Let me refer you to my colleague, Denis Blackmore, in the Dept. of Mathematical Sciences at NJIT. I believe that he will be able to answer your questions. His e-mail address is: [email protected]
I have found the answer to my question in the book of A.D. Polyanin: Handbook of linear partial differnetial equations for engineers and scientists, CRC Press, 2002. This is an excellent reference if you look for a concrete Green function.