Positive harmonically bounded solutions for semi-linear equations

Abstract

For open sets U in some space X, we are interested in positive solutions to semi-linear equations $Lu=\phi (\cdot ,u)\mu$ on U. Here L may be an elliptic or parabolic operator of second order (generator of a diffusion process) or an integro-differential operator (generator of a jump process), μ is a positive measure on U and φ is an arbitrary measurable real function on $U\times R^{+}$ such that the functions $t\mapsto \phi (x,t)$, $x\in U$, are continuous, increasing and vanish at $t=0$.

More precisely, given a measurable function $h\ge 0$ on X which is L-harmonic on U, that is, continuous real on U with $Lh=0$ on U, we give necessary and sufficient conditions for the existence of positive solutions u such that $u=h$ on $X\setminus U$ and u has the same “boundary behavior” as h on U (Problem 1) or, alternatively, $u\le h$ on U, but $u\not\equiv 0$ on U (Problem 2).

We show that these problems are equivalent to problems of the existence of positive solutions to certain integral equations $u+K\phi (\cdot ,u)=g$ on U, K being a potential kernel. We solve them in the general setting of balayage spaces $(X,W)$ which, in probabilistic terms, corresponds to the setting of transient Hunt processes with strong Feller resolvent.