Representation Formulae and Inequalities for Solutions of a Class of Second Order Partial Differential Equations

Let L be a possibly degenerate second order differential operator and let Γη = d^(2−Q) be its fundamental solution at η; here d is a suitable distance. In this paper we study necessary and sufficient conditions for the weak solutions of −Lu ≥ f (ξ, u) ≥ 0 on RN to satisfy the representation formula (R) u(η) ≥ integral RN Γη f (ξ, u) dξ. We prove that (R) holds provided f (ξ, ·) is superlinear, without any as- sumption on the behavior of u at infinity. On the other hand, if u satisfies the condition |u(ξ)|dξ = 0, lim inf − R→∞ R≤d(ξ)≤2R then (R) holds with no growth assumptions on f (ξ, ·).