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

Abstract

LetLLbe a possibly degenerate second order differential operator and letΓη=d2−Q\Gamma _\eta =d^{2-Q}be its fundamental solution atη\eta; hereddis a suitable distance. In this paper we study necessary and sufficient conditions for the weak solutions of−Lu≥f(ξ,u)≥0-Lu\ge f(\xi ,u)\ge 0onRN{\mathbb {R}}^Nto satisfy the representation formula\[(R)u(η)≥∫RNΓηf(ξ,u)dξ.(\mbox R)\qquad \qquad \qquad \qquad \qquad u(\eta )\ge \int _{\mathbb {R}^N} \Gamma _\eta f(\xi ,u) \,d\xi .\qquad \qquad \qquad \qquad \qquad \qquad\]We prove that (R) holds providedf(ξ,⋅)f(\xi ,\cdot )is superlinear, without any assumption on the behavior ofuuat infinity. On the other hand, ifuusatisfies the condition\[lim infR→∞−∫R≤d(ξ)≤2R|u(ξ)|dξ=0,\liminf _{R\rightarrow \infty } {-\!\!\!\!\!\!\int }_{R\le d(\xi )\le 2R}|u(\xi )|d\xi =0,\]then (R) holds with no growth assumptions onf(ξ,⋅)f(\xi ,\cdot ).