Asymptotic behaviour of solution of Kazdan-Warner equations

Question

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 $\Delta=d^*d$.

Answer 1

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.