  In this paper, we study the existence of higher order Poisson type systems.
In detail, we prove a Residue type phenomenon for the fundamental solution of
Laplacian in $\RR^n, n\ge 3$. This is analogous to the Residue theorem for the
Cauchy kernel in $\CC$. With the aid of the Residue type formula for the
fundamental solution, we derive the higher order derivative formula for the
Newtonian potential and obtain its appropriate $\s C^{k, \alpha}$ estimates.
The existence of solutions to higher order Poisson type nonlinear systems is
concluded as an application of the fixed point theorem.
