On the existence of solutions to nonlinear systems of higher order Poisson type

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 ${\text{RR}}^n,n\ge 3$. This is analogous to the Residue theorem for the Cauchy kernel in $\text{CC}$. 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 $\text{s}{C}^{k,\alpha }$ estimates. The existence of solutions to higher order Poisson type nonlinear systems is concluded as an application of the fixed point theorem.