Solving a shifted Laplace equation

Question

I have the Laplace equation $(a-\nabla^{2})\phi(x,y)=0$. I am trying to solve it for the half plane boundary by using methods of images. The first term, constant $a$, is a problem for me. I know without this term I have the Greens function $G(x,y)=-\frac{1}{2\pi}Log[\frac{(x-x_{0})^{2}+(y-y_{0})^{2}}{(x-x_{0})^{2}+(y+y_{0})^{2}}]$ and this satisfy $\nabla^{2}G(x,y)=\delta(x-x_{0})\delta(y-y_{0})$. However this is only satisfied for $a=0$. Any help will be highly appreciated. Thanks.