I'm considering the elliptic PDE with complex Laplacian, for example, write $$ \Delta_c(\cdot):=-g^{i \bar{j}} \partial_i \partial_{\bar{j}}(\cdot), $$ and $$\Delta_c(u)=f,$$ by [P.Gauduchon, Math.Ann, 1984], we know that $$ \Delta_c u=\Delta_{d} u+(d u, \theta)_g, $$ where is the Hodge-Laplacian plus a gradient term, so I wondered, can we consider the solution as the critical point of a functional, then I recall this answer here, which said that the energy functional of $$ -\Delta u+D \varphi \cdot D u=f $$ is $$ I[w]:=\int_U L(D w, w, x) d x $$ with $L(p, z, x)=e^{-\varphi(x)}\left(\frac{1}{2}|p|^2-f(x) z\right)$.
But I was told that I can't do this to the complex laplacian, because the representation $$ \Delta_c u=\Delta_d u+(d u, \theta)_g $$ is 'local', so we could not use variational method, I do not know the reason in detail, could you give me some more explanation if you know the reason?
