GreenFunction Computation for perturbed Laplacian

I am happy computing the Green function for the Laplacian

Gsol := GreenFunction[{-Laplacian[u[x, y], {x, y}]}, u[x, y], {x, y} \[Element] FullRegion[2], {m, n}]

it gives an analytical solution – which is great.

However, if I slightly perturb my operator

Gsol := GreenFunction[{D[u[x, y], {x, 2}] + 2 D[u[x, y], {y, 2}]}, u[x, y], {x, y} \[Element] FullRegion[2], {m, n}]

where I have simply put a 2 infront of the second derivative.

Mathematica is no longer able to provide an analytical or even a numerical approximation to the Green function for this operator. If GreenFunction is not able to handle such an operator, is there another method I can use to do this?