Representation Formulae for Solutions to Some Classes of Higher Order Systems and Related Liouville Theorems

Abstract.

Let m ≥ 1 be an integer and N > 2m. Let μ be a positive Radon measure on ${\mathbf{R}}^N$. We study necessary and sufficient conditions on possible distributional solutions of $(-\Delta)^{m}u = \mu\,{\rm on}\,{\mathbf{R}}^N$, that guarantee the validity of the representation formula $u(x) = l + c(2m) \int_{{\mathbf R}^{N}} {\frac {d\mu(y)} {|x - y|^{N-2m}}}$ a.e. on ${\mathbf{R}}^N$, where $l\,\in\,{\mathbf{R}}$ and c(2m) is a positive constant depending on m and N. Several consequences are derived. In particular we prove Liouville theorems for systems of higher order elliptic inequalities and weighted form of Hardy-Littlewood-Sobolev systems of integral equations.