Let $X$ be a closed manifold. $g:X\rightarrow \mathbb{R}$ be a smooth function ,$\alpha$ a section of a line bundle with discrete zeros and $c>0$ a constant, then Kazdan-Warner's work says that the following equation has an unique solution for$f$:
\begin{align\*}
2\Delta f+\frac{e^g\lvert\alpha\lvert^2}{4}e^{5f}=c 
\end{align\*}
I am interested in the asymptotic behaviour of the solution when we scale $\alpha$ by $\lambda\alpha$ for some constant $\lambda$ and take $\lambda\rightarrow\infty$ and $\lambda\rightarrow 0$. Any idea, answer or reference is most welcome. Here $\Delta=d^\*d$.



Asymptotic behaviour of solution of Kazdan-Warner equations
It's actually quite easy and I completely missed it. If $f$ is the solution of the original equation:
\begin{align\*}
2\Delta f+\frac{e^g\lvert\alpha\lvert^2}{4}e^{5f}=c 
\end{align\*}
and say $f\_\lambda$ is the solution of the perturbed equation:
\begin{align\*}
2\Delta f+\frac{e^g\lvert\lambda\alpha\lvert^2}{4}e^{5f}=c 
\end{align\*}
Then notice just defining $f\_\lambda=f-\frac{2}{5}\text{ln}|\lambda|$ solves the second equation.


