I wonder how I can solve the following 2nd-order PDE on the positive semiplane $\{x>0\}$:
$$(\partial_x^2+\frac{1}{x}\partial_y^2)\phi=\delta(x-x_0)\delta(y).$$
I notice that the l.h.s. is the Laplace-Beltrami operator of a metric of negative curvature (according to my calculations) applied to the function $\phi$. I would appreciate if there is some nice, reasonably general solution to this PDE.