Representation formulae and inequalities for solutions of a class of second order partial differential equations

Abstract

Let L L be a possibly degenerate second order differential operator and let Γ η = d 2 − Q \Gamma _\eta =d^{2-Q} be its fundamental solution at η \eta ; here d d is a suitable distance. In this paper we study necessary and sufficient conditions for the weak solutions of − L u ≥ f ( ξ , u ) ≥ 0 -Lu\ge f(\xi ,u)\ge 0 on R N {\mathbb {R}}^N to satisfy the representation formula \[ ( R ) u ( η ) ≥ ∫ R N Γ η f ( ξ , u ) d ξ . (\mbox R)\qquad \qquad \qquad \qquad \qquad u(\eta )\ge \int _{\mathbb {R}^N} \Gamma _\eta f(\xi ,u) \,d\xi .\qquad \qquad \qquad \qquad \qquad \qquad \] We prove that (R) holds provided f ( ξ , ⋅ ) f(\xi ,\cdot ) is superlinear, without any assumption on the behavior of u u at infinity. On the other hand, if u u satisfies the condition \[ lim inf R → ∞ − ∫ R ≤ d ( ξ ) ≤ 2 R | u ( ξ ) | d ξ = 0 , \liminf _{R\rightarrow \infty } {-\!\!\!\!\!\!\int }_{R\le d(\xi )\le 2R}|u(\xi )|d\xi =0, \] then (R) holds with no growth assumptions on f ( ξ , ⋅ ) f(\xi ,\cdot ) .