $\vec{v}$ (which is known) is irrotational, i.e., $\nabla.\vec{v}=0$, then what is the functional?. The derivative is computed at $u$ and the argument of the derivative is $\phi$.
