It's a well known result by Kazdan and Warner that on a closed Riemannian manifold the pde:
\begin{align*} \Delta f+ge^f=c \end{align*} has a unique solution for $g\geq 0,$ and $c$ a positive constant. So, we can replace $c$ by $c+t$ and for each $t\in(0,\infty)$ we would have a different solution for $f$. I am looking for the limiting behaviour of $f$ when we let $t\rightarrow\infty$. Any idea or reference is most welcome.
