I'm doing a study of surfaces with constant curvature which leads to solving the equation: $$\Delta\phi = -e^{2\phi}K_0$$ for a 2-dimensional metric with constant curvature such that rotation around the origin is an isometry ($\phi = \phi(r)$). A first idea was using polar coordinates, and convert the equation to polar coordinates: $$\frac{1}{r}\frac{d\phi}{dr} + \frac{d^2\phi}{dr} = -e^{2\phi(r)}K_0$$ Which is a second order nonlinear ordinary differential equation. I've used Maple and Mathematica to solve this equation but I'm not happy with the solution, it's quite complicated (Bessel function).
Does anyone have any idea about how to solve the first equation maybe simplifying the procedure with some properties of isometries?
Thanks.
Multiply through by $r$ and note that \begin{equation} \frac{d}{dr}\left( r \frac{d \phi}{dr}\right) = \frac{d \phi}{dr}+r\frac{d^{2} \phi}{d r^{2}} \end{equation} Then you're left with \begin{equation} \frac{d}{dr}\left( r \frac{d \phi}{dr}\right)=-r e^{2 \phi} K_{0} \end{equation}