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

Abstract:

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