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, ·).
