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 R N . We study necessary and sufficient conditions on possible distributional solutions of (−∆) m u = µ on R N , that guarantee the validity of the representation formula u(2m a.e. on R N , 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. Mathematics Subject Classification (2000). Primary 35C15; Secondary 31C10, 45G15.

Liouville-type theorems and decay estimates for solutions to higher order elliptic equations

“…We refer the reader to the paper by Caristi, D'Ambrosio and Mitidieri [5] and references therein (see also [19]). …”

Asymptotic properties of positive solutions of the Hardy–Sobolev type equations

“…By means of the Pohozaev identity in integral forms (cf. [6]), we can deduce that p is the exact critical exponent p = n+α+2σ n−α . This shows that if p is supercritical, u is not integrable.…”

Integral representation of solutions using Green function for fractional Hardy equations

“…It's well-known that the integral representation of solution using Green function of the operator is very useful in studying various properties of solutions. In the case of θ = 0, i.e., for the fractional Laplace operator, integral representation of solution using Green function were used in many papers, to cite few we mention [9], [10], [15], [19], [20]. In our forthcoming paper [5], we establish an asymptotic behaviour of solution for the fractional Hardy equation using the integral representation of the solution.…”