Suppose is the upper half plane. Define the map by the following

One can see that maps onto the open ball . It is easy to verify that looks like

We want to pull the metric of to , that is . Denote . We have

Therefore, is conformal to .

Next, for some , we want to pull the solution of

to . That is defining on . We shall derive the equation satisfy on . Note that the equation means

Let us use the following notations

It follows from the covariant property of conformal laplacian that for any ,

Note that . We turn to calculate

It is not hard to see

Therefore

Next, to handle the term , applying ,

That is

It follows from the transformation formula that

To summarize the above calculation, one the one hand,

on the other hand

Then we get the equation of

Notice that where is the distance of to the center of the the ball . The equation that satisfies is rotationally symmetric with respect to the center of .