Nonlinear PDE from Riemannian Geometry

Question

I am wondering if anyone knows an approach to finding solutions to the following PDE:

$-e^{-2u}\Delta u=\alpha$.

Here $u=u(x,y)$ is an unknown real-valued function of 2 variables and $\alpha$ is a constant. I'd be happy to see a solution on any connected open subset of the plane.

I have 2 basic observations to get us started. The first is that we can reduce to a nonlinear second order ODE by looking for functions of the radius $r$ (or maybe $r^2$ to avoid square roots). Second, the left hand side shows up if one calculates $\Delta (e^{-2u})$. Looking at the result of the substitution, I'm not sure if this helps.

For context, the quantity on the left side of the equation above is the sectional or Gauss curvature of a metric which is conformal by the factor $e^{2u}$ to the standard Euclidean metric on the plane. Thus solving the above would be a simple way to give examples of constant curvature metrics in two dimensions.

Answer 1

Maybe try $\phi (x, y ) =e ^ { -2u}$, then you get $-\Delta\phi=-\alpha$ This is Poisons Equation, Solve This For $\phi $ Substitute In For $u$ And Then $\ln $ It All.

Answer 2

In fact this belongs to a PDE of the form http://eqworld.ipmnet.ru/en/solutions/npde/npde3103.pdf.

The general solution is $u(x,y)=-\ln\dfrac{\sqrt{|\alpha|}\left(1-\text{sgn}(\alpha)f(x+iy)\overline{f(x+iy)}\right)}{2|f_{x+iy}(x+iy)|}$