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

TL;DR

This paper investigates the existence of solutions to higher order Poisson systems, establishing a residue phenomenon for the fundamental solution and deriving derivative formulas and estimates, leading to existence results via fixed point theory.

Contribution

It introduces a residue type formula for higher order fundamental solutions and applies it to prove existence of solutions for nonlinear systems.

Findings

Residue type phenomenon for higher order Laplacian fundamental solutions

Derivative formulas and regularity estimates for Newtonian potentials

Existence of solutions established using fixed point theorem

Abstract

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.