How to solve $\Delta u+\exp(u)=0$ in $R^2$

Question

This is not a good question, since generally this equation is not wellposted. But when I ask for a radially symmetric solution of the type $\phi(r)=u(x)$ where $r=|x|$, this can be handle as following: differentiate both side to translate the original equation into \begin{equation} \phi''(r)+\phi'(r)/r+e^{\phi(r)}=0, \tag{1} \end{equation} and then show that $$\phi(r)=\log\frac{8\mu^2}{(\mu^2+r^2)^2}$$ is a solution.

Then my question is, how the show that this is the (only?) solution, if there have others, how the get them?

Note that $(1)$ is just a ODE (although non-linear), I can't believe it can't solve by hand!

Answer 1

Take a look here in the first link. It seems that without the condition $\int e^{u}<\infty$, things get worse. On the other hand, if you assume this condition, then you have unicity.