I wonder when studying the elliptic PDE on complex manifold, especially studying the existence of solutions, when can we directly study the real case, for example, when studying $$\Delta_c u = f(x,u),$$ on complex manifold $(M,\omega)$ (the discussion about complex laplacian can be found in here), if I use upper and sub-solution method or priori estimate method (if we can now apply maximum principle), it seems that the process has nothing to do with the fact that the manifold is complex.
So I wonder when studying the existence of solutions of elliptic PDE on complex manifold, when will we treat it differently compare with the real case.
