The Kazdan-Warner Type equations I am talking about is the following:


Suppose X be a compact Riemannian manifold(of any dimension) and $A$, $B$, and $w$ to be smooth function, I hope to know more about the known result to the following equation:
$$\Delta u+ Ae^u-Be^{-u}-w=0 $$


Here is my questions:


1. Under some assumptions of A, B and w, is there any good reference for the solvability and uniqueness of this equation?
2. What is the obstruction for the existence of solution to this equation?



Any Good Reference for Kazdan-Warner Type Equations
Not sure what counts as a good reference, but a statement on the solvability of the equation you mention is contained in the paper by Bryant-Wentworth (Lemma 3.4) <http://www.math.ubc.ca/~jbryan/papers/kahlermonos.pdf> 


