Transforming an elliptic PDE into the Laplace equation?

[AKBAR]

For an elliptic PDE Uxx + Uyy + Ux + Uy = -1 in D = {x^2 + y^2 = 1} and U = 0 on the boundary of D = {x^2 + y^2 = 1}
is it possible for me to make a change of variables and eliminate the Ux and Uy and get the Laplace equation Uaa + Ubb = 0?

I tried converting into polar coordinates, but the Ux and Uy don't seem to cancel out. Or am I approaching this the wrong way?

 

[HallsofIvy]

You can't get rid of the "Ux" and "Uy" by changing the independent variables but you can by letting V(x,y)= e^{-(1/2)(x+ y)}U(x,y).