Representation formulae for solutions to some classes of higher order systems and related Liouville theorems

Let m>=1 be an integer and N > 2m. Let μ be a positive Radon measure on RN. We study necessary and sufficient conditions on possible distributional solutions of (−")mu = μ on RN, that guarantee the validity of the representation formula u(x) = l + c(2m) ! RN dμ(y) |x − y|N−2m a.e. on RN, where l # 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.