Existing work on $\Delta u=c-h e^{u}$ on compact manifold with dimension n, I have read J.Kazdan's work, the condition $c > 0, n \ge 3$ is not solved

Question

I'm reading Prof. Kazdan's lectures

At page 69, Prof. Kazdan describes the research on the $\Delta u=c-h e^{u}$ PDE on a compact $n$-dimensional manifold before 1983. (Here $c$ is a constant while $h$ is a smooth function).

In this lectures, the condition $c > 0$ is not fully solved, and we only have some results for $n = 2$: the reason we can't find here results for higher dimensions is that he uses the variational method and Moser-Trudinger inequality for dimension $2$.

I tried to find recent works dealing with the conditions $c > 0$ and $n \ge 3$, but I didn't get any good result. Does anyone know about this ?